Algebra 2 2023 Virginia SOL - Student Edition by Mathspace

1 Analyzing Functions Big ideas • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • Functions provide a representation for how related quantites vary. This makes functions a good way to represent many real-world situations. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06 1.07

Characteristics of functions Function families Function transformations Piecewise functions Absolute value equations Compound inequalities Absolute value inequalities

4 25 37 52 63 71 79

• Consider the value(s) of x so that f (x) = 4

The 4 represents the y-value. To ﬁnd the x-value we will start at 4 on the y-axis and ﬁnd its corresponding x-value(s). Therefore, f (x) = − 2, and f (4) = 2.

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

The inputs and outputs of a function can also be described as the domain and range, which can be represented in different notations.

Domain −3

−2

−1

2 1

0 −1

−3

−2

−1

0 −1

−2 −3

−4

−5

Domain: set of all possible inputs

y x 1

Range

Range: set of all possible outputs

Two common styles of notation for characteristics of functions include interval notation and set notation. Interval notation uses brackets and an interval to show the set of all numbers which lie between two values. Set notation speciﬁes a set of elements that satisfy a set of conditions. The following are graphs of functions with their domain and range written in both interval notation and set notation: 4

Domain

Interval Notation: (−∞, 3)

Set Notation: {x∣x < 3}

2 1

−4 −3 −2 −1 −1

Range Interval Notation: [−2, ∞)

Set Notation: {y∣y ≥ −2}

−2 −3 −4

Domain

Interval Notation: (−∞, ∞)

Set Notation: {x∣x ∈ } or {x∣ −∞ < x < ∞}

2 1 −4 −3 −2 −1 −1 −2

x 1

Range Interval Notation: [0, ∞) Set Notation: {y∣y ≥ 0}

−3 −4

1.01 Characteristics of functions mathspace.co

Example 1 Consider the curve on the grap.h below.

y 6 4 2 x −7−6 −5−4 −3−2 −1

1 2 3 4 5 6 7

−2 −4 −6

a State the domain of the function.

Create a strategy The domain is the set of x-coordinates of all the points on the curve. Notice the graph extends inﬁnitely in both the positive and negative directions along the x-axis (to the left and right).

Apply the idea Looking at the graph we see the function reaches all x-values, so our domain is all real numbers. In interval notation we can write this as: Domain = (−∞, ∞)

Reﬂect and check If we were to write our domain using set notation, we would have {x∣ −∞ < x < ∞}.

b State the range of the function.

Create a strategy The range is the set of y-coordinates of all the points on the curve. This parabola has an absolute maximum at the vertex. There is no minimum y-value.

Apply the idea Looking at the graph, the range of this function is all of the real numbers that are less than or equal to 2. This can be written in interval notation as: Range = (−∞, 2]

Reﬂect and check If we were to write the range using set notation, we would have {y∣ −∞ < y ≤ 2}. Notice that 2 is included in the range because the graph of the function exists at that value. But we never include positive or negative inﬁnity because it’s not a number the graph can ever reach.

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Example 2 Consider the function shown in the graph:

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

a State the domain.

Create a strategy Domain is the set of all possible x-values. We will look at the graph from left to right to identify all x-values for which the graph exists.

Apply the idea The function reaches every x-value between −2 and 2, including −2 (notice the ﬁlled point) but not including 2 (notice the unﬁlled point). We can show this in interval notation as: Domain: [−2, 2)

Reﬂect and check Let’s now represent our domain using set notation. Set Notation: {x∣ −2 ≤ x < 2}

b State the range.

Create a strategy Range is the set of all possible y-values. We will look at the graph from bottom to top to identify all y-values for which the graph exists.

Apply the idea The function reaches every y-value between 0 (the lowest point on the graph, at the x-intercepts) and 4 (the highest point on the graph, at the vertex). We can show that in interval notation as: Range: [0, 4]

Reflect and check The range in set notation is: Set Notation: {y∣0 ≤ y ≤ 4} Notice that 0 is included in the range because one of the points is filled. Even though the other point is unfilled, you only need one filled point on a graph to be able to include that value.

1.01 Characteristics of functions mathspace.co

Example 3 Consider the function shown in the graph:

y 3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6 7

−2 −3 −4 −5

a Evaluate the function for f (−2).

Create a strategy We want to ﬁnd the y-value when x = − 2. We will start at −2 on the x-axis and move vertically until we hit the function. Then we will move horizontally to the y-axis and ﬁnd the y-value there.

Apply the idea y 3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6 7

−2 −3 −4 −5

When x = − 2, we can see on the graph the y-value is − 4.

Reﬂect and check If we knew the equation of the function, we could check by substituting −2 into the equation for x and conﬁrming that the result when we evaluate is −4.

b Determine the value of x such that f (x) = − 1.

Create a strategy We want to ﬁnd the x-value for when y = − 1. We will start at −1 on the y-axis and move horizontally until we hit the function. Then we will move vertically to the x-axis and ﬁnd the x-value there.

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Apply the idea y 3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6 7

−2 −3 −4 −5

We can see from the graph that when the y-value is −1, the x-value is 7.

Reﬂect and check This means that, f (7) = − 1.

Example 4 Let f (x) represent the height of a growing plant, f, in inches, where x represents the time since it was planted in days.

Plant Growth 20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

a Interpret the meaning of f (10) = 8.

Create a strategy We can use the units of the given information and the graph to help with the interpretation.

Apply the idea We’re given that f (x) represents the height of a growing plant in inches, so to interpret f (10), we need to determine what an input of x = 10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted. We also know that all of this is equal to 8. This is the output, or what our function f (x) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall. Based on the graph, when x = 10, y = 8 so f (10) = 8 is represented by the ordered pair (10, 8) on the graph. The plant has a height of 8 inches 10 days after being planted.

1.01 Characteristics of functions mathspace.co

b Interpret the meaning of f (6).

Apply the idea

Reﬂect and check

We know that x represents the time in days since the plant was planted and x = 6. So this means that 6 days have passed since the plant was planted.

Using the graph, we can ﬁnd the actual height of the plant after 6 days.

Since f (x) represents the height of a growing plant in inches, f (6) represents the height of the plant 6 days after being planted.

Plant Growth 20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

c Interpret the meaning of f (x) = 12.

Apply the idea

Reﬂect and check

We know that f (x) represents the height of a growing plant in inches, so if f (x) = 12, then the height of the plant is 12 inches x days after being planted.

By using the graph, we can ﬁnd the number of days when the height of the plant is 12 inches. Plant Growth 20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

Idea summary Domain: set of all possible inputs Range: set of all possible outputs Interval Notation: [−4, ∞) Set notation: {x∣ −4 ≤ x < ∞}

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More characteristics of functions In addition to domain and range, characteristics for identifying and describing functions include: x-intercept

y-intercept

A point (x, 0) where a line or graph crosses the x-axis. The point represents the value of the domain where f (x) = 0. A function can have any number of x-intercepts.

A point (x, 0) where a line or graph crosses the y-axis. The point represents the value of the range where f (0) = y. A function can have at most one y-intercept.

An x-intercept can also be called a zero.

y x x

y-intercept

x-intercept

• Absolute maximum: The point with the largest y-value across the domain • Absolute minimum: The point with the smallest y-value across the domain • Relative maximum: The point with the largest y-value in a region of the domain • Relative minimum: The point with the smallest y-value in a region of the domain

7 y 6

Absolute maximum

5 4 3

Relative maximum

2 1 −5 −4 −3 −2 −1

−1

x 1

Relative −2 minimum −3

Asymptote

End behavior

A line that a curve approaches as one or both of the variables in the equation of the curve approach inﬁnity.

Describes the trend of a function or graph at its left and right ends; speciﬁcally the y-value that each end obtains or approaches.

Asymptote x

1.01 Characteristics of functions mathspace.co

End behavior as x → −∞, f (x) → ∞

3 2 1

−4 −3 −2 −1 −1

x 1

−2 −3 −4

End behavior as x → ∞, f (x) → −∞

Sections of functions can also display certain properties. A connected region in the domain over which the output values become higher as the input values become higher is known as an increasing interval. Similarly, a connected region in the domain over which the output values ( y-values) of a function become lower as the input values (x-values) become higher is known as a decreasing interval. 5 4

1 1

−5 −4 −3 −2 −1 −1

−2

Increasing intervals

Decreasing intervals

−5 −4 −3 −2 −1 −1

x 1

−2

−3

−4

−5

Increasing intervals: (−5, −2) ∪ (0, 4)

Decreasing interval: (−2, 0)

Note that we did not use square brackets to include the endpoints of the intervals. This is because the function is not increasing or decreasing at the points of change between increasing and decreasing intervals. At these points, the function is considered to have a rate of change of zero. If a function is neither increasing nor decreasing for part of its domain, we have a constant interval. In the blue, left most, portion of the graph shown, the function is not increasing or decreasing. The y-value never increases or decreases over that interval, but instead remains the same. This is a constant interval.

y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

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Constant Interval: (−∞, −5)

Example 5 Consider the function shown in the graph:

y 6 4 2 −4 −3 −2 −1 −2

x 1

−4 −6 −8 −10

a Determine the coordinates of the absolute maximum or minimum.

Create a strategy For this function, the vertex is the absolute minimum. Note the values on the x-axis change by 1, and the values on the y-axis change by 2.

Apply the idea

Reﬂect and check

The absolute minimum is the point (1, −4).

Recall the axis of symmetry is a characteristic of quadratic functions. The axis of symmetry is the line that passes through the middle of the parabola and the x-value of the vertex. So, the axis of symmetry for this parabola is x = 1.

b Determine the intervals where the function is increasing or decreasing.

Create a strategy The increasing intervals are the domain values for which the values of f (x) increase as x increases. The decreasing intervals are the domain values where f (x) decreases as x increases.

Apply the idea

Reﬂect and check

The function is increasing on the interval (1, ∞).

Since the graph changes directions at a domain value of 1, and the graph does not have end points, we do not use square brackets for the intervals. Square brackets would only be used for deﬁned endpoints of functions over a set domain.

The function is decreasing on the interval (−∞, 1).

c Write the domain and range of the function in interval notation.

Create a strategy Remember that the domain of the function is the set of all possible input values, which are the x-values that correspond to points on the graph. Similarly, the range of the function is the set of all possible output values, which are the y-values that correspond to points on the graph.

1.01 Characteristics of functions mathspace.co

Apply the idea If we were to continue extending both ends of the function indefinitely, it would stretch upwards towards positive infinity on both sides. In addition, the ends would continue indefinitely toward the left and toward the right. In part (a) we stated that the absolute minimum is the point (1, −4), this is also the vertex of the function. So, the domain is “all real values” and the range is “all values greater than or equal to −4”. In interval notation, this is: • Domain: (−∞, ∞) • Range: [−4, ∞)

Reﬂect and check In set notation, we would write the domain and range as follows: • Domain: {x∣x ∈ R} or {x∣ −∞ < x < ∞} • Range: {y∣y ≥ −4}

Example 6 Consider the function shown in the graph:

y 7 6 5 4 3 2 1 −4 −3 −2 −1

x 1

−1

a Determine the equation of the asymptote.

Create a strategy Notice how the function approaches the x-axis without ever reaching it:

y 7 6 5 4 3 2 1 −4 −3 −2 −1

−1

x 1

Apply the idea

Reﬂect and check

This means that the x-axis is an asymptote for the function. The equation of the horizontal asymptote is y = 0.

Asymptotes are lines, so they should always be stated as an equation. The equation for horizontal asymptotes is always of the form y = c, and the equation for vertical asymptotes is always of the form x = c where c is any real number.

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b Identify the intercepts.

Create a strategy In part (a), we found there is a horizontal asymptote on the x-axis. This means there will be no x-intercepts, so we only need to identify the y-intercept.

Apply the idea The y-intercept of the function is at (0, 1).

Reﬂect and check Recall from Algebra 1 that all the exponential functions of the form f (x) = bx will have a y-intercept at (0, 1). This is because f (0) = b0 = 1. Later in this unit, we will begin transforming functions, so we will see exponential functions that do not have a y-intercept at (0, 1).

c Describe the end behavior of the function.

Create a strategy The end behavior refers to the left and right “ends” of the graph. We want to know what happens to the output values as the input values get increasingly smaller (to the left) and what happens to the output values when the input values get increasingly larger (to the right).

y 7 6 5 4 3 2 1 −4 −3 −2 −1

−1

x 1

Apply the idea As the input values decrease indefinitely, the function output values continue to increase indefinitely. So as x → −∞, f (x) → ∞. As the input values increase indefinitely, the function output values approach the asymptote at y = 0. So as x → ∞, f (x) → 0.

1.01 Characteristics of functions mathspace.co

Example 7 Consider the shown function graph: 8

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

a Determine the increasing and decreasing intervals.

Create a strategy To determine the increasing and decreasing intervals, we need to ﬁrst ﬁnd the x-values of the turning points. These values will be the endpoints of the increasing and decreasing intervals. The turning points occur at x = − 3.5, −1, 1.5, 4, 6.

Apply the idea Tracing the graph from the smallest x-values toward the largest x-values, we can see that the graph switches between increasing and decreasing at each of the turning points. Increasing intervals: (−∞, −3.5) ∪ (−1, 1.5) ∪ (4, 6) Decreasing intervals: (−3.5, −1) ∪ (1.5, 4) ∪ (6, ∞)

Reﬂect and check Notice that we are not intersted in the y-values when listing the increasing and decreasing intervals. If we used the y-values, some of them would be listed in multiple intervals which would make the notation confusing. By only using the domain values, the notation is clear and the x-values do not appear in multiple intervals.

b Determine any absolute and relative maxima and minima.

Create a strategy We have already identiﬁed the x-values of the turning points in part (a). Now, we need to identify the corresponding y-values and classify each as an absolute maximum, an absolute minimum, a relative maximum, or a relative minimum.

Apply the idea The point with the largest y-value is (−3.5, 6). This is the absolute maximum. The relative maxima are (1.5, 0.5) and (6, 0). The relative minima are (−1, −3) and (4, −4). There is no absolute minimum since the end behavior of the function tends toward negative inﬁnity.

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Example 8 For the given graph: Graph 1 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Graph 2

5 4 3 2 1

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

a Compare the zeros of the two functions.

Create a strategy

Apply the idea

The zeros occur when the y-value is 0. In other words, the zeros are the x-values where the graph crosses the x-axis.

Looking at Graph 1, we can see the y-value is 0 at the x-values 4 and −4, so the zeros are: x = − 4, 4 Looking at Graph 2, since it’s a cube root function, it only has one zero at x = − 4. So, both functions share a zero of x = − 4.

b Which graph has a domain of (−∞, ∞)?

Create a strategy

Apply the idea

Remember that the domain of a function is the set of all possible input values, (x-values)

Looking at Graph 1, we see the function reaches all x-values, so its domain is all real numbers. Graph 2, being a cube root function, also reaches all x-values, since cube root functions are deﬁned for all real numbers. Both graphs have a domain of (−∞, ∞).

c Which function is only increasing?

Create a strategy

Apply the idea

The increasing intervals are parts of the domain where, when the y-values increase the x-values also increase.

The function in Graph 1 is increasing from (3, ∞), constant from (−3, 3), and decreasing from (−∞, −3). Looking at the function in Graph 2, as we observe the y-values as we move from left to right, the y-values are always increasing. The function is increasing over the interval (−∞, ∞). Graph 2 is only increasing.

1.01 Characteristics of functions mathspace.co

Idea summary The characteristics of functions include: • • • • • • •

domain and range x- and y-intercepts maximum or minimum value(s) zeros end behavior increasing, decreasing, and constant intervals asymptotes

Practice What do you remember? 1

For each of the functions identify the: i

Domain

Range

iii

End Behavior

Function m (x)

Function n (x)

y 8

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

−4

−6

−8

x 2

The table of values for the function P and for the function Q are provided. Function P : x −2 −1 0 1 2 y 9 6 3 0 −3

Function Q: x y

0 6

1 3

2 2

Determine what type of functions Function P and Function Q are.

Graph the functions on the same coordinate plane.

As x → ∞, determine which function will have the greater value.

Determine if each function is increasing or decreasing on the domain x < 2.

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3 3

4 6

Use the graph of the function f (x) to find each of the following values a

f (0)

f (4)

Find the value(s) of x such that f (x) = 0

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

Let’s practice 4

For each function: i

Determine the domain, using set and interval notation.

Determine the range, using set and interval notation.

b 8

2 −6 −4 −2

x 2

−4

−6

−8

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−8 −6 −4 −2 −2

−2

d 8

6 4

x 1 2 3 4 5

2 −8 −6 −4 −2 −2

−4 −6 −8

1.01 Characteristics of functions mathspace.co

For each graph shown: i

Determine the zero(s) of the function.

Determine the y-intercept(s) of the function.

4 2

2 −16 −14 −12 −10 −8 −6 −4 −2 −2

−8 −6 −4 −2 −2

2 4

7 6 5 4 3 2 1

2 1

x 2

−2 −3 −4 −6

x 1 2 3

For each graph shown: i

Identify any increasing, decreasing, or constant intervals using interval notation.

Determine the end behavior as x → ∞ and as x → −∞.

4 3 2 1

−8 −6 −4 −2−1

3 x 2

−2 −3 −4 −5 −6 −7 −8 −9

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3

−5

−8

−6

−2 −1 −1

−4

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2 1 −4 −3 −2 −1 −1 −2 −3 −4

x 1

3 2 1 −4 −3 −2 −1 −1

x 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−2 −3 −4

5 4 3 2 1

1 2 3 4 5

For each graph shown: i

Identify the ordered pairs of any relative maxima or minima.

Identify the ordered pairs of any absolute maxima or minima.

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

1 −4 −3 −2 −1 −1

x 1

4 3 2

−2

−3

−7−6−5−4−3−2−1 −1

−4

x 1 2 3 4 5 6 7

−2

1.01 Characteristics of functions mathspace.co

Evaluate each function for the given values of x or f (x). i

x=4

x=0

iii

x = −1

iv b

y 10

−1

−2 −1 −2

−2

−4

x 1 2 3 4 5 6 7 8

Which of the following key features are true regarding both of the functions below? Select all that apply. a

y 6

−6 −4 −2

−2

−4

−6 −5 −4 −3 −2 −1 −1

−6

−2

x 1

Both functions have an intercept at (0, 1).

Both functions have a domain of (−∞, ∞).

Both functions have an increasing and decreasing intervals.

The family of functions that each of these functions belong could have no zeros.

The family of functions that each of these functions belong to will always have exactly one y-intercept.

Compare and contrast the following key features for functions g (x) and h (x). i

Domain

Range

iii

Intercepts

Function g (x)

Function h (x)

y 6 5 4 3 2 1 −2 −1 −1

x 1

−2

f (x) = 1

−2

f (x) = 3

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7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

End Behavior

x 1 2 3 4

Consider the function f (x) = − 2− x shown in the graph. 2

Emily claims that the function has the following features:

• One x-intercept at (0, 0), which is also the y-intercept • A domain of (∞, ∞) • Decreasing across the whole domain • A range of [0, ∞) • Has a horizontal asymptote at y = 0 • Evaluated the function for x = 2 to be f (2) = 0.25

−4 −3 −2 −1 −1

−2 −3 −4 −5

Determine what error(s) Emily has made, and improve her answer.

−6

The height of a tennis ball after it is thrown is shown in the graph. a

Determine the coordinates of the y-intercept.

Interpret the y-intercept in this context.

Determine the coordinates of the zeros.

Interpret the zeros in this context.

Determine the range of the function, in context, that represents this graph, using set notation.

height

18 16 14 12 10 8 6

Determine the domain of the function, in context, that represents this graph, using set notation.

4 2

seconds 1

Identify the following key features for the graphed functions. i

Domain, in set notation.

Range, in set notation.

iii

Zeros

Intercepts

End Behavior

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

b 12 10

8 6

x 1 2 3 4 5

4 2 −5 −4 −3 −2 −1 −2

x 1 2 3 4 5

−4 −6

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Let’s extend our thinking 14

Sketch a graph that has the following features: • Domain of −∞ < x < ∞ • Increasing for x > 1 • x-intercepts at (−2, 0) and (4, 0) • A y-intercept below the x-axis

Sketch a graph that has the following features: • Domain of [0, ∞) • A y-intercept at (0, 4) • No x-intercepts • Decreasing across the whole domain

Sketch a graph that has the following features: • Increasing on the domain intervals (−∞, −2) and (4, ∞) • Decreasing on the domain interval (−2, 4) • Passes through the point (−2, 6) • Exactly two x-intercepts

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1.02 Function families After this lesson, you will be able to... • identify the family of a function given the graph or equation. • describe the general characteristics of each function family. • identify the family of a function given a description of the function’s characteristics.

Function families There are many types of functions, and we can group them into categories called function families. Function families consist of a parent function and all transformations of the parent function. Let’s look at the key features of several parent functions we will see in this course. Square root function A radical function with an index of 2. The square root parent function is represented by the form f (x) = x ≥ 0.

, where

• Domain: x ≥ 0 • Range: y ≥ 0 • Has an absolute minimum at (0, 0) • x-intercept at (0, 0) • y-intercept at (0, 0) • Increasing over its domain

y 8 7 6 5 4 3 2 1

x 1

Cube root function A radical function with an index of 3. The cube root parent function is represented by the form f (x) = y 2 1 x −2

−1

• Domain: all real numbers • Range: all real numbers • No absolute maximum or minimum • No relative maximum or minimum • x-intercept at (0, 0) • y-intercept at (0, 0) • Increasing over its domain

−1 −2

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Rational function A quotient of polynomials in which the denominator has a degree of at least 1. The rational parent functions are f (x) =

and f (x) =

where x ≠ 0 and y ≠ 0.

The rational parent function with a linear denominator is f (x) = . 4

3 2 1

−4 −3 −2 −1 −1

• Domain: x ≠ 0 • Range: y ≠ 0 • No absolute maximum or minimum • No relative maximum or minimum • No x-intercept or y-intercept • Asymptotes at the x-axis and y-axis • Decreasing over its domain

−2 −3 −4

The rational parent function with a quadratic denominator is f (x) = 4

3 2 1

−4 −3 −2 −1 −1

−2

• Domain: x ≠ 0 • Range: y ≠ 0 • No absolute maximum or minimum • No relative maximum or minimum • No x-intercept or y-intercept • Asymptotes at the x-axis and y-axis • Increasing for x < 0 • Decreasing for x > 0

−3 −4

Exponential function A function where the independent variable is in the exponent. An exponential function can be written in the form f (x) = abx where a ≠ 0 and b > 0

• Domain: all real numbers • Range: y > 0 • Horizontal asymptote at the x-axis • Increasing when b > 1 • Decreasing when 0 < b < 1 • No absolute maximum or minimum • No relative maximum or minimum • No x-intercept • y-intercept at (0, 1)

7 6 5 4 3 2 1 −4 −3 −2 −1

x 1

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Logarithmic function A function f (x) that represents the exponent to which b must be raised to get x. The logarithmic parent function is represented by the form f (x) = logb x, where x > 0, b > 0, and b ≠ 1.

3 2 1

−4 −3 −2 −1 −1

−2 −3

• Domain: x > 0 • Range: y ∈  • Increasing when b > 1 • Decreasing when 0 < b < 1 • No absolute maximum or minimum • No relative maximum or minimum • x-intercept at (1, 0) • No y-intercept • Asymptote at the y-axis

−4

Absolute value function A function that contains a variable expression inside absolute value bars; a function of the form f (x) = a∣x − h∣ + k

3 2 1 −4 −3 −2 −1 −1

x 1

• Domain: all real numbers • Range: y ≥ 0 • Has an absolute minimum at (0, 0) • x-intercept at (0, 0) • y-intercept at (0, 0) • Has a vertical line of symmetry separating the increasing and decreasing intervals

−2 −3 −4

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We have previously studied other types of functions known as polynomial functions. Some examples of polynomial functions are shown. 4

−4 −3 −2 −1 −1

−2

−3

−4

Constant function (degree = 0) 4

Linear function (degree = 1)

−4 −3 −2 −1 −1

1 −4 −3 −2 −1 −1

−2

−3

−4

Quadratic function (degree = 2)

x 1

Polynomial function (degree > 3)

Example 1 Identify the function family represented by each graph. a 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Create a strategy Use the shape and characteristics of the graph to determine the function family represented. Characteristics to consider include: domain, range, x-intercept, y-intercept, absolute and relative minimums or maximums, and increasing/decreasing intervals.

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Apply the idea This graph represents a cube root function because it has a domain of all real numbers, has a range of all real numbers, and increases over its domain.

Reﬂect and check Note that the graphs of transformed functions may not have the same exact key features as the parent function. This is because the key features of a function may change when it is transformed. For example, the y-intercept of the parent cube root function is at (0, 0), but the y-intercept of this cube root function is at (0, 3). Looking at the shape of the graph will be important for helping us determine the function family. Cube root functions will increase at a decreasing rate, then in the center, it changes and increases at an increasing rate. The point in the center is known as an inﬂection point, and we will learn more about this later.

b 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Apply the idea This graph represents a rational function because it has no x or y-intercepts, and it is decreasing over its domain. Additionally, it appears to have both a vertical and a horizontal asymptote because its domain is x ≠ 0 and its range is y ≠ 1.

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c 1

y x

−1 −1

−2 −3 −4 −5 −6 −7

Apply the idea

Reﬂect and check

This graph represents a logarithmic function because it appears to have an asymptote at the y-axis, it has a domain of x > 0 and a range of all real numbers, and it is increasing over its domain.

Square root functions and logarithmic functions appear to have a similar shape. Both functions increase at an increasing rate over their domain. The main difference between these functions is their range. A logarithmic function has a range of all real numbers, but a square root function’s range does not include all real numbers.

Example 2 Determine the type of function represented by each table. a

4 2

f (x)

5 3

8 4

13 5

20 6

Create a strategy To determine the type of function represented by a table, we can either plot the points on a coordinate plane and note the shape of the graph or examine how the function values increase or decrease.

Apply the idea 10 9 8 7 6 5 4 3 2 1

Plotting and connecting the points on a graph, it appears the points represent a square root function. • Domain: x ≥ 4 • Range: y ≥ 2 • Has an absolute minimum at (4, 2) • Increasing over its domain • No asymptotes

x 2 4 6 8 10 12 14 16 18 20

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0 8

f (x)

1 12

2 18

3 27

4 40.5

Create a strategy We can begin by plotting the given points on the coordinate plane and observing the characteristics. It is diﬃcult to identify characteristics with the limited domain provided by the table, and there could be a few options of functions this graph is representing. It could be: • A rational function • An exponential function • A polynomial function

y 40 35 30 25 20

From here, will we try to ﬁnd a pattern in the way the y-values increase.

15 10 5

x 1

Apply the idea An exponential function increases at a constant factor. Since the x-values increase by 1 each time, we can ﬁnd the constant factor by dividing the outputs. 12 ÷ 8 = 1.5 18 ÷ 12 = 1.5 27 ÷ 18 = 1.5 40.5 ÷ 27 = 1.5 Since the ratio of the outputs is the same, this function is exponential.

Example 3 Determine the type of function represented by each equation. a g (x) =

Create a strategy Use the structure of the equation to determine the function family represented. Notice that for the function families we have learned so far, the independent variable (usually x) is inside different symbols or located in different parts of the parent equations.

Apply the idea Since the independent variable (x) is in the denominator, this is a rational function.

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Reﬂect and check 4

Graphing the equation can also help us determine what type of function is represented by the equation.

This graph appears to have a vertical and horizontal asymptote at y = 0 and x = 1, so it is easy to identify that it is rational.

2 1

−4 −3 −2 −1 −1

−2 −3 −4

b y=

Create a strategy Use the structure of the equation to determine the function family represented.

Apply the idea Since the independent variable (x) is located inside a radical with an index of 2, this is a square root function.

Idea summary Function families consist of a parent function and all transformations of the parent function. The graphs of the parent functions within each family are shown. Linear

Quadratic

Square root

Cubic

y x

Absolute value

y x

Logarithmic

Rational-Linear

Rational- Quadratic

Exponential

y x

Cube root

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y x

Practice What do you remember? 1

Determine which family each function belongs to: a

f (x) = 8x + 7

y = 2x

g(x) =

p (x) = 5 + 4x + x2

y = 2x + 4 − 3

y − 2 = 4x +

x=6

Identify the function family for each of the graphs. a

x −5

−5

5 x

−5

x −5

5 −5

For each table of values: i

Use the table of values to sketch the graph on a coordinate plane.

Determine the function family described by the table of values.

x −1 0 1 2 3 y 7 4 1 −2 −5

x −1 0 1 2

y e

−12

−11

−10

x y

−2 18

−2

−9 4

−8

−7

−1 6

0 2 −1

1 6

2 18

−1

−6 7

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Let’s practice 4

Write the equation of the parent function for each graph: a

2 −8 −6 −4 −2 −2

−4

−6

−8

x 2

−8 −6 −4 −2 −2

−4

−6

−8

The graph of f (x) is shown. Determine whether each function belongs to the same function family as f (x). a

g(x) =

j(x) =

h(x) = 3 log2 x

k(x) =

−8 −6 −4 −2 −2

−4

−8 −6 −4 −2 −2

x 2

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

The table shows the approximate total number of patent applications in the U.S. for a number of years since 1900: a

Create a scatterplot using the data in the table.

Determine the function family described by the table of values. Explain your answer.

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Year 2010 2005 1995 1980 1950 1900

Applications (in thousands) 480 370 260 170 80 40

The graph of f (x) is shown. Determine whether f (x) appears to be a square root function or logarithmic function and explain your thinking.

9 8 7 6 5 4 3 2 1

−2 −1 −1 −2

The graph of g(x) is shown. Determine whether g(x) appears to be a rational function or cube root function and explain your thinking.

9 8 7 6 5 4 3 2 1 −2 −1 −1 −2

1 2 3 4 5 6 7 8 9

x 1 2 3 4 5 6 7 8 9

Consider the shown graph on which the logarithmic function y = log2 x has been sketched. Which of the statements below is correct? A

y = log2 x has a vertical asymptote at x = 2.

Because y = log2 x is always increasing, y = log2 x has no vertical asymptotes.

y = log2 x has a vertical asymptote at x = 0.

3 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Consider the shown graph of a rational function. a

State the equations of the vertical asymptote(s).

State the equations of the horizontal asymptote(s).

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

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What type of curve is shown on the graph? 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Compare and contrast characteristics of the parent functions of logarithmic and rational functions.

Let’s extend our thinking 13

Consider the list of function families: • Square Root • Rational • Exponential • Logarithmic

Determine which parent functions are only increasing on the interval {x∣x > 0}.

Identify the domain of each parent function in set notation.

Determine possible parent functions described below by considering its key features. a • b • Domain {x∣x ∈ } and range {y∣y > 0} Domain {x∣x ≠ 0} and range {y∣y ≠ 0} • Horizontal asymptote at y = 0 • Asymptotes at the x-axis and y-axis • It has a y-intercept at (0, 1) • No intercepts • As x increases, f (x) increases at an increasing rate • Decreasing over (−∞, 0) and (0, ∞) c • Domain {x∣x ∈ } and range {y∣y ∈ } • No asymptotes • Its intercepts are located at (0, 0) • As x decreases, f (x) decreases at a constant rate • As x increases, f (x) increases at a constant rate

Harry and Skye each have $2000 in savings to invest. • Harry chooses to put his savings in a fund that generates a return of $60 each month. • Skye chooses to put her savings in a fund that generates a return of 2% each month. a

Create a model to represent Harry and Skye’s investments.

What function family represents the value of Harry’s investment? Explain your answer.

What function family represents the value of Skye’s investment? Explain your answer.

Determine whose investment increased the most in value in the ﬁrst few months. Explain your answer.

Choose two or three families of functions and compare and contrast the characteristics of their parent functions.

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Translation A transformation in which every point in a ﬁgure is moved in the same direction and by the same distance Translations can be categorized as horizontal (moving left or right, along the x-axis) or vertical (moving up or down, along the y-axis), or a combination of the two. Vertical translations can be represented algebraically by g (x) = f (x) + k where k > 0 translates upwards and k < 0 translates downwards. 4

g(x)

f (x)

−4 −3 −2 −1 −1

1 2 f (x)

−4 −3 −2 −1 −1

−2

−3

g(x)

−4

Vertical translation of 4 units upwards: g (x) = f (x) + 4

−4

Vertical translation of 4 units downwards: g (x) = f (x) − 4

Similarly, horizontal translations can be represented by g (x) = f (x − h) where h > 0 translates to the right and h < 0 translates to the left. 4

1 −4 −3 −2 −1 g(x) −1

f (x) 1

x −3 −2 −1

−1

−2

−3

−4

Horizontal translation of 3 units left: g (x) = f (x + 3)

x 1

4 5 g(x)

Horizontal translation of 3 units to the right: g (x) = f (x − 3)

Vertical compression

Vertical stretch

A transformation that scales all of the y-values of a function by a constant factor towards the x-axis

A transformation that scales all of the y-values of a function by a constant factor away from the x-axis

Compressions and stretches are more generally called dilations. Vertical dilations can be represented algebraically by g (x) = af (x) where 0 < ∣a∣ < 1 corresponds to a compression and ∣a∣ > 1 corresponds to a stretch. 38

f (x)

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1.5

f (x)

g (x)

1 g(x)

0.5 −4 −3 −2 −1 −0.5

f (x)

−4 −3 −2 −1 −1

−1

−2

−1.5

−3

Vertical compression with scale factor of 0.5: g (x) = 0.5f (x)

Vertical stretch with a scale factor of 2: g (x) = 2f (x)

Horizontal compression

Horizontal stretch

A transformation that scales all of the x-values of a function by a constant factor toward the y-axis

A transformation that scales all of the x-values of a function by a constant factor away from the y-axis

Horizontal dilations can be represented algebraically by g (x) = f (bx) where ∣b∣ > 1 corresponds to a compression and 0 < ∣b∣ < 1 corresponds to a stretch. For horizontal stretches and compressions, b= y 3

y g(x) f (x)

f (x)

3 2

g(x)

x −1

x −2

Horizontal compression with a scale factor of 0.5: g (x) = f (2x)

−1

Horizontal stretch by a scale factor of 2: g (x) = f (0.5x)

When performing multiple transformations at once, we use the standard function notation a ⋅ f [b (x − h)] + k with the correct values of a, b, h and k to apply transformations to f (x). When given a transformed function, we must convert it back to standard notation to correctly identify the transformations applied to the parent function. We can use the relationship between an equation and its transformations to write equations and sketch graphs.

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Example 1 Consider the graph of the parent function f (x) and the transformed function g (x). Write the equation of g (x).

y 6 5 4 g(x)

3 2 1

−4 −3 −2 −1 −1

x 1

−2

Create a strategy We can ﬁrst note the transformations that have been applied to the parent function. We can see that g (x) has been reﬂected across the y-axis and translated 1 unit left. In function notation, this means b = − 1 and h = − 1 which corresponds to g (x) = f [− (x + 1)].

Apply the idea Because f (x) =

, we will substitute −(x + 1) for x underneath the radical. g (x) = f [−(x + 1)] g (x) =

The equation of the transformed function is g (x) =

or g (x) =

Reflect and check When identifying transformations that have been applied to a function, it is important to identify them in the correct order. We should always look for reflections and dilations first, then translations. 4

If we had written the translation ﬁrst and the reﬂection second, we would have mistakely found the equation to be g (x) = .

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

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As shown in the graph, this is not the same function as the given one. Additionally, notice that the coeﬃcient of x is factored out when written in function notation g (x) = f [b (x − h)] That helps us see the function is equivalent to g (x) = which represents a reﬂection across the y-axis and a translation to the right 1 unit.

Example 2 Point A (−3, 9) lies on the graph of f (x). Determine the coordinates of the corresponding point on the graph of g (x) =

⋅ f (x + 4).

Create a strategy We can use the given expression to determine the transformations from f (x) to g (x), then apply these transformations to the point A.

Apply the idea

Reﬂect and check

In the expression ⋅ f (x + 4), the +4 inside of function f indicates a horizontal translation of 4 units to the left,

We can also think about these transformations algebraically.

while the

outside of function f indicates a vertical

shrink by a factor of .

The only point we know on the graph of f is A, which tells us that f (−3) = 9. We can rewrite this to be in the correct form for g (x) as follows: f (−3) = 9

Shifting the point left 4 units takes the point to (−7, 9), and shrinking it vertically by 3 requires multiplying the

f (−7 + 4) = 9

y-coordinate by .

⋅ f (−7 + 4) = 3

9⋅

= 3 gives us the point (−7, 3).

g (−7) = 3

Known point Rewrite −3 in the form ⬚ + 4 Multiply both sides by

Use deﬁnition of g (x)

So, the corresponding point is (−7, 3).

Example 3 The graph of a function f (x) is shown.

f (x) 4 3 2 1 x −2

−1

a Determine the equation after the function has been horizontally dilated by a factor of 2, then translated 6 units to the right.

Create a strategy The graph is a parabola, which tells us that it is a quadratic function. The parent quadratic function is f (x) = x2. We need to give the transformed function a new name, like g (x), since we are creating a new function. A horizontal dilation and a horizontal translation are represented by b and h in the function notation g (x) = f [b (x − h)]. Remember b =

for horizontal stretches and compressions.

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Apply the idea In this case, the stretch factor is 2, so b = . To translate the function 6 units right, h = 6. To ﬁnd the equation after it has been horizontally translated and stretched, we need to ﬁnd

Horizontal dilation and translation

Since f squares its input

Distribute the square and evaluate the coefficient

The equation of the transformed function is g (x) =

(x − 6)2, or g (x) =

x2 − 3x + 9 in standard form.

Reﬂect and check Since the original function, f (x), is a quadratic function, we can graph it from its expanded form by factoring or using the axis of symmetry. However, for new function families, we may not want to expand the equation to standard form, and instead leave it as is after the substitution of a, bh, or k so it is easier to see the transformation.

b Graph g (x) and f (x) on the same coordinate plane.

Create a strategy A horizontal stretch will make the graph wider, and horizontal translation of 6 units to the right will move the vertex to (6, 0). We will stretch the graph horizontally to determine the correct shape of the graph, then we will shift it to the right.

Apply the idea Starting with a table of values for f (x) we can transform the x-values to account for the transformations. −2 4

x f (x) 2x

−1 1

0 0

1 1

For the parent function f (x), we can see five clear points where it crosses the grid, so can put these in the table of values.

2 4

−4

−2

When these points are dilated horiontally by a factor of 2, we multiply all x-values by 2 and leave the y-values as is. This means the curve will be stretched away from the y-axis.

takes the doubled x-values and “undoes” the doubling, so that is why the y-values stay as is and

Notice that the

why we take the reciprocal of the horizontal dilation when writing out the equation. 2x + 6

This gives that: x g (x)

2 4

4 1

6 0

8 1

10 4

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When these points are translated 6 units to the right we add 6 to all of the x-values and leave the y-values as is.

Using the tables to graph both functions, we can see we have stretched f (x) horizontally by 2. 9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1

2 4

x 1 2 3 4 5

Now, we simply need to shift all the points on the stretched graph to the right 6 units. 9 8 7 6 5 4 3 2 1 −2 −1 −1

g(x)

1 2 3 4 5 6 7 8

A better scale shows how the two graphs are related more clearly.

y 14 12 10 8 6 4 2 −2

g(x) 2

x 8

10 12

Reﬂect and check In general, when the transformations are both horizontal transformations (or a reflection across the y-axis), apply the reflections, stretches, and compressions first. Apply the translations last. The same rule applies when the transformations are both vertical (or a reflection across the x-axis).

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Example 4 Describe how g (x) = 3x − 5 − 6 has been transformed from its parent function, f (x) = 3x.

Create a strategy When we compare f (x) and g (x), we can see that there are numbers that have been added and subtracted. Addition and subtraction represent translations.

Apply the idea 5 is subtracted from the input values and 6 is substracted from the output values. In function notation, it can be represented as: g (x) = f (x − 5) − 6 The graph of g (x) is the graph of f (x) after it has been translated right 5 units and down 6 units.

Reﬂect and check When we graph the functions, we can see that f (x) has been shifted right 5 units and down 6 units to obtain the graph of g (x). y

f (x)

g(x)

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Idea summary The reﬂections and translations can be summarized as follows:

af [b(x − h)] + k

Vertical

Horizontal

a > 0: No reflection a < 0: Reflection across x-axis | a | > 1: Vertical stretch 0 < | a | < 1: Vertical compression

b > 0: No reflection b < 0: Reflection across y-axis | b | > 1: Horizontal compression 0 < | b | < 1: Horizontal stretch

k > 0: Vertical translation up k < 0: Vertical translation down

k > 0: Horizontal translation right k < 0: Horizontal translation left

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Practice What do you remember? 1

Match each transformation with its description: i

All y-coordinates of points on a graph are multiplied by a factor between 0 and 1.

Vertical Reﬂection

Changes the sign of the y-coordinates of all points.

Horizontal Reﬂection

iii

A shift of a graph to the left or right on the coordinate plane.

Vertical Stretch

Changes the sign of the x-coordinates of all points.

Vertical Compression

All y-coordinates of points on a graph are multiplied by a factor greater than 1.

A shift of a graph up or down on the coordinate plane.

Vertical Translation

Horizontal Translation

Describe each transformation of f (x). a

g(x) = f (x) + 4

g(x) = f (x + 6)

g(x) =

g(x) = − 3f (x)

g(x) = − f (x − 5)

g(x) = f (−x) − 7

g(x) = − 0.75f (x)

g(x) = 2f (x) +

Identify the equation of the parent function for each of the following graphs: a

−2

−3

−4

x 1

2 1 x

−3 −4

−2

−4 −3 −2 −1 −1

−1

−4 −3 −2 −1 −1

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1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

3 2

1 −4 −3 −2 −1 −1

x 1

−2

−3 −4 −3 −2 −1

−4

For each function, describe the transformation required to produce the graph shown: a

f (x) = x

f (x) = 2x

2 1 −4 −3 −2 −1 −1

x 1

−2

−3

x −2 −1

−4

f (x) = x2

f (x) =

y 2

1 −4 −3 −2 −1 −1

x 1

−2

3 2 1

−3

−4 −3 −2 −1 −1

−4

−2

−5

−3

−6

−4

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x 1

The graph of y = is translated horizontally 6 units to the right and vertically 5 units upwards. Write the new equation after the transformations are applied.

Let’s practice 6

Consider the table of values of f (x) = log2 x and the graph of g (x): a

Describe the type of transformation needed to get from f (x) to g (x):

Determine the equation of g (x).

1 0

f (x)

2 1

4 2

8 3

16 4

32 5

y 3 2 1 −1

g(x) 2 4 6 8 10 12 14 16 18

−2 −3 −4 −5

Determine the equation that corresponds to each graph after the transformation has been applied: a

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Vertically compress by a factor of c

5 4 3 2 1

x 1 2 3 4 5

Translate 1 unit to the left

1 2 3 4 5

Translate 5 units upwards

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Translate 2 units to the left

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y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Vertically stretch by a factor of 2 SOL

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Translate 4 units to the right

The function f (x) = x2 is transformed to create g(x) as shown in the graph. Determine which transformations to f (x) produced g(x). a

First, f (x) was ⬚.

B vertically stretch by a factor of 2 C reﬂected across the x-axis

−2 −3 −4

g(x)

The result was then ⬚.

−6 −5 −4 −3 −2 −1 −1

D reﬂected across the y-axis b

A translated right 2 units

−5

A translated left 2 units

−6

B vertically stretch by a factor of 2

−7 −8

C reﬂected across the y-axis D translated down 4 units 9

Consider the graph of the cube root function f (x) = and its transformed function g(x). Which of the following is the equation of the transformed function g(x)? A

y 1 x −4 −3 −2 −1

1 −1

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Consider the graph of various parent functions f (x) and their transformations g(x). i

Write g(x) in terms of f (x).

Write the equation of g(x). y

g(x)

1 −4 −3 −2 −1 −1

x 1

−2

g(x)

−3 −4

y 3

g(x)

g(x) x 1

−1 −2

−2

f (x) = log2 x

−3

−4 −3 −2 −1 −1

−4

g(x)

f (x) = x

1 f (x) = 2x

x −2

−1

g(x)

2 1

−2

−4 −3 −2 −1 −1

−4 −3 −2 −1

x 1

f (x) = log2 x

g(x)

3 2 1

−3 −4

x 1

2 3 4 5 6 7 8 9

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Consider the function f (x) = x2 − 5. Write the transformation(s) of f required to obtain the each function: a

h (x) = 16x2 − 5

k (x) = 3x2 − 15

If the x-intercepts of the graph of y = f (x) are (−5, 0) and (6, 0), determine the x-intercepts of each function: a

g (x) = x2 − 7 y = f (x + 4)

y = f (x − 4)

y = 3f (x)

y = f (−x)

For each statement, point A lies on the graph of f (x). Determine the coordinates of the corresponding point after the given transformation: a

Point A is (−3, −1). The transformation is f (x − 5) − 2.

Point A is (0, 3). The transformation is

Point A is (1, −2). The transformation is f (3x) − 2.

Point A is (12, 3). The transformation is 6f (6x).

Write an equation for each description and describe how the domain and range changed from the parent function. a

The parent function y = log2 x is vertically translated by 2 units up and horizontally translated by 4 units right.

The parent function y = 2x is reﬂected across the y-axis, vertically translated by 3 units down and then reﬂected across the x-axis.

The parent function y = x2 is horizontally translated by 6 units left and vertically translated by 1 unit up.

A landscape architect is designing a path in a park. The width of the path at different points, in meters, can be modeled by the square root function. Initially, the function modeling the path’s width is f (x) = , where x is the distance in meters from the start of the path. To fit the design specifications, the path needs to be widened. The modified width function, g(x), includes a vertical stretch by a factor of 3 and a horizontal shift 2 meters to the left. Write the equation of the transformed function, g(x), that models the width of the path.

A company manufactures a special type of liquid soap dispenser. The rate of soap released is initially modeled by the function f (t) =

, where t is the time in days since the last reﬁll.

To adjust for a more eﬃcient soap usage, the rate function needs to be recalibrated. The modiﬁed rate function, g(t), includes a vertical compression by a factor of

and a horizontal shift 3 days to the right. Write the equation

of the transformed function, g(t), that models the new rate of soap release. 17

Each expression represents one or more transformations of h(x) = x2. For each expression: i

Describe the effect of the transformation(s) on the graph of h(x).

Draw a graph of h(x) = x2 and its transformation on the same coordinate plane.

h(x) − 3

h(x − 3)

−3h(x)

The profit, P (x), in thousands of dollars, for a company that sells handmade crafts online can be modeled by a quadratic function based on the number of crafts sold, x. The profit function is given by P (x) = − (x − 50)2 + 144. Using graphing knowledge of transformations, sketch the graph of the profit function on a coordinate plane.

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Let’s extend our thinking 19

Consider the graph of f (x) = 6x and the graph of g (x) = 6x − 2: a

Find the y-intercept of each function.

Find the horizontal asymptote of each function.

For each type of transformation, explain whether or not each key feature change. If they change, explain how. • Intervals where the function is increasing or decreasing • Zeros of the function • End behavior

Horizontal translations

Reﬂections across the x-axis

Stretches and compressions

Consider the graphs of y = f (x) and y = g (x) shown. a

Describe a series of transformations to get from f (x) to g (x).

Phoebe thinks she has two different sets of transformations to get from f (x) to g (x), but she isn’t sure which one is correct.

Can you ﬁnd a second series of transformations that is different to your answer in part (a)?

−8 −6 −4 −2 −2

x 2

f (x)−4 −6 g(x) −8

Crystal jogs at a constant speed of 5 mph. a

Determine the type of function that could be used to represent Crystal’s distance traveled over time.

If Crystal starts her jog half a mile from her house, describe the transformation that this represents.

Yasushi and Peter are looking at the graphs of a few types of functions. a

Yasushi claims that a translation of a units to the right and a translation of b units upwards can be performed in either order and still get the same result. Determine whether or not Yasushi is correct. Support your answer with an example or counter-example.

Peter thinks that the same might be true for other types of transformations, and that the order in which transformations is applied doesn’t matter. Determine whether or not Peter is correct. Support your answer with an example or counter-example.

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1.04 Piecewise functions After this lesson, you will be able to... • identify the domain, range, zeros, and intercepts of a piecewise function given its equation or graph. • identify discontinuities of a piecewise function. • compare and contrast the characteristics of piecewise functions with other functions. • identify increasing, decreasing, and constant intervals of piecewise functions. • identify relative and absolute maxima and minima. • find the value of f (x) given x, and vice versa. • describe the end behavior of a piecewise function.

Piecewise functions When more than one function is needed to create a model for a given situation, we can use a piecewise function to connect the different pieces. A domain is given for each individual function which makes up the piecewise function. Each piece has a domain where the given function applies Piecewise function name

f (x) =

x−3

x < −2

−4

−2 ≤ x < 0

x2 − 4

x>0

Functions

Domains

Each piece is on a separate line

Brace

To graph piecewise functions, it can be helpful to first find all of the endpoints, then determine what the graph will look like between them. The inequalities for the domains help us determine whether the endpoints are filled (closed) or unfilled (open). Filled (closed): • for ≤ or ≥ Unfilled (open): ∘ for < or > 5 4 3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6

For the piecewise function above: • The line for f (x) = x − 3 is drawn for x-values less than −2. • The line for f (x) = − 4 is only drawn for x-values between −2 and 0, including −2 but not 0. • The parabola for f (x) = x2 − 4 is drawn for x-values greater than 0.

−2 −3 −4 −5 −6 −7

The domain of the function is all real numbers except for 0 because 0 is the only value where there is no solid graph. The range is all real numbers except for those between −4 and −5. −4 is included in the domain because there is solid graph at a y-value of −4 but −5 is not included in the range.

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A discontinuity occurs when there are values missing in the domain or range. Graphically, a discontinuity appears as a hole or a gap in the graph. For example, the graph above has a discontinuity at x = − 2 and at x = 0.

Example 1 Consider the piecewise function:

a Evaluate the function at f (−7), f (−4), f (−2), f (1).

Create a strategy For each value of x, we can determine which function from the piecewise function to evaluate based on the given domain.

Apply the idea Since x = − 7 is in the domain of x + 6: f (−7) = (−7) + 6 = − 1 Since x = − 4 is in the domain of 4: f (−4) = 4 Since x = − 2 is not in the domain, f (−2) is undefined Since x = 1 is in the domain of (x + 2)2 + 5: f (1) = (1 + 2)2 + 5 = 32 + 5 = 14

b Graph the piecewise function.

Create a strategy The three functions have their own place on the coordinate plane, so we can draw each function and erase the part of the function in the domain that the line does not belong to.

Apply the idea

Reﬂect and check Note that this is still considered a function because it passes the vertical line test, or rather, each input has a single, unique output.

y 14 12 10 8 6 4 2 −7 −6 −5 −4 −3 −2 −1

x 1

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c Describe the key features of the function graphed in part (b).

Create a strategy Recall that key features may include domain, range, intercepts, minimum and maximum points, increasing and decreasing and constant intervals, and end behavior.

Apply the idea

Reﬂect and check

Domain: (−∞, −2) ∪ (−2, ∞)

Notice that the domain is not continuous because there are two gaps in the range.

Range: (−∞, 2) ∪ [4] ∪ (5, ∞) x-intercept: (−6, 0) y-intercept: (9, 0) Increasing on (−∞, −4) and (−2, ∞) Constant on (−4, −2) End behavior: As x → −∞, y → −∞ and as x → ∞, y → ∞

Example 2 Write the piecewise-deﬁned function based on the graph shown:

y 10 8 6 4 2 −4 −3 −2 −1 −2

x 1

−4 −6

Create a strategy By covering up each part of the graph or imagining that we can extend each line or curve, we can identify the types of graphs that are in the piecewise function.

Apply the idea 10

For x ≤ 0, the graph is a straight line. We can determine its equation by ﬁnding the y-intercept and counting the rise and run to determine the slope.

8 6

The equation of the line is y = 4x + 3 for x-values less than or equal to 0.

4 2 −4 −3 −2 −1 −2

x 1

−4 −6

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For x > 0, the graph appears to be quadratic or exponential. By ﬁnding the ratio of the y-values, we can see it is an exponential function.

y 10 8 6 4 2 −4 −3 −2 −1 −2

x 1

The equation of the curve is y = 3x for x-values greater than 0.

−4 −6

The piecewise-deﬁned function for the given graph is

Example 3 Determine the types of functions that are in this piecewise function.

Create a strategy There are 3 types of functions that make up this piecewise function: • y=x+4 • y = −2 • y = 12 − x2 We can use the structure of each equation to determine the function family it belongs to.

Apply the idea The first equation is of the first degree and in the form f (x) = mx + b. In this case, m = 1 and b = 4. Therefore, the function that defines the interval x < 0 is linear. The second equation is in the form f (x) = c. In this case, c = − 2. Therefore, the function that defines the interval 0 ≤ x < 4 is constant. The third equation is of the second degree and in the form f (x) = ax2 + bx + c. In this case, a = − 1, b = 0, and c = 12. Therefore, the function that defines the interval x ≥ 4 is quadratic.

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Reﬂect and check 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

If we had looked at the graph alone, we may have mistakenly classiﬁed the functions. Note that the linear function could have been part of an absolute value function, and the piece of the quadratic that is shown also appears linear. This is why it is important to consider multiple features of the function instead of relying on a graph alone.

1 2 3 4 5 6 7

Example 4 Consider the functions shown: • Function 1:

• Function 2:

g(x) 20 15 g(x)

10 5

x −4 −3 −2 −1

a Determine which function is increasing if x > 0.

Create a strategy To visualize where Function 1 is increasing, we can graph it and compare it to the graph of Function 2.

For x-values to the left of the y-axis, the graph is an upside down parabola translated 5 units up. The y-intercept will be a closed circle.

For x-values to the right of the y-axis, the graph is a line with a slope

Function 1 y

2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

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and a y-intercept of −3. The y-intercept will be an open circle.

It is now easy to see that Function 1 is decreasing when x > 0.

Apply the idea If x > 0, Function 2 is increasing.

Reﬂect and check We also could have just looked at the equation for Function 1 when x > 0, and identiﬁed it as a linear function with a slope of

, meaning the function would be decreasing over that interval.

b Determine whether or not each function has a maximum or minimum value.

Create a strategy Use the graph of the piecewise function from part (a) and the given graph of Function 2 to analyze them more easily.

Apply the idea Function 1 has a maximum point at the point (0, 5), which is the vertex of the quadratic portion of the piecewise function.

Function 1 y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Function 2 is exponential, and has a horizontal asymptote at y = 5. Because the function never actually reaches 5 as x approaches negative inﬁnity, we can not call this a minimum value.

Function 2 g(x) 20

There are an inﬁnite number of smaller and smaller fractions that get close to, but never quite reach, 5.

As x approaches positive inﬁnity, the exponential function will also approach positive inﬁnity.

g(x)

Therefore, Function 2 does not have a minimum or maximum value. 5 x −4 −3 −2 −1

Idea summary A piecewise function is made up of two or more functions. The domain of a piecewise function is visible in both the piecewise-deﬁned function and the graph.

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Practice What do you remember? 1

Explain what a piecewise function is. Include an example in your explanation.

For the piecewise function:

Find: a 3

f (−2)

f (0)

f (1)

f (3)

Complete the domain of the piecewise function based on the graph.

f (x) 2 1 x −4 −3 −2 −1

2 3 4 5

−1 −2

Consider the graph of y = f (x) shown: a

Find f (−2).

State the domain of f (x) in interval notation.

State the range of f (x) in interval notation.

State the coordinates of any zeros of f (x).

State the coordinates of any y-intercepts of f (x).

8 7 6 5 4 3 2 1

−12−10−8 −6 −4 −2 −1

2 4 6 8 10 12

−2

For the graph of y = f (x): a

Identify the increasing interval(s) of the function in set notation.

Identify the decreasing interval(s) of the function in set notation.

Identify the constant interval(s) of the function in set notation.

Identify the end behavior of the function as x → ∞.

Identify the end behavior of the function as x → −∞.

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4 3 2 1 −8 −6 −4 −2−1

−2 −3 −4 −5 −6 −7 −8 −9

x 2

Given the piecewise function: 4

f (x)

3 2 1

Explain why the value x = − 1 is part of the domain, but the value x = 1 is not part of the domain of this function.

−4 −3 −2 −1 −1

2 3 4 5

−2 −3 −4

Let’s practice 7

Identify the key features in interval notation for each of the piecewise functions graphed. i

Domain and Range

Increasing, decreasing, and constant interval(s)

iii

End behavior

4 3

2 1

−8 −6 −4 −2 −1

−3 −4

y 7 6 5 4 3 2 1

−6 −5 −4 −3 −2 −1 −1

−2 −3 −4 −5 −6 −7

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−2

5 4 3 2 1

1 2 3 4 5

y 4 3 2 1

x 1 2 3 4 5 6 7

−8 −6 −4 −2 −1

x 2

−2 −3 −4

1.04 Piecewise functions mathspace.co

For the graph of y = f (x): a

Write the domain of f (x) in interval notation.

Write the range of f (x) in interval notation.

Describe the end behavior of y = f (x).

Identify the intervals where the function is increasing, decreasing, or constant.

10 9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

x 1 2 3 4 5

Identify the ordered pairs for each of the piecewise functions graphed. i

Relative and Absolute Maxima

Relative and Absolute Minima

iii

x-intercept and y-intercept

4 3 2 1 −8 −6 −4 −2 −1

x 2

−3 −4

4 3 2 1 x −4 −3 −2 −1

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5 4 3 2 1 −5−4−3−2 −1 −1 −2 −3 −4 −5

−2

x 1 2 3 4 5 6 7

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

• Function A:

Consider the following pair of functions: a

Determine which function has the smallest x-intercept.

Determine which function has the smallest minimum.

Determine for which function y is increasing most rapidly over x > 0.

• Function B: 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

For the piecewise function:

Find:

f (2)

f (−6)

f (−2)

f (−4)

f (−1)

f (6)

f (8)

f (0)

For the graph of y = f (x), ﬁnd: a

f (−2)

f (4)

f (0)

x when f (x) = − 2

x when f (x) = 2

−4 −3 −2 −1 −1

x 1

−2 −3 −4

Horacio goes out for a run. From rest, he accelerates to a desired speed and maintains that speed for some time. Feeling exhausted, his velocity drops until he’s back at rest. The speed S in kilometers per hour after t seconds is given by the following piecewise relationship:

Find the speed in kilometers per hour after the following times have elapsed: a

2 seconds

4 seconds

150 seconds

250 seconds

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Let’s extend our thinking 14

A piecewise function is deﬁned as

3 2

The graph of f (x) is shown.

Write the domain of the function in interval notation.

Does the function have a discontinuity? Explain your answer.

−4 −3 −2 −1 −1

−2 −3 −4

Without graphing, decide whether the piecewise function has a discontinuity. Justify your answer.

For the function f (x) deﬁned as:

Find the value of the function when x = 5.

Determine the value of x if the function f (x) is equal to 9.

Describe the end behavior of y = f (x).

Sketch the graph of the function.

Write the piecewise function deﬁnition of each graph: a

9 8 7 6 5 4 3 2 1

6 4 2 −3 −2

−1

x 1

−2 −4 −6

−5 −4 −3 −2 −1 −1

x 1 2 3 4 5

Sketch the graph of a piecewise function that satisﬁes the following requirements: • Domain is (−∞, ∞) • Range is [6, −∞) • Contains both linear and quadratic pieces.

• f (−4) = 6 • f (0) = − 3 • f (4) = − 1

Label each piece with its equation. 19

Write two contexts where piecewise functions would be appropriate to model the behavior. Explain your reasoning.

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The solutions to absolute value equations can be veriﬁed graphically. Once the absolute value expression is isolated, we can create a table of values for the corresponding absolute value function. In the previous example, the isolated absolute value expression is ∣x − 2∣, so the corresponding function is y = ∣x − 2∣. x f (x)

−1 3

0 2

1 1

2 0

3 1

4 2

We can create a table of values by choosing x-values and substituting them into the function to ﬁnd the corresponding y-values.

5 3

Next, we can plot these points on a coordinate grid, and connect the points to graph the function.

y 5

The solution(s) to the absolute value equation will be the x-value(s) of the points where the output of the absolute value function is equal to 3.

4 (−1, 3)

(5, 3)

To help us ﬁnd these points, we can graph the constant function y = 3.

2 1

This veriﬁes that the solutions to ∣x − 2∣ = 3 are x = − 1 and x = 5.

x −2 −1

Example 1 Solve each absolute value equation. a ∣x∣ = 10

Apply the idea The solutions to this equation will be any values of x that have an absolute value of 10. The solutions are x = 10 and x = − 10 or {−10, 10}.

Reﬂect and check Notice that this equation has different solutions from the equation ∣x − 10∣ = 0 which has only a single solution of x = 10. This shows that adding or subtracting terms in or out of the absolute value bars changes the equation, similar to parentheses.

b ∣2.5x∣ = 5

Create a strategy Since the absolute value expression is isolated, we create and solve the following two equations: 2.5x = 5 2.5x = − 5

Apply the idea Solve the ﬁrst equation: 2.5x = 5 = x=2

First equation Divide both sides by 2.5 Evaluate the division

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Solve the second equation: 2.5x = − 5

Second equation

Divide both sides by 2.5

x = −2

Evaluate the division

The solutions to this equation will be any values of x that make 2.5x have an absolute value of 5. The solutions are x = 2 and x = − 2 or {−2, 2}.

Reflect and check We can verify that the solutions are viable solutions to the absolute value equation by graphing the function y = ∣2.5x∣ and y = 5 and finding the points of intersection. To graph the function y = ∣2.5x∣, we will begin by creating a table of values and substituting the x values into the function to find the corresponding y values. x f (x)

−2 5

−1 2.5

0 0

1 2.5

2 5

Next, we will plot the points on the coordinate plane and connect the points to graph the function. To help us ﬁnd the solutions, we can graph the constant function y = 5 The constant line intersects with the absolute value function at (−2, 5) and (2, 5) which veriﬁes that the solution set {−2, 2} is correct. Both x = − 2 and x = 2 are viable solutions. y 5 4 3 2 1 x −4 −3 −2 −1

c ∣4x − 8∣ = − 12

Apply the idea The absolute value of an expression can never be a negative number since absolute value is a distance. In this case, there are no solutions.

Reﬂect and check We can attempt to solve this equation, but checking our solutions will reveal that both are extraneous. 4x − 8 = − 12

First equation

4x − 8 + 8 = − 12 + 8

Add 8 to both sides

4x = − 4

Evaluate the addition

Divide both sides by 4

x = −1

Evaluate the division

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When we substitute x = − 1 in ∣4x − 8∣ = − 12, we get ∣4 (−1) − 8∣ = ∣−4 − 8∣ = ∣−12∣ = 12. The solution is not equal to −12, therefore this solution is extraneous. 4x − 8 = 12

Second equation

4x − 8 + 8 = 12 + 8

Add 8 to both sides

4x = 20

Evaluate the addition

Divide both sides by 4

x=5

Evaluate the division

When we substitute x = 5 in ∣4x − 8∣ = − 12, we get ∣4 (5) − 8∣ = ∣20 − 8∣ = ∣12∣ = 12. The solution is not equal to −12, so this solution is also extraneous.

d 2∣x + 1∣ + 3 = 21

Create a strategy First, we need to perform inverse arithmetic operations to isolate the absolute value expression. From there, we can create two separate equations and solve them.

Apply the idea 2 ∣x + 1∣ + 3 = 21

Original equation

2 ∣x + 1∣ + 3 − 3 = 21 − 3

Subtract 3 from both sides

2 ∣x + 1∣ = 18

Evaluate the subtraction

Divide both sides by 2

∣x + 1∣ = 9

Evaluate the division

Now that the absolute value expression is isolated, we can solve the following two equations: x+1=9 x + 1 = −9 Solve the ﬁrst equation: x+1=9

First equation

x+1−1=9−1

Subtract 1 from both sides

x=8

Evaluate the subtraction

Solve the second equation: x + 1 = −9 x + 1 − 1 = −9 − 1 x = − 10

Second equation Subtract 1 from both sides Evaluate the subtraction

The solutions are x = 8 and −10. In set notation, this is written as {−10, 8}.

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Reﬂect and check We can verify that the solutions to the absolute value equation are viable solutions by using technology to graph the function y = ∣x + 1∣ and the line y = 9. Then, we will look for the x-values of the points of intersection.

The constant line intersects with the absolute value function at (−10, 9) and (8, 9) which veriﬁes that the solution set {−10, 8} is correct.

Both x = − 10 and x = 8 are viable solutions.

y 14

8 6 4 2 −10 −8 −6 −4 −2

x 2 4 6 8

Example 2 A machine is used to ﬁll each of several bags with 16 ounces of sugar. After the bags are ﬁlled, another machine weighs them. The bag can only weigh 0.3 ounces heavier or lighter than the desired weight, otherwise, the bag is rejected. Write the equation for the heaviest and lightest bag the machine will approve.

Create a strategy In general, we can determine the minimum and maximum weight allowances for a bag of sugar based on the problem. The minimum and maximum weight allowances are the solutions to the absolute value equation, so the equation we write should lead to those two solutions.

Apply the idea A bag of sugar can weigh 16 − 0.3 or 15.7 ounces at its lightest, and 16 + 0.3 or 16.3 ounces at its heaviest. On a number line, we can see that the weights are 0.3 from 16: 15 oz

15.5 oz

16 oz

16.5 oz

17 oz

The absolute value equation that represents the difference between the bag’s weight, 16 ounces, and its weight allowances is 0.3 ounces. Therefore, the equation that represents the minimum and maximum weight allowances for a bag of sugar is ∣x − 16∣ = 0.3.

Reﬂect and check Using a number line to identify the solutions and distance from a particular value on the number line for a problem in context helps us visualize the problem.

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Idea summary An absolute value equation is an equation where at least one expression contains an absolute value. Consider the absolute value equation ∣ax + b∣ = k When k > 0, an absolute value equation has two solutions. When k = 0, an absolute value equation has one solution. When k < 0, an absolute value equation has no solutions. Viable solutions for an absolute value equation also depend on context. If a solution is mathematically valid but does not make sense in the context then we say it is non-viable.

Practice What do you remember? 1

Describe what the absolute value of −4.5 represents on a number line.

Find the possible values of the variable for each of the following and explain you reasoning. a

∣a∣ = 9

∣b∣ = 4.5

∣d∣ = − 7

Write an absolute value equation that represents each of the following: a

All real numbers x that are 8 units away from 0.

All real numbers x that are 5 units away from −2.

Let’s practice 4

Solve each of the following equations: a

3∣m ∣ = 9

e 5

1.5 = 0.5 ∣ p ∣

Solve each of the following equations: a

4 = ∣3u + 1∣

∣−4x∣ = 12

−10 = ∣y + 2∣

c g

0 = ∣z + 1.2∣

Compare and contrast the solutions to the equations: 2y + 3 = 11 and ∣2y + 3∣ = 11

Solve each of the following equations: a

2 + ∣h − 1∣ = 5

e 8

10 = −5∣p∣

∣w − 0.75∣ = 2.25

∣5x − 3∣ = −7

c g

−6∣5x∣ = 15

Solve each of the following equations: a e

−7∣x + 6∣ − 14 = 20

2 ∣n + 1∣ + n = 6

5∣z − 3∣ = z − 25

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5 − ∣3m + 1.5∣ = 1.5

How many solutions does the equation 3 ∣p + 2∣ − 1 = 3p + 11 have?

For each of the following: i

Solve the absolute value equation and identify which solutions are extraneous and which are valid.

Verify your solutions graphically.

∣x + 6∣ = 2x

∣2.4x − 10∣ = 4x + 6.8

∣3x + 2∣ = 4x + 5

∣x − 1∣ = 5x + 10

−2∣x∣ = 6 − 2x

Amora is solving an absolute value equation. Her ﬁrst step was to write the equations 5x − 2 = 3 and 5x − 2 = − 3. What was the original aboslute value equation?

For each of the following relations:

Write an equation for the relation, using w to represent the unknown number.

Solve the equation for w.

The size of the difference between a number and 3 is equal to ﬁve more than two times the number.

The size of the difference between twice a number and 5 is equal to eight less than three times the number.

The size of the sum of two-thirds of a number and 2 is equal to 4 minus the number.

The minimum and maximum lengths of AA batteries, b in millimeters, can be represented with the absolute value equation ∣b − 49.9∣ = 0.6. a

Solve the absolute value equation.

Interpret your solution in the context of the scenario.

Baby clothing is rated using a TOG rating for what temperatures it is designed for to avoid the baby being too cold or overheating. A sleep sack with rating 2.5 TOG is designed for 65 °F, plus or minus 4 °F. Write an absolute value equation that represents the minimum and maximum temperatures for that sleep sack.

Roxanne performed the following steps to solve the equation 4 ∣x∣ = − 2: 1

4 ∣x∣ = − 2

4x = ± 2

Show that neither of these solutions is valid and explain where her error was. 16

For each absolute value equation: a

Solve the equation and justify your steps

Verify your solutions graphically.

∣2x − 3∣ = 7

∣3x + 1∣ − 5 = 0

2 ∣x − 4∣ + 3 = 11

Compare and contrast the process for solving each equation and state their solutions. • Equation A: ∣3x∣ + 4 = 5 • Equation B: ∣3x + 4∣ = 5

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Consider the following equation m = ∣kx − 3∣ + 7. Samara rearranges the equation to isolate x by considering two separate cases. They start by considering the case where kx − 3 ≥ 0 as follows: 1

m = (kx − 3) + 7

m = kx + 4

m − 4 = kx

Combine like terms ⬚

Division property of equality

⬚

Determine the property that justiﬁes Samara’s third step of work.

Determine the property that justiﬁes Samara’s last step of work.

Find the solution for the case where kx − 3 < 0. Justify each step of work.

Determine the values of m that will lead to the equation having no valid solutions.

Let’s extend our thinking 19

Describe why the absolute value of a number cannot be negative. Give an example of an absolute value equation with no solution.

Determine the value(s) of b for which the equation ∣x∣ = b has only one solution.

Compare and contrast the process for solving each equation and state their solutions. Assume a, b > 0. • Equation A: ∣ax∣ = b • Equation B: a ∣x∣ = b

Sayuri swims two laps of a 25 meter pool. Her distance, d meters, from the end where she started after t seconds, is given by:

Write an equation that could be used to solve for t.

Find the value(s) of t when d = 23.

Sayuri needs to start her turn 2 meters from the end of the pool, and wants to ﬁnd the time to start her turn. Explain whether or not the solutions found in part (b) are reasonable solutions in this context.

Consider the equation: ∣kx − 9∣ = y + 6. a

Write an equation that could be used to solve for x.

Determine the values of y that will lead to the equation having no valid solutions.

Clem and Finn are waiting in line at a petting zoo. Clem is 6 spots in line away from Finn. Consider an absolute value equation that represents Clem’s possible position, compared to Finn’s position. a

Describe the positions for Finn where there are two possible positions for Clem.

Describe the positions for Finn where there is only one possible position for Clem.

Describe the positions for Finn where there is no possible position for Clem.

Kaydence is running to meet her friend at the local market. She knows that her average speed is 7.5 mph (0.125 miles per minute). She tells her friend that she will be there in 30 minutes, plus or minus 6 minutes depending on the route she ends up taking. Remember that:

Write and solve an equation for the minimum and maximum distances for her run. 70

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1.06 Compound inequalities After this lesson, you will be able to... • write compound inequalities to represent real-world situations. • solve linear compound inequalities in one variable. • represent solutions to compound inequalities in set and interval notation. • represent solutions to compound inequalities on a number line.

Compound inequalities A compound inequality is a conjunction of two or more inequalities. The set of solutions for a compound inequality are the values that make all of the inequalities true. −5 −4 −3 −2 −1 0 1 2 3 4 5

• We use “and” to indicate that a value must satisfy both inequalities in order to be in the solution set. For example: x < 3 and x ≥ −2 • We can also write this compound inequality more simply as −2 ≤ x < 3

−5 −4 −3 −2 −1 0 1 2 3 4 5

• We use “or” to indicate that a value need only satisfy at least one inequality in order to be in the solution set. For example: x > 3 or x ≤ −2

Interval

Interval notation

A set of numbers that lie between two values

A way to represent a solution set or interval as a pair of numbers using a combination of square brackets and parentheses. Example: Inequality notation 3≤x Interval notation [3, ∞)

We use square brackets if the endpoint is included and parentheses if the endpoint is not included. We always use parentheses for inﬁnity. We can join two sets together using the union symbol ∪. −5 −4 −3 −2 −1 0 1 2 3 4 5

• The solution in interval notation is written as [−2, 3) • The solution in set notation is written as {x∣−2 ≤ x < 3}

−5 −4 −3 −2 −1 0 1 2 3 4 5

• The solution in interval notation is written as (−∞, −2] ∪ (3, ∞) • The solution in set notation is written as {x∣x ≤ −2, x > 3}

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Example 1 Consider the graphed solution of a compound inequality on the number line shown: −5

−4

−3

−2

−1

a Write a compound inequality to represent the solution set.

Create a strategy Describing the compound inequality in words can help us write the solution to the compound inequality algebraically. Description: x is less than −1 or x is greater than or equal to 2

Apply the idea Compound inequality: x < −1 or x ≥ 2

b Write the solution to the compound inequality in interval notation.

Create a strategy Since the solution to the inequality occurs in two intervals, we should write the two intervals connected with the union symbol.

Apply the idea

Reﬂect and check (−∞, −1) ∪ [2, ∞)

The unﬁlled circle at −1 indicates that −1 is not in the solution set of the compound inequality, which is stated with the inequality x < −1. The ﬁlled circle at 2 indicates that 2 is in the solution set of the compound inequality, which is stated with the inequality x ≥ 2.

Example 2 Solve and graph each compound inequality. a −2.9 ≤ −2.4x + 7.3 < 3.7

Create a strategy An “and” compound inequality can be solved simultaneously as long as we apply each operation to all parts of the inequality.

Apply the idea −2.9 ≤ −2.4x + 7.3 < 3.7 −2.9 − 7.3 ≤ −2.4x + 7.3 − 7.3 < 3.7 − 7.3 −10.2 ≤ −2.4x < −3.6 ≥

4.25 ≥ x > 1.5

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Original inequality Subtraction property of inequality Evaluate the subtraction Division property of inequality Evaluate the division

By the division property of inequality, division by a negative number requires reversing the inequality symbols. We can write the inequality in order from its minimum value to its maximum value by the symmetric property of inequality: 1.5 < x ≤ 4.25. In set notation, this would be {x∣ − 1.5 < x ≤ 4.25}. In interval notation, this would be (1.5, 4.25]. Finally, we can graph it on the number line with an unﬁlled endpoint at 1.5 because the inequality is not an “equal to” inequality and a ﬁlled endpoint at 4.25 because the inequality is an “equal to” inequality. 0

Reﬂect and check We could have also written the compound inequality as two inequalities and solved them separately. Solve the ﬁrst inequality: −2.9 ≤ −2.4x + 7.3

First inequality

−2.9 − 7.3 ≤ −2.4x + 7.3 − 7.3

Subtraction property of inequality

−10.2 ≤ −2.4x

Evaluate the subtraction

≥

Division property of inequality

4.25 ≥ x

Evaluate the division

Solve the second inequality: −2.4x + 7.3 < 3.7

Second inequality

−2.4x + 7.3 − 7.3 < 3.7 − 7.3

Subtraction property of inequality

−2.4x < −3.6

Evaluate the subtraction

Division property of inequality

x > 1.5

Evaluate the division

When writing the final compound inequality, we can graph the solutions on the number line, then write the compound inequality to match the solution set. First, plot a filled circle at 4.25 and an unfilled circle at 1.5. Then, determine values of x that make both 4.25 ≥ x and x > 1.5 true. 0

Based on the number line, the values of x that satisfy both inequalities are between 1.5, (exclusive) and 4.25, (inclusive). The solution to the inequality is 1.5 < x ≤ 4.25.

+ 8 < 2 or 12x + 9 ≤ −15

Create a strategy We can solve the two inequalities separately using properties of inequality and graph the solution on the same number line.

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Apply the idea Solve the ﬁrst inequality: First inequality

Subtraction property of inequality

Evaluate the subtraction

Multiplication property of inequality

Evaluate the multiplication

Solve the second inequality: 12x + 9 ≤ −15

Second inequality

12x + 9 − 9 ≤ −15 − 9

Subtraction property of inequality

12x ≤ −24

Evaluate the subtraction

≤

Division property of inequality

x ≤ −2

Evaluate the division

The solution to the compound inequality is x > 12 or x ≤ −2. In set notation, this would be {x∣x ≤ −2, x > 12}. In interval notation, this would be (−∞, −2] ∪ (12, ∞). The solution set is shown on the number line: −15

−10

−5

c 3.5x + 0.75 < 13 or −6.25x − 8 ≥ 4.5

Apply the idea Solve the ﬁrst inequality: 3.5x + 0.75 < 13

First inequality

3.5x + 0.75 − 0.75 < 13 − 0.75

Subtraction property of inequality

3.5x < 12.25

Evaluate the subtraction

Division property of inequality

x < 3.5

Evaluate the division

Solve the second inequality: −6.25x − 8 ≥ 4.5

Second inequality

−6.25x − 8 + 8 ≥ 4.5 + 8

Addition property of inequality

−6.25x ≥ 12.5

Evaluate the addition

≤

Division property of inequality

x ≤ −2

Evaluate the division

When graphing the solutions to the inequality, we see an overlap in the solution sets since the division property of inequality by a negative number required us to reverse the inequality symbol for the second inequality. −5

−4

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−3

−2

−1

Since the statement for the compound inequality is x < 3.5 or x ≤ −2 and the solution set is a single set of values that are strictly less than 3.5, the solution to the inequality is x < 3.5 since its solutions will satisfy either inequality.

Reﬂect and check We could also have written the answer using interval notation or set notation. • Interval notation: (−∞, 3.5) • Set notation: {x∣x < 3.5} or {x : x < 3.5} In set notation, we can use a vertical line or a colon before writing the inequality; both notations read “the set of x such that x is less than 3.5.”

Example 3 To earn a ﬁnal grade of a B in her social studies class, Malia must have a test score average of an 82 to an 87 on 4 tests. Suppose that Malia scored 89, 91, and 80 on her ﬁrst three tests. a Write a compound inequality to solve for the possible scores Malia can earn on her 4th test in order to earn a B.

Create a strategy Since Malia’s scores must be an average of an 82 to an 87, we can write an “and” compound inequality statement to determine the range of test scores she could earn.

Apply the idea

Reﬂect and check

Let x = the final test score. Since the minimum score she After writing a compound inequality to model a real-world needs is an 82 and the maximum score she needs is an situation, it helps to review the inequality written and 87, we can write our expression representing the average conﬁrm that it is reasonable for the situation. of the four test scores in between.

b Solve the compound inequality from part (a) and interpret its meaning in context.

Apply the idea 82 ≤ 82 ≤ 82 ⋅ 4 ≤

≤ 87

Original inequality

≤ 87

Evaluate the addition

⋅ 4 ≤ 87 ⋅ 4

Multiplication property of inequality

328 ≤ 260 + x ≤ 348 328 − 260 ≤ 260 + x − 260 ≤ 348 − 260 68 ≤ x ≤ 88

Evaluate the multiplication Subtraction property of inequality Evaluate the subtraction

Since x can be anywhere from 68 to 88, (inclusive) and we know that x represents Malia’s score needed on her ﬁnal test, the solution means that Malia needs anywhere from a 68 to an 88 on her ﬁnal test in order to earn a B in social studies.

Reﬂect and check In interval notation, this solution can be written as [68, 88]. In set notation, this solution can be written as {x∣68 ≤ x ≤ 88}.

1.06 Compound inequalities mathspace.co

c Determine whether x = 75 is a viable solution.

Create a strategy Substituting the given value into the inequality will determine if the solution is a solution to the inequality, then making sense of the solution in context will further our decision.

Apply the idea If x = 75, then 68 ≤ 75 ≤ 88 is a true statement. x = 75 means a score of 75 on the test, and since we know the score can be from a 68 to an 88, we can conﬁrm that x = 75 is a viable solution.

Idea summary Remember both types of compound inequalities: • • •

“And” statements: a ≤ x ≤ b written in interval notation as [a, b] “Or” statements: x < a or x ≥ b written in interval notation as (−∞, a) ∪ [b, ∞) Set Notation: a ≤ x ≤ b written in interval notation as [a, b] The solution in set notation is written as {x∣a ≤ x < b}

Practice What do you remember? 1

For each pair of inequalities, plot the solution to the compound inequality that is Inequality 1 and Inequality 2 on a number line. a

Inequality 1: x ≥2

Inequality 1: x >3

Inequality 2: x ≤7 Inequality 2: x ≤7 2

For each pair of inequalities, plot the solution to the compound inequality that is Inequality 1 or Inequality 2 on a number line. a

Inequality 1: x ≤3

Inequality 1: x <8

Inequality 2: x >8 Inequality 2: x >3 3

For the compound inequalities: i

Rewrite the compound inequality as a pair of inequalities joined by either and or or.

Plot the solution to the compound inequality on a number line.

iii

Write the solution in interval notation.

−5 < x < 8

−7 < x ≤ 5

Which of the sets of inequalities represent the solution for x in 1 ≥ x − 4 > −6? A

x > −2 and x ≥ 5

x > −2 or x ≥ 5

x > −2 and x ≤ 5

x > −2 or x ≤ 5

Freshwater temperature can affect the development of various ﬁsh species including salmon. The optimum temperature for salmon ranges from 33 °F to 68 °F. Which of the sets of inequalities represent the temperatures where salmon will thrive? A

T ≥ 33 or T ≤ 68

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T ≥ 33 and T ≤ 68

T ≤ 33 and T ≥ 68

T ≤ 33 or T ≥ 68

Which of the number lines represents the solution for m + 2 < −5 or m − 2 ≥ −5? A

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Which of the number lines represents the solution for 9 − 2p > −6 and 12 < 3p? A

B 9

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

D 9

For each number line, identify if the solution set graphed could represent an ‘and,’ ‘or,’ or either type of compound inequality: a

b −7−6−5−4−3−2 −1 0 1 2 3 4 5 6 7

−7−6−5−4−3−2 −1 0 1 2 3 4 5 6 7

d −7−6−5−4−3−2 −1 0 1 2 3 4 5 6 7

−7−6−5−4−3−2 −1 0 1 2 3 4 5 6 7

let’s practice 9

For each pair of inequalities, state the solution to the compound inequality that is Inequality 1 and Inequality 2. a

Inequality 1: 4x + 3 ≥ −17

Inequality 2: −4x − 2 ≥ 26

Inequality 2: 3x − 4 < 5

For each pair of inequalities, state the solution to the compound inequality that is Inequality 1 or Inequality 2. a

Inequality 1: −3x + 2 ≤ −13

Inequality 1: 5x − 7 < −22

Inequality 1: −6x + 8 <26

Inequality 2: −6x + 8 < −28

Inequality 2: 5x − 4 ≤26

Consider the pair of inequalities: Inequality 1: 2.9x + 4.8 < 25.1 Inequality 2: 20.7 − 8.2x < −90

Determine the solution to the compound inequality that is Inequality 1 and Inequality 2.

Determine the solution to the compound inequality that is Inequality 1 or Inequality 2.

Explain the similarities and differences in the solutions for Inequality 1 and Inequality 2 compared to Inequality 1 or Inequality 2.

For the compound inequalities: i

Solve for x.

Write the solution in interval notation.

16 < 3x + 7 ≤ 37

e g

−34 < −5.6x + 2.4 ≤ 22

−52 < −5x + 3 ≤ −22

8 < 2x − 5 (x − 7) < 41

21 < 4.5x − 6 ≤ 33

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For the compound inequalities, test if each of the values of x are in the solution set: i

x=0

x=5

iii

x = −2

12 < 4x + 5 ≤ 32

−5 < −4.2x − 3 ≤ 8.5

−10 < 5 − 6x ≤ 16

x = 8.5

For each of the scenarios, determine whether the compound inequality would use and or or: a

Francesco solves a compound inequality and expresses the solution in interval notation as (−∞, ∞).

Zenaida solves a compound inequality and ﬁnd that there are no possible values for x.

Dora solved a compound inequality and expressed their solution as the interval (5, 13]. a

Rewrite this solution as a pair of inequalities joined by either and or or.

Write a compound inequality which Dora could have solved, using at least two steps, to produce this solution.

Carlton tried to solve the following compound inequality, but made a mistake in his work: Step 0: −8 ≤ 10 − 3x < 4 Step 1: −18 ≤

−3x < −6

Step 2:

6≤

3x < 18

Step 3:

2≤

x<6

Determine which step is incorrect and explain the error. 17

Skyler tried to plot the solution to the compound inequality 4x − 1 ≤ 11 and their answer is incorrect. −5 −4 −3 −2 −1 0

> 5 on a number line, however

2 3 4 5 6 7 8 9 10 11 12 13 14 15

Identify the error(s) and explain how to rectify them.

Let’s extend our thinking 18

The velocity, in feet per second, of a tennis ball t seconds after it is projected directly upward is v = 85 − 32t. a

Write a compound inequality to represent when the ball has a velocity faster than 37 ft/sec but slower than 69 ft/sec.

Solve the inequality for the range of time t at which the ball will be faster than 37 ft/sec but slower than 69 ft/sec.

The formula for converting temperatures from Fahrenheit to Celsius is C =

(F − 32).

During a recent year, the temperatures in Moscow ranged from −20 °C to 30 °C.

Write the given range of temperatures as a compound inequality.

Solve for the corresponding range of values of F.

To get a final grade of B, a student must have an average score on five tests that is greater than or equal to 80 and less than 90. Maria’s grades on the first four tests were 96, 87, 78 and 94. a

Write a compound inequality that models the situation for the score, x, Maria must obtain on her ﬁfth test to get a ﬁnal grade of B.

Solve the compound inequality for x.

Describe the solution regarding Maria’s score.

Determine if x = 55 is in the solution set. Explain what this means in the given context.

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1.07 Absolute value inequalities After this lesson, you will be able to... • write absolute value inequalities to represent real-world situations. • solve absolute value inequalities in one variable. • represent solutions to absolute value inequalities in set and interval notation. • represent solutions to absolute value inequalities on a number line. • verify solutions to absolute value inequalities algebraically, graphically, and with technology. • justify the reasonableness of solutions to absolute value inequalities justify the steps in solving an absolute value inequality . • interpret solutions in terms of a context.

Absolute value inequalities Recall that the absolute value of a number is a measure of the size of a number, and is equal to its distance from 0, which is always a non-negative value. Absolute value is sometimes called the magnitude. An absolute value inequality is an inequality containing the absolute value of a variable expression.

Exploration Plot the range of values that satisfy each of the following inequalities on number lines. Form ∣x∣ > a: ∣x∣ > 1, ∣x∣ > 2, ∣x∣ > 5, ∣x∣ > 100 Form ∣x∣ < a: ∣x∣ < 1, ∣x∣ < 2, ∣x∣ < 5, ∣x∣ < 100 1.

What do you notice about absolute value inequalities in the form ∣x∣ > a?

What do you notice about absolute value inequalities in the form ∣x∣ < a?

Solutions to absolute value inequalities usually involve multiple inequalities joined by one of the keywords “and” or “or”. Solutions with two overlapping regions joined by “and” can be rewritten as a single compound inequality. −5 −4 −3 −2 −1 0 1 2 3 4 5

∣x∣ ≥ 4 is read as the absolute value of some value, x, is 4 or more spaces from zero. The inequality has solutions of x ≤ −4 or x ≥ 4. Solutions can also be written in: • Interval notation: (−∞, −4] ∪ [4, ∞) • Set notation: {x∣x ≤ −4 or x ≥ 4}

−4 −3 −2 −1

∣x∣ < 3 is read as the absolute value of some value, x, is within 3 spaces of zero. The inequality has solutions of x > −3 and x < 3. Solutions can also be written in: • Interval notation: (−3, 3) • Set notation: {x∣ −3 < x < 3}

To solve an absolute value inequality, we begin by isolating the absolute value expression. Once the expression is isolated, we write and solve two separate inequalities without the absolute value bars. This is similar to solving an absolute value equation, but the second inequality’s symbol will be reversed.

1.07 Absolute value inequalities mathspace.co

In general, for an algebraic expression p (x) and k > 0, we have: • ∣p (x)∣ < k is within k spaces of 0 and can be written as −k < p (x) < k • ∣p (x)∣ > k is more than k spaces of 0 and can be written as p (x) < −k or p (x) > k Similar to absolute value equations, the solutions of absolute value inequalities can be veriﬁed graphically or with technology. Once the absolute value expression is isolated, we can create a table of values for the corresponding absolute value function. For example, to verify the solutions to the inequality ∣x − 3∣ > 2, the isolated absolute value expression is ∣x − 3∣, so the corresponding function is y = ∣x − 3∣. x f (x)

0 3

1 2

2 1

3 0

4 1

5 2

We can create a table of values by choosing x-values and substituting them into the function to ﬁnd the corresponding y-values.

6 3

Next, we can plot these points on a coordinate grid, and connect the points to graph the function. The solution set will be the x-value(s) of the points where the output of the absolute value function is strictly greater than 2.

y 5 4 3 2 1 −2 −1 −1

There are two different sections of the graph where the outputs are greater than 2: when x < 1 and when x > 5. This is an “or” inequality so the solutions can be written as: • Interval notation: (−∞, 1) ∪ (5, ∞) • Set notation: {x∣x < 1 or x > 5}

1 2 3 4 5 6 7 8

−2

Example 1 Consider the inequality ∣x∣ > 2. a Represent the inequality ∣x∣ > 2 on a number line.

Create a strategy This inequality represents values of x which are “more than 2 units away from 0.” To plot this, we will need to use two regions. Also, note that this inequality doesn’t include the endpoints, so we will use unﬁlled points to show this.

Apply the idea −5 −4 −3 −2 −1 0 1 2 3 4 5

Reﬂect and check We can give a quick check of the answer by thinking about whether this is an “and”-type inequality or an “or”-type inequality. In this case, the inequality ∣x∣ > 2 has a solution of x < −2 or x > 2. Our solution on the number line has two distinct parts, which matches what we expect for an “or”-type inequality.

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b Rewrite the solution to ∣x∣ > 2 in interval notation.

Create a strategy We can use either the number line or x < −2 or x > 2 to help write this in interval notation.

Apply the idea The x < −2 part can be written as (−∞, −2). The x > 2 part can be written as (2, ∞). Since this is an “or” inequality we will ﬁnd the union of these two sets to give: (−∞, −2) ∪ (2, ∞)

c Rewrite the solution to ∣x∣ > 2 in set notation.

Create a strategy In the previous parts, we determined that the inequality is an “or”-type because it uses the greater than symbol (>) and there are two distinct sections of the solution.

Apply the idea Since this is an “or” inequality, we will write the answer in set notation as: {x∣x < −2 or x > 2}

Example 2 Consider the inequality

a Solve the inequality for x. Express your solution using interval notation.

Create a strategy In order to solve this inequality, we need to first remove the absolute value by rewriting the inequality as a compound inequality. In this case, the inequality is of the form ∣p (x)∣ ≥ k, so this will be an “OR” compound inequality.

Apply the idea If we consider the two possibilities for the absolute value, we get:

Since these two inequalities are disjoint, this is an “OR” inequality and needs to be solved separately.

1.07 Absolute value inequalities mathspace.co

First, solve for when

Next, solve for when

Use the division property of inequality

So the solution set is “all values of x between less than or equal to express this using interval notation as:

or greater than or equal to 6 (inclusive)”. We can

b Represent the solution set on a number line.

Create a strategy The solution set for this inequality consists of two intervals, so it will involve two regions on the number line. The endpoints are also included this time, which we represent using filled points.

Apply the idea −1

Reﬂect and check We can give a quick check of the answer by thinking about whether this is an “AND”-type inequality or an “OR”-type inequality. In this case, we found the solution to the inequality the number line has two regions.

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to an “OR”-type inequality, which matches the fact that

c Determine whether x = 2.5 is a valid solution to the absolute value inequality.

Create a strategy Plot the solution set together with the indicated point to check if the point lies inside or outside the solution set. Or test algebraically by substituting the value into either the original or rearranged inequality and checking if the resulting statement is true.

Apply the idea Graphically: Consider the plot of the solution set together with the point at x = 2.5. −1

The point x = 2.5 lies on a section of the line that is not in the solution set, indicated by the green interval, and therefore is not a valid solution. Algebraically: Substitute x = 2.5 into the original inequality. Original inequality

Substitute x = 2.5

Evaluate the multiplication

Evaluate the subtraction

Evaluate the absolute value

Considering

≈ 1.67 this statement is true and x = 2.5 is a valid solution.

Example 3 Consider the inequality

1.07 Absolute value inequalities mathspace.co

a Solve the inequality.

Create a strategy In order to solve this inequality, we need to isolate the absolute value expression and then rewrite the inequality as a compound inequality. We can then solve by applying the properties of inequality.

Apply the idea Original inequality

Subtraction property of inequality

Multiplication property of inequality

Now, we can write ∣x + 2∣ < 14 as a compound inequality and solve. −14 < x + 2 < 14

Rewrite as a compound inequality

−16 < x < 12

Subtraction property of equality

b Represent the solution on a number line.

Create a strategy The solution set for this inequality consists of a single interval, so it will only need one region on the number line. Because neither inequality has an “equal to” the endpoints are not included in the solution, which we represent using unﬁlled points.

Apply the idea −20

−15

−10

−5

c Verify the solution graphically.

Create a strategy In part (a), we found the isolated absolute value expression to be ∣x + 2∣ < 14. First, we will create a table of values for the absolute value function y = ∣x + 2∣. Then, we will plot the values from the table and look for the x-values of the points where y < 14.

Apply the idea To create a table of values for y = ∣x + 2∣, we can use the solution from part (a) when choosing x-values to substitute. The solution tells us that the x-values should range from x = − 16 to x = 12. The vertex of the function will be exactly halfway between these values, so it will be at x = x f (x)

−16 14

−12 10

−8 6

−2 0

4 6

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8 10

12 14

= −2

Next, graph the points on the coordinate grid, and look for the section of the graph where the outputs are strictly less than 14. 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 −20−16−12 −8 −4

The outputs are less than 14 when −16 < x < 12. At x = − 16 and x = 12, the outputs are equal to 14, so these values are not included in the solution set.

This veriﬁes our solution from part (a).

x 4 8 12 16

Reﬂect and check We can write our solution in interval notation as (−16, 12) and in set notation as {x∣ − 16 < x < 12}.

Example 4 In a survey, 76% of people asked stated they would likely vote yes for new neighborhood park plans. The margin of error is 4 percentage points. a Write and solve an absolute value inequality that represents the scenario.

Create a strategy A margin of error indicates that the percentages for the poll could be off by 4 percentage points in either direction of 76%. It may be helpful to ﬁrst graph possible solutions to the inequality on a number line and then use the number line to help us write an inequality.

Apply the idea 70

Similar to writing and solving absolute value equations, understanding the distance of the possible solutions from the middle or average value will help us write our statement. Let v = the percentage of people who said they will vote yes. If the solutions are up to 4 percentage points from 76%, an inequality with the possible solutions of 72% to 80% is ∣v − 76∣ ≤ 4.

1.07 Absolute value inequalities mathspace.co

Reﬂect and check By solving the inequality we wrote, we can conﬁrm that our inequality makes sense in context and that its solution matches the number line solution set. ∣v − 76∣ ≤ 4

Original inequality

−4 ≤ v − 76 ≤ 4

Rewrite as a compound inequaltiy

−4 + 76 ≤ v − 76 + 76 ≤ 4 + 76 72 ≤ v ≤ 80

Addition property of inequality Evaluate the addition

b Determine what the viable solutions to the inequality mean in context.

Apply the idea The margin of error would mean that anywhere from 72% to 80% of people would likely vote yes on the survey.

Idea summary For absolute value inequalities with an algebraic expression p (x) and k > 0. • •

∣p (x)∣ < k is within k spaces of 0 and can be written as −k < p (x) < k ∣p (x)∣ > k is more than k spaces of 0 and can be written as p (x) < −k or p (x) > k

For the absolute value inequality a ∣p (x)∣ + b < c, the expression ∣p (x)∣ must be isolated using inverse operations before rewriting the inequality as a compound inequality.

Practice What do you remember? 1

Rewrite the following inequalities without using absolute value: a

∣x∣ ≥ 5

∣x∣ > 13

∣ x∣ ≤ k, for a positive value of k

∣x∣ < 7

∣x∣ ≤ 9

−0.6 ≤ 2x + 3 ≤ 0.6

Represent the following inequalities on a number line: a

∣x∣ > 6

∣x∣ ≠ 3

∣x∣ ≥ 5

∣x∣ > −5

Rewrite the following as absolute value inequalities: a

∣x∣ < 2

−7 ≤ x ≤ 7

x < −8 or x > 8

For the absolute value inequalities below, test if each of the following is in the solution set: i

x=0

x=5

iii

x = −3

∣4x − 6∣ ≤ 12

∣1.2x + 4∣ ≤ 6.5

∣8 − 5x∣ ≥ 11

Write an absolute value inequality that represents each of the following: a

All real numbers x that are less than 8 units away from 0.

All real numbers x that are more than 8 units away from 0.

All real numbers x that are at least 2 units away from 8.

All real numbers x that are at most 5 units away from −2.

Mathspace Virginia SOL Algebra 2 mathspace.co

x = 2.5

Let’s practice 6

Solve each of the following inequalities: a

2 ∣m∣ > 8

4 ∣r∣ ≤

3 < ∣2u + 2∣ u>

2.5 ≥ 0.75 ∣p∣

∣w − 1∣ < 3

∣ − 3x∣ ≥ 9

3 ∣x − 1∣ ≥ 9

∣s∣ >

b u<

SOL

∣n∣

Solve each of the following inequalities: a

−5 ≤

0 ≥ ∣z + 0.8∣

For each of the following inequalities: i

Express the solution using set notation.

Express the solution using interval notation.

iii

Represent the solution on the number line.

∣2.5x∣ ≤ 10

∣x − 5∣ ≥ 2

∣4.8x + 7.2∣ < 24

∣5x + 4∣ + 3 > 2

This graph best represents the solution to which inequality? −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

A 10

2 + ∣h − 2∣ < 6

b f

2 ∣n − 2∣ + n > 5

≥ 6.5

6 − ∣2m + 1.5∣ < 3

Solve each of the following inequalities and express the solution using interval notation. a

Solve each of the following inequalities and express the solution using set notation. a

≤ 6.75

∣p − 4∣ −

For the absolute value inequalities below, test if each of the following is in the solution set: i

x=1

x=6

iii

x = −4

2 + ∣3x − 5∣ < 10

3 − ∣2.3x + 5∣ > 7.5

5 − ∣7 − 4x∣ ≤ 14

x=3 +x≥4

In a certain company, the measured thickness, m, of a helicopter blade must not differ from the standard, s, by more than 0.17 millimeters. The manufacturing engineer expresses this as the inequality ∣m − s∣ ≤ 0.17. Find the range of values that m can take if s is 17.92 millimeters.

1.07 Absolute value inequalities mathspace.co

Cell phone cases have dimension requirements to ensure the phone will ﬁt properly in the case. The manufacturing engineer has written a speciﬁcation that the new length, n, of the case can differ from the previous length, p, by at most 0.04 centimeters. a

Write an inequality to represent the dimension requirements for cell phone cases.

Find the range of values for the new length of a cell phone case if the previous length was 18.9 centimeters.

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. A coin is deemed to be unfair if h, the number of outcomes that result in heads, satisﬁes:

Solve the inequality.

Interpret your solution in the context of the problem.

A freediver holds his breath and dives below the surface of the water. His depth, d in meters, after t seconds, is given by: d=

∣t − 105∣ − 120

Where d = 0 means the diver is at the surface and d = − 20 means the diver is 20 meters below the surface.

When the diver is 40 meters or less below the surface he is accompanied by a safety diver. Write an absolute value inequality to model the time when the diver is without a safety diver.

Solve for the range of times during the dive when the diver is without a safety diver.

Determine if t = 180 is a valid solution for the absolute value inequality. Explain what this means in the given context.

For each of the following, write two distinct absolute value inequalities that have the given solution: a

−2 < x < 2

x ≤ −9 or x ≥ 9

1≤x≤7

−5 −4 −3 −2 −1 0 1 2 3 4 5

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

x < −4 or x > 0

−5 −4 −3 −2 −1 0 1 2 3 4 5

0 1 2 3 4 5 6 7 8 9 10

An Algebra 2 student attempted to solve the absolute value inequality ∣3x − 9∣ ≤ 6 and provided the following solution: Case 1

Case 2

3x − 9 ≤ 6

−(3x − 9) ≤ 6

2 3x ≤ 15

−3x + 9 ≤ 6

3 x ≤ 5

−3x ≤ −3

x≤1

The student concludes that the solution to the inequality is x ≤ 1 or x ≤ 5. Identify and explain the error in the student’s solution and provide the correct solution to the inequality. 19

Consider the absolute value inequality ∣2x + 1∣ − 4x > 5. Solve the inequality and justify each line of your work to solve the absolute value inequality.

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Let’s extend our thinking 20

Use the graph of f (x) = ∣2x − 4∣ and g (x) = 8 to solve the following equation and inequalities: a

∣2x − 4∣ = 8

∣2x − 4∣ ≤ 8

∣2x − 4∣ > 8

12 10 8 6 4 2 −8 −6 −4 −2 −2

x 2 4 6 8 10

−4

Toya claims that −5 < x < −1.4 is the solution to 3 ∣x + 2∣ > 2x + 1. Explain why her answer is incorrect.

Determine the solution(s) to the inequality ∣2x − 3∣ ≤ 1 − x. Explain your answer.

The speedometer in Najah’s car is displaying 35 mph. They know that the speedometer is accurate to within 10% of the actual speed.

Write an absolute value inequality to represent the situation.

The road Najah is driving down has a speed limit of 40 mph. Is it possible for Najah to be driving over the speed limit?

Floyd measures the distance between two cities on a map, which has a scale factor of 1 : 50 000. He determines that the cities are approximately 16 inches apart, with a possible error of measurement of up to half an inch. a

If x represents the actual distance between the two cities, write an absolute value inequality to represent the situation.

The actual distance between the two cities is 12.9 miles (to one decimal place). Determine if Floyd’s measurement was accurate.

If x represents the temperature in Charlottesville, Virginia, describe a possible interpretation of the inequality ∣x − 84∣ ≤ 7.

1.07 Absolute value inequalities mathspace.co

2 Quadratic Functions Big ideas • A standard algorithm can be applied to rewrite many different kinds of expressions. • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. • The degree of an equation indicates the number of solutions that it has, however, the solutions may not be unique or real. • An equals sign indicates an equivalent relationship between two expressions. • An inequality symbol indicates the way that the expression on one side is unequal to the expression on the other side.

Chapter outline 2.01 2.02 2.03 2.04 2.05 2.06 2.07

Factor polynomials Quadratic functions Quadratic equations with real solutions Complex numbers and operations Quadratic equations with complex solutions Quadratic inequalities Quadratic systems

92 99 108 118 131 142 155

2.01 Factor polynomials After this lesson, you will be able to... • factor polynomials in one or two variables. • use the structure of a polynomial to rewrite it in factored form.

Factor polynomials Factoring a polynomial is a process of expressing a polynomial as a product of its factors. In other words, it is the inverse process of multiplying polynomials. A polynomial is considered completely factored when none of its factors can be written as a product of polynomials with a lower degree. We have learned several different strategies, including a few identities, that can help us factor a polynomial. • Factoring using Greatest Common Factor (GCF): A greatest common factor from each term of a polynomial is factored out ax + ay + … = a(x + y + …) • Factoring by grouping: A method for factoring an expression containing at least four terms, by grouping the terms in pairs and taking out common factors ax + ay + bx + by = a (x + y) + b (x + y) = (x + y) (a + b) • Factoring quadratic trinomials: A trinomial that can be expressed as the product of two binomials ax2 + bx + c = (mx + p) (nx + q) where mn = a, pq = c and np + mq = b • Perfect square trinomials: A trinomial that is formed by multiplying a binomial by itself a2 + 2ab + b2 = (a + b) 2 or a2 − 2ab + b2 = (a − b)2 • Difference of two squares: The result of a perfect square being subtracted from another perfect square a2 − b2 = (a + b) (a − b)

Example 1 Factor the polynomials using an appropriate method. a x2 − 17x + 60

Create a strategy First, we always check for a GCF. As x does not have a coefficient, there is no GCF here. Because this is a trinomial where the leading coefficient is 1, we want to factor by finding two numbers that add to −17 and multiply to 60.

Apply the idea Since we know we want it multiply to positive number and add to a negative number, we know both factors will be negative.

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−1, −60 −61

Factors of 60 Sum

−2, −30 −32

−3, −20 −23

−4, −15 −19

−5, −12 −17

−6, −10 −16

We need to choose factors of 60 that have sum of −17. Looking at our table above, we see the factors are −5 and −12. x2 − 17x + 60 = (x − 5) (x − 12)

Reﬂect and check We can check that we factored correctly by multiplying the binomials and conﬁrming we get original expression. (x − 5) (x − 12) = x2 − 12x − 5x + 60 2

= x − 17x + 60

Multiply using the distributive property Combine like terms

We can conﬁrm our expression is the same as original which shows we factored correctly.

b 8xy4 − 16x3y2 + 4xy3

Create a strategy We will ﬁrst ﬁnd the GCF of 8xy4, −16x3y2, and 4xy3. We can then rewrite each expression as a product of the GCF and any remaining factors.

Apply the idea The GCF of 8xy4, −16x3y2 and 4xy3 is 4xy2. So we will divide each term by 4xy2:

Divide the coeﬃcients

Quotient of powers

The remaining terms do not have any other common factors, and the linear term does not contain both variables. Therefore, this expression is fully factored.

Reﬂect and check We can check the answer by distributing the multiplication and using the product of powers rule for exponents: 4xy2(2y2 − 4x2 + yz4) = 4xy2(2y2) − 4xy2(4x2) + 4xy2( y) = 8xy4 − 16x3y2 + 4xy3 Notice that, if no mistakes have been made, these are the same steps, just in reverse.

c 121m2 − 64

Create a strategy

Apply the idea

The expression has 2 terms that are both perfect squares, and the terms are subtracted. This means we can factor using the difference of two squares identity:

Since (11m)2 = 121m2 and 82 = 64, we can use A2 − B2 = ( A + B) ( A − B), where A = 11m and B = 8.

121m2 − 64 = (11m + 8) (11m − 8)

A − B = ( A + B) ( A − B)

2.01 Factor polynomials mathspace.co

d x2 − 12xy + 36y2

Create a strategy Notice that there are no common factors, and the ﬁrst and last terms are perfect squares. Let’s check if this expression meets the criteria for a perfect square trinomial: ( A − B)2 = A2 − 2AB + B2 We can see that A2 = x2, which means A = x. Next, we see that B2 = 36y2, so B = 6y. Now, let’s check whether −2AB = − 12xy: −2 ⋅ x ⋅ 6y = − 12xy This means the expression is a perfect square trinomial, so we can use that identity to factor the expression.

Apply the idea Using the perfect square trinomial identity ( A − B)2 = A2 − 2AB + B2 with A = x and B = 6y: x2 − 12x + 36y2 = (x − 6y)2

Reﬂect and check We can check the answer by expanding it again. This can also help us notice patterns when factoring expressions with two variables. (x − 6y)2 = (x − 6y) (x − 6y)

Expand

= x − 6xy − 6xy + 36y 2

= x − 12xy + 36y

Multiply using the distributive property Combine like terms

e 18m3 − 2mn2 + 9m2n − n3

Create a strategy First, we want to see if all terms share a common factor. These terms do not, so we can check for factoring by grouping next because the expression has 4 terms.

Apply the idea 18m3 − 2mn2 + 9m2n − n3 = (18m3 − 2mn2) + (9m2n − n3) 2

Group based on common factors

= 2m (9m − n ) + n (9m − n ) 2

= (2m + n) (9m − n ) 2

Factor out each GCF (2m and n) Factor out the common binomial factor

Observe that 9m − n = (3m) − (n) . This means that 9m − n is a difference of two squares, so we use the formula in factoring: a2 − b2 = (a − b) (a + b)

Formula of difference of two squares

Substitue a = 3m and b = n

(3m) − (n) = (3m − n) (3m + n) 2

If we substitute 9m − n = (3m − n) (3m + n), we get 18m3 − 2mn2 + 9m2n − n3 = (2m + n) (3m − n) (3m + n)

Reﬂect and check Alternatively, we can group 18m3 and 9m2n and −2mn2 and −n3 together and get the same answer. 18m3 − 2mn2 + 9m2n − n3 = (18m3 + 9m2n) + (−2mn2 − n3) 2

Group based on common factors

= 9m (2m + n) + (−n ) (2m + n)

Factor out each GCF (9m2 and −n2)

= (9m2 − n2) (2m + n)

Factor out the common binomial factor

= (3m + n) (3m − n) (2m + n) Apply the formula for difference of two squares

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6x4 − 10x3 − 24x2

Create a strategy First, we look for the greatest common factor for all three terms and factor it out. Then we look at the simpler trinomial to see if it factors further. If it does, we ﬁnd the values of r and s that multiply to ac and add to b. After ﬁnding those values, we write the trinomial in the form ax2 + rx + sx + c and factor it by grouping.

Apply the idea The greatest common factor of the three terms is 2x2. We factor out 2x2 and get 2x2 (3x2 − 5x − 12) For 3x2 − 5x − 12, we need to ﬁnd two numbers that have a product of ac = (3) (−12) = − 36 and sum of b = − 5. The factors of −36 include ±1, ±2, ±3, ±4, ±6, ±9, ±18 and ±36. Among these factors, 4 and −9 are the only numbers that add to −5 and multiply to −36. We can let r = 4 and s = − 9. Next, we write 3x2 − 5x − 12 in the form ax2 + rx + sx + c. Substituting the values for r and s, we get 3x2 + 4x − 9x − 12 Now, we factor this expression by grouping 3x2 + 4x − 9x − 12 = (3x2 + 4x) + (−9x − 12)

Group based on common factors

= x(3x + 4) − 3 (3x + 4)

Factor out each GCF (x and −3)

= (x − 3) (3x + 4)

Factor out the common binomial factor

Therefore, 6x4 − 10x3 − 24x2 = 2x2 (x − 3) (3x + 4).

Reﬂect and check We can check the answer by multiplying the factored form 2x2 (x − 3) (3x + 4). 2x2 (x − 3) (3x + 4) = 2x2 (x (3x + 4) − 3 (3x + 4))

Distributive property

= 2x (3x + 4x − 9x − 12) 4

= 6x + 8x − 18x − 24x 2

= 6x − 10x − 24x

Distributive property Combine like terms

Idea summary The factoring methods and identities we can use to fully factor polynomials are • • • • •

Factoring using the GCF Factoring by grouping Factoring quadratic trinomials Perfect square trinomial identity Difference of two squares identity

2.01 Factor polynomials mathspace.co

Practice What do you remember? 1

SOL

Multiply the polynomials and simplify. a

(x + 1) (x − 8)

(4x + 3) (2x + 5)

(a + 3)2

(5a + 4)2

4x + 20

22y − y2

3ax2 + 6ax − 6a

2x3 − 16x2 + 2x

2ax3 + 2ax2 + 4ax

3 (9x − 8y) (9x + 8y)

16 (4x + 3y)2

5xy2 + 10xy + 15x

3 3

pq − p q − p q

5y( y − 10) + 4 (10 − y)

8x (10y + 7) − 9 (10y + 7)

24(x − 3)2 + 18(x − 3) (x + 2)

7a2 (4a2 − 3) (a + 1)

4x + x2 − 68y − 17yx

x2 − y2 − x − y

2 2

What is the complete factorization of 28a5 − 7a3 + 21a2? 7a2 (4a3 + 1)

7a2 (4a3 + 3a)

7a2 (4a3 − a + 3)

Factor the expressions completely by grouping: 12x2 − 21x − 20x + 35

m − 9m − 5m + 45

c 5

Factor the expressions completely:

A 4

(3a + 2b) (3a − 2b)

Using the diagram shown: a

Find the areas of the smaller rectangles A, B, C and D.

Add the smaller areas together to get the total area.

Explain why in the product, (2y + 3)2 = (2y)2 + 2(2y) (3) + (3)3, there is a coeﬃcient of 2 on the middle term.

Let’s practice 6

Consider the expression a2 − 16 + 2a + 8. a

Factor a2 − 16.

Factor 2a + 8.

Describe the similarities and differences between the factored expressions for part (a) and part (b).

Now, factor a2 − 16 + 2a + 8.

Factor the quadratic trinomials completely. x2 + 6x – 7

10x + 5x – 30

e SOL

x2 + x – 12 2

3y + 28y + 9

c g

3x2 − 21x – 54 2

2x − 11x – 40

Select the correct answers. When factored completely, identify the factors of this polynomial. 4x2 − 10x − 24 2 4

2x – 8 4x + 6

2x – 3 2x + 3

x−4 x+4

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4x2 + 40x + 100

12t2 − 13t − 4

Factor completely: a

x2y2 − 49

81x2 − 16y2

7x2 − 63

x2 + 6x + 9

64 + 16d + d2

5q2 + 10qt + 5t2

64r2 + 48r + 9

x2 − y2 − x − y

x2 − 6x + 9 − y2

Factor completely by grouping strategically: a

3a + 15 + ab + 5b

x2 − x2y + xy − x

9x2 − 24x + 16 − 4y2

2y − x + 8xy − 16y2

For each expression: i

Determine an appropriate strategy or combination of strategies for factoring. Explain your choice.

Factor the expression completely.

x2 + 17x + 72 2

4x2 − 9

4x + 13x − 12

18ab − 6ac − 15b2 + 5bc

9x2 + 6x + 1

n3 − 121n

4x + 45y + 36xy + 5

(x + 16)2 − y2

Factor completely using appropriate techniques: a

−8t2 + 18

3y2 − 36y + 108

x3 + 8x2y + 16xy2

16a2 − 25b2

2 2

4a b − 81c d

3x2 + 24x + 48

56xy + wx + 56y + w

9a2 + 24ab + 16b2

x2y2 − 36x2

x2 − x2y + xy − x

a − x + 8xy − 16y

6x2 − 36x + 48

m 15x2y + 50xy − 40y

8p ( p2 − 100) − 5 ( p2 − 100)

Factor completely: a

16 − 9y2

2 2

The side length of a regular pentagon is S = 2x2 + 21x + 49. a

Determine the perimeter of the pentagon in terms of x in expanded form.

Determine the perimeter in fully factored form. S

A cube has a surface area of (6x2 + 36x + 54) square units, where x > 0. a

Factor 6x2 + 36x + 54 completely.

Write an expression for the length of a side of the cube.

Let’s extend our thinking 16

A student wrote (a + b)2 = a2 + b2. Explain why they are incorrect and write the correct answer.

Verify the identity: a2 − 2ab + b2 = (a − b)2.

Betty and her family sell beautiful and intricate beadwork crafts to tourists in the Tampa Bay area. Her proﬁt, P (in thousands of dollars), can be modelled by P (x) = − x3 + 5x2 + 36x, where x is the number of items sold (in thousands). a

Betty’s family business is proﬁtable. State the domain constraint for this model. Explain your answer.

Betty can afford to upgrade her beading loom and expand her business if she makes more than $180 thousand dollars. If she sells 6 thousand crafts she can do this. Determine a lesser number of crafts she can sell and make a proﬁt of $180 thousand dollars. 2.01 Factor polynomials mathspace.co

Are these two expressions equivalent? Explain your reasoning: [(5t + 9) − 6r] [(5t + 9) + 6r]

and

(5t + 9)2 − 36r2

Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a trinomial that can be factored, and provide the factored form: 4x2 + ⬚⬚x + ⬚.

Quadratic trinomials can be factored using the identity:

Find the value of k that will make 16x2 − 24x + k a perfect square trinomial.

ax2 + bx + c = where m + n = b and mn = ac. Find the values m and n for the quadratic 4x2 − 14x + 12. 23

Factor completely. Consider using laws of exponents to rewrite higher powers. a

x2 + 8x + 16 − x4

x4 − x2 − 16x − 64

1 − n4

80x4 + 92x3 + 24x2

−2x4 + 2x3 + 24x2

The area model represents the expression: 8x3 + 20x2 − 14x + ⬚ + ⬚ + 35 = (2x − ⬚) (4x2 + ⬚ + ⬚)

Fill in the missing values of the area model.

Using the area model or otherwise, ﬁll in the blanks in the expression.

Fill in the blanks to make the statements true. 4x3 + ⬚x2 + 34x + 57 = (x + 3) (⬚x2 + ⬚x + ⬚)

Mathspace Virginia SOL Algebra 2 mathspace.co

x6 − 64

4x2 2x

8x3

20x2

−14x

2.02 Quadratic functions After this lesson, you will be able to... • graph quadratic functions from an equation. • write equations of quadratic functions from a graph. • identify characteristics of quadratic functions (domain, range, increasing, decreasing, end behavior, max, min, zeros, intercepts). • find the value of f (x) given x, and vice versa from a graph. • interpret characteristics in terms of a context.

Quadratic functions When graphing parabolas and solving quadratic equations it is often useful to have the function written in a particular form, depending on the context and what characteristics we are interested in. Vertex form

Standard form 2

f (x) = a (x − h) + k

f (x) = ax2 + bx + c

(h, k) are the coordinates of the vertex (of the quadratic function)

c is the y-intercept of the graph

is the equation of the axis of symmetry y c

(h, k)

Factored form f (x) = a (x − x1) (x − x2) x1 and x2 are the x-values of the x-intercepts y

In all of the above forms, the value of a is the scale factor of the quadratic function, and indicates the direction of opening of the graph. If a > 0 then the parabola will open upwards, and if a < 0 then the parabola opens downwards. This also means a ≠ 0.

2.02 Quadratic functions mathspace.co

If we want to reveal different characteristics of a parabola, we can rewrite the quadratic function in different forms. This can be useful when sketching a graph of a quadratic function where we want to show all the characteristics: • x-intercepts • y-intercept • vertex (absolute maximum or minimum) • axis of symmetry • increasing, decreasing and constant intervals • end-behavior In addition to this, we can also use the context of a quadratic function to determine if there is an appropriate domain and range, or interpret what the characteristics represent in the context.

Example 1 Consider the graph.

4 2 −6 −4 −2 −2

x 2 4 6 8 10

−4 −6 −8 −10

a State the coordinates of the vertex of the parabola.

Create a strategy The vertex lies on the axis of symmetry of the parabola, and is the maximum value in this case.

Apply the idea The coordinates of the vertex are (4, 6).

b Write the equation of the parabola in vertex form.

Create a strategy We want to write the equation of the parabola in the form y = a (x − h)2 + k, and have already identified the vertex, so we know the equation will be y = a (x − 4)2 + 6, for some value of a. To find a, we can substitute the values of any other point on the parabola into the equation and solve for a. The intercepts are not easily identifiable, but we can see the parabola passes through the point (2, 2), so we can use this point.

Apply the idea Substituting x = 2, and y = 2 into y = a (x − 4)2 + 6: 2 = a (2 − 4)2 + 6

Substitute x = 2, and y = 2

2 = 4a + 6

Evaluate the parentheses

−4 = 4a

Subtract 6 from both sides

−1 = a

Divide both sides by 4

The equation of the parabola in vertex form is y = − (x − 4)2 + 6

100

Mathspace Virginia SOL Algebra 2 mathspace.co

Reﬂect and check Let’s use our knowledge of transformations to check our equation. 6

4 2 −6 −4 −2 −2

x 2

8 10

Because the graph is a parabola, we know the parent function is f (x) = x2. Identifying transformations from the parent function, we see the graph has been: • Reﬂected over x-axis (ﬂipped vertically) • Translated right 4 units (shifted horizontally) • Translated up 6 units (shifted vertically)

−4 −6 −8 −10

Vertex form of a quadratic equation is the same as the tranformation form, f (x) = a(x − h)2 + k. The reﬂection corresponds to a = − 1, the horizontal translation corresponds to h = 4, and the vertical translation corresponds to k = 6. Therefore, our equation y = − (x − 4)2 + 6 is correct.

Example 2 A golf ball is hit into the air and its height h feet above the ground at time t seconds after being hit is given by h = − 16t2 + 128t. a Assuming the ball starts at a height of 0 feet, determine when it will hit the ground.

Create a strategy The ball will hit the ground when h = 0, so we want to solve the equation 0 = − 16t2 + 128t. The values of t that will solve this equation correspond with the zeros of the equation when written in factored form, so one approach to solving would be to write the equation in factored form.

Apply the idea Writing the given equation in factored form, we get: 0 = − 16t(t − 8) To ﬁnd the possible solution(s), we must set each factor equal to 0 and solve. −16t = 0 t=0 t−8=0 t=8

Set the ﬁrst factor equal to 0 Divide both sides by −16 Set the second factor equal to 0 Add 8 to both sides

The solution t = 0 represents 0 seconds after the ball was hit, which is the inital time. Because we know that the ball started on the ground, we can discount the solution of t = 0. This means the solution is t = 8. The ball will hit the ground after 8 seconds.

2.02 Quadratic functions mathspace.co

101

Reﬂect and check We can check our solutions by graphing with technology. Recall that the solution(s) to an equation of the form f (x) = 0 are the x-value(s) of the x-intercepts.

280 h 260 240 220 200 180 160 140 120 100 80 60 40 20

Here, we are using t for the independent variable rather than x. The graph intercepts the t-axis at t = 0 and t = 8 which corresponds to the solutions we found.

t 1

2 3 4 5 6 7 8 9

b Find the greatest height the ball reaches above the ground.

Create a strategy Since the vertex of a parabola lies on the axis of symmetry, the greatest height the ball will reach is exactly halfway between the two x-intercepts, when being hit and when hitting the ground. As we know, the ball hits the ground after 8 seconds, so it reaches its greatest height after exactly 4 seconds. To ﬁnd this height, we can substitute t = 4 into the initial equation and solve for h.

Apply the idea h = − 16t2 + 128t 2

State the given equation

h = − 16 (4) + 128 (4)

Substitute t = 4

h = 256

Simplify

The maximum height the ball reaches is 256 feet.

Reﬂect and check 280 h 260 240 220 200 180 160 140 120 100 80 60 40 20

Another way to ﬁnd the axis of symmetry is using x =

given equation is in standard form. Since a = − 16 and b = 128:

which gives us x = 4. Then, we would still ﬁnd the height by substituting x = 4 into the equation.

2 3 4 5 6 7 8 9

We could also have rearranged the equation to be in vertex form: Factor out −16 y = − 16 (t2 − 8t) y = − 16 (t2 − 8t + 16) + 256

Complete the square

y = − 16 (t − 4) + 256 Factor the perfect square trinomial We can see from this that the vertex is at (4, 256).

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since the

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c Find the domain constraint for h, so it ﬁts the restrictions of hitting the golf ball. Give your answer using interval notation.

Create a strategy The domain is constrained by two things, the fact that time starts at t = 0 and that the ball hits the ground after 8 seconds. After this time the quadratic equation will not model the height of the ball.

Apply the idea

Reﬂect and check

As the boundary times of t = 0 and t = 8 are included in the domain, we will use square brackets to indicate that they are included.

The domain can also be written using set notation, as shown: {t∣0 ≤ t ≤ 8}

The domain is [0, 8].

Example 3 Consider the quadratic function: y = 2x2 + 4x − 30 a Rewrite the quadratic equation in a form that allows us to identify the x-intercepts.

Create a strategy To identify the x-intercepts we can rewrite the equation in factored form.

Apply the idea To rewrite the function in factored form, we want to first factor out the scale factor. This will give us: y = 2 (x2 + 2x − 15) We then want to find two values that have a product of −15 and a sum of 2. If we check all the factor pairs of −15, we can find that −3 and 5 satisfy these requirements. So the factored form of the quadratic function is: y = 2 (x − 3) (x + 5) b Rewrite the quadratic equation in a form that allows us to identify the coordinates of the vertex.

Create a strategy To identify the vertex coordinates we can rewrite the equation in vertex form.

Apply the idea To rewrite the function in vertex form, we again want to factor out the scale factor, and then use the complete the square method. y = 2 (x2 + 2x − 15)

Factor out the scale factor

y = 2 (x2 + 2x + 1 − 1 − 15)

Add and subtract

y = 2 ((x + 1)2 − 1 − 15)

Factor the perfect square

Distributive property of multiplication

Simplify

y = 2 (x + 1) − 2 – 30 y = 2 (x + 1) − 32

So the vertex form of the quadratic function is: y = 2 (x + 1)2 − 32

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c Sketch the graph of the quadratic function, labeling the x- and y-intercepts, and the vertex.

Create a strategy We can identify the y-intercept from the standard form. We can identify the x-intercepts from the factored form. And we can identify the coordinates of the vertex from the vertex form.

Apply the idea From the standard form of a quadratic equation, the y-intercept is (0, c). In this case, it will be (0, −30).

(−5, 0)

(3, 0)

From the factored form of a quadratic equation, the x-intercepts are (x1, 0) and (x2, 0). In this case, they will be (3, 0) and (−5, 0). From the vertex form of a quadratic equation, the vertex coordinates are (h, k). In this case, they are (−1, −32). So we can sketch the quadratic function, labeling all the key features: (0, −30) (−1, −32)

Reﬂect and check When sketching a quadratic function, we do not need to include a scale for the axes if we label all the characteristics, since we only need those points to determine the equation of the parabola.

Idea summary Quadratic functions could be in the following forms. •

Vertex form

f (x) = a (x − h)2 + k (h, k) •

are the coordinates of the vertex (of the quadratic function)

Factored form

f (x) = a (x − x1) (x − x2) x1 and x2 are the x-values of the x-intercepts •

Standard form

f (x) = ax2 + bx + c c

is the y-intercept is the equation of the axis of symmetry

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Practice What do you remember? 1

Consider the equation y = 16 + (x + 5)2. a

Determine whether the vertex is a maximum or a minimum.

Find the coordinates of the vertex.

For each of following quadratic functions: i

Find the coordinates of the y-intercept.

Find the coordinates of the x-intercepts.

iii

Find the coordinates of the vertex.

Draw a graph of the function.

y = x (6 − x)

y = (x + 2)2

y = x + 4x − 5

y = (1 − x) (x + 5)

y = − (x − 1)2 + 16

g (x) = − x2 + 4

y = − x + 8x − 7

For each of the following quadratic equations: i

Describe how the graph of g (x) be obtained from the graph of f (x) = x2.

Draw a graph of g (x).

g (x) = x2 + 1

g (x) = (x − 5)2

g (x) = 2x2

g (x) = (x + 3)2

g (x) = − x2

Consider the graph of the function y = f (x). a

State coordinates of the vertex

State the range of f (x).

State the interval of the domain where f (x) is increasing.

State the interval of the domain where f (x) is decreasing.

8 7 6 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2

x 1 2 3 4

Let’s practice 5

For each of the following quadratic functions: i

Explain whether it is more eﬃcient to complete the square to rewrite in vertex form or factor to rewrite in factored form.

Rewrite in your chosen form.

iii

Sketch a graph of the quadratic function, clearly marking the intercepts and vertex.

y = x2 − 6x + 5

y = x2 − 2x + 4

y = x2 + 6x

y = − 2x2 − 4x + 16

y = − x2 + 8x − 18

Consider the following functions: i

State the coordinates of the vertex.

State the set of x-values where the function is increasing in interval notation.

iii

State the set of x-values where the function is decreasing in interval notation.

Determine whether the vertex is an absolute minimum or an absolute maximum.

Find the domain of the function in interval notation.

Find the range of the function in interval notation.

y = 4 (x − 4)2 − 1

y = − 3 (x + 5)2 + 4

y = 0.5 (x − 1) (x + 3)

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Consider the graph.

State the coordinates of the vertex of the parabola.

Write the equation of the parabola in the form y = a (x − h)2 + k.

14 12 10 8 6 4 2

−6 −4 −2 −1

106

8 10

Consider the graph. i

Find the coordinates of the y-intercept of the graph.

Describe the end behavior as x → ∞.

iii

Write the equation of the parabola in the form y = a(x − m) (x − n).

9 8 7 6 5 4 3 2 1

−6 −5 −4 −3 −2 −1

−1

y 3 2 1 −5 −4 −3 −2 −1 −1

x 1

−2 −3 x 1

−4 −5

A parabola has equation of the form f (x) = (x − a) (x − b). a

Write down the equation of the parabola if it has x-intercepts at x = − 1 and x = − 5.

Find the y-value of the y-intercept.

Find f (1).

Sketch a graph of the equation.

A toy is launched into the air and its height h feet above the ground at time t seconds after being kicked is given by h(t) = − t2 + 10t. a

Calculate and interpret the meaning of h(0), the h-intercept.

Determine and interpret when h(t) = 16.

Determine when the toy will hit the ground.

Find the greatest height the toy reaches above the ground.

Find the domain constraint for h, so it ﬁts the restrictions of launching the toy. Give your answer using interval notation.

A parabola has its vertex at x = − 4 and one of the x-intercepts is (1, 0). a

Find the coordinates of the other x-intercept.

If the parabola has a y-intercept at (0, −18), write down the equation in factored form.

Find the coordinates of the vertex.

Mathspace Virginia SOL Algebra 2 mathspace.co

Let’s extend our thinking SOL

Which of the following describes the end behavior of y = x2 + bx − c as x approaches either positive or negative inﬁnity? A

y approaches positive inﬁnity

y approaches negative inﬁnity

y approaches c

y approaches

When an object is thrown into the air, its height above the ground is given by the equation h = 3 + 14d − d2, where d is its horizontal distance from where it was thrown. Both h and d are measured in feet. a

Determine the horizontal distance the object travels when it reaches its greatest height above the ground.

Find the maximum height reached by the object.

Sketch a graph of y = 3 + 14x − x2 and label all intercepts and key features.

Explain why in this context the left-most x-intercept on the graph in part (c) is not relevant.

The given graph shows the revenue function for a hair dryer based on the unit price. Revenue ($) 65000 60000 55000 50000 45000 40000 35000 30000 25000 20000 (0, 17 850) 15000 10000 5000 5

(39, 63 480)

(85, 0)

Unit price($)

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105

Identify and interpret the restricted domain and range for this context.

Bessie says the equation that best models the proﬁt is R(x) = − 29x(x − 85). If she is correct, explain why. If she is incorrect, explain and correct her error.

Identify and interpret the intercepts. Explain whether or not they make sense in the context.

Explain whether or not the graph of y = x2 + 6 has x-intercepts.

Consider the quadratic equation f (x) = a (x − m) (x − n), where a, m, and n are real numbers. a

Interpret f (0) for the graph of f (x).

Interpret f (m) for the graph of f (x).

Interpret

Sketch f (x) if:

for the graph of f (x).

a > 0, m > 0, and n > 0

a > 0, m > 0, and n < 0

iii a < 0, m = n, and m, n < 0

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If we can rewrite an equation by completing the square, then we can solve it using square roots. For quadratic equations where a = 1, we can write them in perfect square form by following these steps: 1

Quadratic equation in standard form

Subtract c from both sides

Rewrite the x coeﬃcient

Add

Factor the perfect square trinomial

to both sides

If a ≠ 1, we can first divide through by a to factor it out. Method 4: Graphing If the solutions are integers, drawing the graph of the corresponding quadratic function and finding the x-intercepts is an efficient way to find the solutions. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Method 5: Using the quadratic formula If we are unable to solve the quadratic easily using one of the previously stated methods, the quadratic formula is often the best approach since it can be used to solve any quadratic equation once it’s written in standard form. Quadratic formula A formula that can be used to ﬁnd solution(s) to any quadratic equation of the form f (x) = ax2 + bx + c

a, b, c the coeﬃcients and constant from the equation f (x) = ax2 + bx + c

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Example 1 For the following quadratic equations, determine an appropriate strategy for solving, explaining your choice, and then solve for x. a

x2 + x + 12 = 0

Create a strategy If we want to solve this quadratic equation by factoring, we will need the trinomial to have a leading coeﬃcient that is an integer. We can factor out a GCF of

, so that the equation becomes

(x2 − 2x − 24) = 0.

Since there are no common factors for the remaining three terms, we proceed with ﬁnding the value of two integers that multiply to ac = (1) (−24) = − 24 and add up to b = − 2. After ﬁnding these integers, we use them to rewrite the middle term −2x as a sum of two terms. Lastly, we can factor the trinomial by grouping and solve the equation using the zero product property.

Apply the idea The factor pair whose sum is −2 is −6 and 4. We can use this to rewrite the trinomial and factor by grouping as follows: (x2 − 2x − 24) =

(x2 − 6x + 4x − 24)

Rewrite polynomial with four terms

[x(x − 6) + 4 (x − 6)]

Factor each pair

(x − 6) (x + 4)

Factor out the GCF of (x − 6)

There are no more common factors to be divided out, so the fully factored form of the polynomial is

(x − 6) (x + 4).

This leads to the equation (x − 6) (x + 4) = 0 We can then solve the equation by setting each factor equal to zero, giving us x − 6 = 0 and x + 4 = 0, which gives the solutions x = 6 and x = − 4.

b 3 (x − 5)2 − 27 = 0

Create a strategy As this is written in vertex form, we can solve it using square roots and inverse operations to solve for x.

Apply the idea 3 (x − 5)2 − 27 = 0

Given equation

Addition property of equality

Division property of equality

3 (x − 5) = 27 (x − 5) = 9 x − 5 = ±3 x=5±3

Square root property of equality Addition property of equality

Giving us two solutions x = 2, x = 8.

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Reﬂect and check One way to check our solutions is by graphing. We can graph the corresponding quadratic function y = 3 (x − 5)2 − 27 and look for the places where the y-values equal 0. y 15 10 5 −1 −5

x 1 2 3 4 5 6 7 8 9

−10 −15 −20 −25

Recall, the solution(s) to a function occur at the x-value(s). For this equation, we want to ﬁnd the x-values when y = 0. This corresponds to the points where the graph crosses the x-axis. Looking at the graph, we can conﬁrm our two solutions of x = 2, x = 8.

c 3x2 − 5x + 12 = 0

Create a strategy In general, if the leading coefficient is not 1 then factoring is not likely to be efficient. In this particular case the most efficient method would be to use the quadratic formula. We can also calculate the discriminant b2 − 4ac, to identify if there are two, one or no real solutions.

Apply the idea

State the quadratic formula

Substitute values for a, b, c

Evaluate the operations

We can see that the discriminant is equal to −119. As it is less than zero, we know that the quadratic equation has no real solutions.

Reﬂect and check Using the quadratic formula will always be an appropriate method, and has the advantage of identifying the number and type of solutions, whether they are real or non-real, or rational or non-rational. If you cannot quickly and easily identify a way to solve it using one of the other methods, then using the quadratic formula is always suitable.

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d 9x2 − 12x − 2 = 0

Create a strategy In this example, we would need to factor the trinomial by rewriting it as four terms, where the coefficients of the linear terms have a product of a ⋅ c = − 18 and a sum of b = − 12. Since we do not have factors that add to −12, we can solve the quadratic using the quadratic formula or by completing the square. We can complete the square. The coefficient of x2 is 9, so we will need to divide this coefficient out before completing the square. Once 9 has been divided out, the coefficient of x will be so this is the value that completes the square.

. Taking half of

and squaring it gives us

Apply the idea Given equation

Divide by 9 on both sides

Add

Complete the square

Factor the left side, evaluate the right side

Square root property

This leaves us with two equations: Next, we add

to both sides

and

to solve both equations, and we find that the solutions are

and

Reﬂect and check As stated in part (c), the quadratic formula is always a suitable method. Solving the quadratic equation using the quadratic formula, we have State the quadratic formula

Substitute values for a, b, c

Evaluate the operations

The solutions to the quadratic equation are x =

and x =

. By simplifying the solutions we found by

completing the square and using the quadratic formula, we can conﬁrm that the solutions are equivalent.

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Example 2 A sculpture includes a cast iron parabola, coming out of the ground, that reaches a maximum height of 2.25 m, and has a width of 6 m. Let the position of the start of the parabola be (0, 0). Let x be the horizontal distance and y be the height of the sculpture above the ground. Determine an appropriate quadratic function that will model the shape of the parabolic sculpture.

Create a strategy The vertex lies half way between the two x-intercepts and has a height of 2.25 so has coordinates (3, 2.25). As we know the vertex we will write the function in vertex form f (x) = a (x − h)2 + k. Using another known point we can solve for a. In this case we know that (0, 0) lies on the parabola.

Apply the idea

Vertex form

Substitute the vertex (3, 2.25)

Substitute (0, 0)

Evaluate the exponent

Subtraction property of equality

Division property of equality

Simplify the fraction

The parabola can be modeled by the function

Reﬂect and check As we knew the x-intercepts of the parabola, we could have also written the function in factored form, using a similar method. Starting with f (x) = a (x − 6) x and the substituting in the values of the vertex we would ﬁnd the equation in factored form is

Idea summary Below is a list of the easiest method to use and the form of the quadratic equation for which we should use it: Graphing Factoring Square root property Completing the square Quadratic formula

Easiest equation form: Any form is fine when using technology ax2 + bx + c = 0 where a, b, c are small x2 = k or a (x − h)2 = k x2 + bx + c = 0 where b is even ax2 + bx + c = 0 where a, b, c are large

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Practice What do you remember? 1

Match each term with its mathematical representation. a

Factored form

If a ⋅ b = 0, then a = 0 or b = 0

Zero product property

If x2 = k, the

Quadratic formula

iii

b2 − 4ac

Discriminant

(2x − 3) (5 − x)

Square root property

Solve each quadratic equation using the zero product property. a

0 = − 3x(2x + 6)

0 = (2x − 9)2

0 = (2x + 4) (x − 5)

3x2 + 4x + 1 = 0

3x2 − 7x + 2 = 0

0 = (8x − 5) (3x − 7)

h2 + 3h − 28 = 0 2

4x + 12x + 9 = 0

b f

2x2 + 3x − 2 = 0

−5x + 9x + 2 = 0

Solve using inverse operations, including square roots. Give your answer as an exact value. a

4y2 = 100

−x2 + 121 = 0

(x + 3)2 − 49 = 0

−(3y + 9)2 + 100 = 0

−4 = 6

(2 − x)2 = 81

3x2 + 9x − 4 = 0

For each quadratic equation: i

Solve using the quadratic formula, rounding to three decimal places if necessary.

Verify using technology.

x2 − 7x + 9 = 0

e 6

Solve by factoring and then using the zero product property. a

(x + 5) (x − 10)

5x − 15x + 2 = 0

b f

x2 − 5x − 2 = 0

−2x2 − 15x − 4 = 0

−5x − 15x + 3 = 0

Would the discriminant be positive, negative, or zero for these scenarios? a

No real solutions

One real solution

One x-intercept

Two x-intercepts

Two distinct real roots d

A rectangular swimming pool is 50 ft long and 20 ft wide. It is surrounded by a pebble path of uniform width x ft. The area of the path is 200 ft2. Create an equation for the area of the path in terms of x. Do not solve it.

Non-real roots

x x

20 ft 50 ft x

114

Are the following statements true or false? a

Every quadratic equation can be solved by factoring over the real numbers and using the zero product property.

To solve an equation in the form (x − h)2 + k = 0, we can use inverse operations, including square roots.

Completing the square is a method used to solve linear equations.

Factoring is a method used to ﬁnd the roots of a quadratic equation by rewriting it as a product of two binomials.

The quadratic formula can be used to ﬁnd real solutions of any quadratic equation.

A quadratic equation always has exactly two real solutions.

Mathspace Virginia SOL Algebra 2 mathspace.co

Let’s practice 9

For each of the following equations: i

Determine an appropriate strategy for solving and explain your choice.

Solve for x.

x2 + 4x − 21 = 0

2 (x + 3)2 − 8 = 0

x2 +

x2 −

x2 − 3x − 40 = 0

x2 − 20x + 99 = 0

−15 + 22x + 5x2 = 0

−3.5x2 − 7.5x + 1 = 0

(x + 9)2 = 121

x2 = 6

+1=0

Solve the following equations for x: a

x2 = 144

0.5x2 = 9.5

−2x2 − x + 5 = 0

(x + 7)2 = 0

64x2 = 25

(4x + 5)2 = 64

0.05x2 − 6.25 = 0

12x2 + 7x − 45 = 0

m 3x2 − 21x + 30 = 0 11

Solve each quadratic equation using two different methods and comment on the eﬃciency of both methods: a

x2 − 10 = 15

x + 24x + 63 = 0

x2 − 7x + 6 = 0

x − 7x = 0

c g

25y2 = 36 2

5k − 17k + 13 = 0

4x2 + 5x + 1 = 0

x2 + 9x + 20 = 0

Solve the quadratic equation x2 + 2x − 8 = 0 and verify the solution by substitution. Explain your process.

A square has 3 in added to its length and 9 in added to its width. The area of the new rectangle is 280 in2.

Write an equation that could be used to ﬁnd the dimensions of the original square.

Solve to ﬁnd the dimensions of the original square.

Explain whether or not your answer in part (b) is reasonable.

The length of a rectangular mat is twice its width and the diagonal length is 55 cm. a

Write an equation that could be used to ﬁnd the the width of the mat.

Solve to ﬁnd the width of the mat, rounded to the nearest centimeter.

Explain whether or not your answer in part (b) is reasonable.

A rectangle has one side 3 cm longer than the other side, and its area is 28 cm2. a

Write an equation that could be used to ﬁnd the length of the shorter side of the rectangle. Explain your thinking.

Determine the length of the shorter side of the rectangle. Show your work.

Verify your solution. Explain how you checked your answer.

Mae throws a stick vertically upwards. After t seconds, its height h meters above the ground is given by the formula h(t) = 25t − 5t2. a

What do the solutions to h(t) = 10 represent?

At what time(s) will the stick be 30 m above the ground?

How long after it is thrown does the stick hit the ground?

A t-shirt is ﬁred straight up from a t-shirt cannon at ground level. After t seconds, its height above the ground is h feet, where h(t) = − 16t2 + 48t. a

For what values of t is the t-shirt 11 ft above the ground?

How long was the t-shirt at least 11 ft above the ground?

How long was the t-shirt at least 35 ft above the ground?

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Sarah attempted to solve the quadratic equation x2 + 15x + 54 by factoring method. Identify the errors in Sarah’s solution and explain what Sarah should have done instead. Sarah’s solution:

x2 + 15x + 54 = 0

Given

x2 + 15x + 54 = (x + 9) (x + 6)

Factoring the quadratic equation

x = 9 and x = 6

Finding the solutions

Three students, Alice, Grace and Bob, are solving the quadratic equation x2 − 7x + 10 = 0. Alice uses factoring, Grace completes the square, and Bob uses the quadratic formula. a

Will they arrive at the same solutions? Explain your reasoning.

Explain which method would be the most eﬃcient and why.

A frisbee is thrown upward and away from the top of a cliff that is 48 ft above the ground. The height h(x), in feet, of the frisbee at time x, in seconds, is given by: h(x) = − 4x2 + 16x + 48 The graph of this relationship is shown. a

Identify and interpret when f (x) = 0.

Determine when the frisbee reaches a height of 55 feet above the ground.

The frisbee reaches a maximum height of 64 feet above the ground. Determine when it reaches the maximum height.

For the equation k = − 4x2 + 16x + 48, determine a value of k which would give no real solutions. Explain your reasoning.

Let’s extend our thinking 21 22

A quadratic equation has solutions x = ± . Find the equation: x2 = ⬚.

An equation of the form 0 = x2 + bx + c has solutions of x = − 1 and x = − 5. Determine the original quadratic equation.

Consider the graph shown. 8

Select the quadratic equation that represents the graph.

y = − (x + 1) (x − 4)

y = − (x − 1) (x − 4)

y = (x + 1) (x + 4)

y = (x − 1) (x − 4)

−8 −6 −4 −2 −2

−4 −6 −8

Write the quadratic function that is represented by the given table of values: x y

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−2 0

−1 −4

0 −4

1 0

2 8

x 2

A ball is thrown into the air at an angle. The height y (in feet) of the ball at time x (in seconds) is modeled by the equation y = − 10x2 + 39x + 4. The graph of this relationship is shown. y 50 45 40 35 30 25 20 15 10 5

x 1

Identify three different strategies for ﬁnding when the ball would hit the ground. Explain which strategy you would use and why. 26

A science project involved giving carrots different concentrations of nitrogen fertilizer. Five different concentrations were used and the lengths of those carrots was recorded. A quadratic model was created where the maximum length of 396.05 mm occurred using a 5.9 g/mL concentration of nitrogen. Carrot length (mm) 400

(5.9, 396.05)

350 300

(1.5, 300)

(11.3, 249.9)

250 200

(0, 222)

150 100

(14, 69)

Nitrogen (g/mL) 2

Using technology or otherwise, solve for how much nitrogen would lead to i

Carrots that are 210 mm long.

Killing the carrots.

In general, root vegetables need less nitrogen than other vegetables. Assess the reasonableness of your answers to part (a).

Consider the quadratic equation x2 + bx + c = 0. a

Solve for x by completing the square, then using square roots. Give your answer in terms of b and c.

What connections can you see to the quadratic formula?

We can solve some quartic equations as a quadratic equation by using substitution. a

Solve the equation x4 − 13x2 + 36 = 0. Show your solution.

Solve x4 − x2 − 20 = 0. How many solutions do you get? Explain.

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When we raise i to some power, we notice a pattern unfolds: i1 = i i2 = − 1 i3 = − i i4 = 1

i5 = i i6 = − 1 i7 = − i i8 = 1

i9 = i i10 = − 1 i11 = − i i12 = 1

We can see that there is a cycle of 4, and we can use this pattern to determine the value of any power of i. • When the exponent of i is a multiple of 4, the expression simplifies to 1. • When the exponent is one more than a multiple of 4, the expression simplifies to i. • When the exponent is 2 more than a multiple of 4, the expression simplifies to −1. • When the exponents is 3 more then a multiple of 4, the expression simplifies to − i. We can summarize this algebraically like so: i4n + 1 = i i4n + 2 = − 1 i4n + 3 = − i i4n = 1

This shows that we can write any exponent in the form 4n + r where n is any integer and r can be 0, 1, 2, or 3.

Example 1 Express the following in terms of i: a

Create a strategy If we use the multiplication property of radicals, we can separate the radicand into −1 ⋅ 49 to deﬁne with i.

Apply the idea Factor out −1

Product of square roots

Deﬁnition of i

Commutative property of multiplication

Create a strategy We will use the same reasoning as the example above.

Apply the idea Factor out − 1

Product of square roots

Deﬁnition of i

Since the radical cannot be simpliﬁed further, this is the ﬁnal answer.

Reﬂect and check This answer can also be written as

, but it needs to be clear that the i is not underneath the radical.

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119

Create a strategy After writing the expression in terms of i, this radical can be simpliﬁed which we can do using properties of radicals. Our goal is to ﬁnd the largest perfect square factor of 32.

Apply the idea Factor out − 1

Product of radicals

Substitute

Factor 32

Product of radicals

Evaluate the radical

Reﬂect and check As you become more familiar with imaginary numbers, you can use fewer steps to simplify:

Example 2 Simplify the following expressions: a i15

Create a strategy We can use properties of exponents and the fact that i4 = 1 to make the process simpler. If we can ﬁnd out how many times 4 goes into 15, then we only need to know the ﬁrst 3 powers of i to simplify.

Apply the idea Using the fact that 15 = 4 ⋅ 3 + 3, i15 = i4⋅3 + 3 43

Rewrite the exponent 3

= (i ) ⋅ i 3

Rewrite using power and product property

= (1) ⋅ − i

Substitute i4 = 1 and i3 = − i

= −i

Evaluate the exponent

Therefore, i = − i.

Reﬂect and check We can use fewer steps by dividing the exponent by 4 and determining the remainder. 15 ÷ 4 = 3 with a remainder of 3 The remainder will be the new exponent of i that we will use to evaluate the expression. i3 = − i which means i15 = − i.

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b i−6

Create a strategy The deﬁnition of negative exponents says i − m =

Apply the idea Deﬁnition of negative exponents

Product of powers

Substitute i4 = 1 and i2 = − 1

Evaluate the multiplication and division

Therefore, i

−6

= −1

Reﬂect and check If we extend the cyclic pattern into the negative exponents, we see that this still follows the pattern i, −1, −i, 1: i−7 i

i−6 −1

i−5 −i

i−4 1

i−3 i

i−2 −1

i−1 −i

i0 1

i1 i

i2 −1

i3 −i

i4 1

c 5i2 − 2i4 + 3i7

Create a strategy We have already calculated these powers of i above: i2 = − 1 i4 = 1 i7 = − i

Apply the idea 5i2 − 2i4 + 3i7 = 5 (−1) − 2 (1) + 3 (−i)

Substitute i2 = − 1, i4 = 1, and i7 = − 1

= − 5 − 2 − 3i

Evaluate the multiplication

= − 7 − 3i

Evaluate the subtraction

Reﬂect and check Because one term has an i and one does not, these are not like terms and cannot be combined.

Idea summary Imaginary numbers are used to deﬁne square roots of negative numbers. The imaginary unit i is deﬁned as . The powers of i follow a cyclic pattern of 4: i1 i

i2 −1

i3 −i

i4 1

… …

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Complex numbers Previously, all the numbers we knew fell under the umbrella of real numbers. Now, we have learned about imaginary numbers which are not real numbers, so we need to introduce a new type of number that encompasses both real and imaginary numbers. Complex number The set of all real and imaginary numbers. a ± bi where a and b are real numbers, and i =

A complex number consists of both real (a) and imaginary (bi) but either part can be 0. By this deﬁnition, all numbers are complex numbers. Sometimes, a is referred to as the real part and bi is called the imaginary part. Both parts together make up one complex number. A real number is a complex number where b = 0. For example, −5 = − 5 + 0i. A pure imaginary number is a complex number where a = 0. For example, 2i = 0 + 2i. Complex Numbers Real Numbers Rational Integers Whole Natural ... −4 −3 −2 −1 0 1 2 3 4 ... −3.12

Imaginary Numbers i −5i

1.3 Irrational 3 + 4i

−2 −7i

We perform the algebraic operations for complex numbers the same way we perform operations for rational algebraic expressions, except we sometimes have an extra step to account for the powers of i. Addition: add in the same way as binomials with like terms General example Numerical example

(a + bi) + (c + di) = (a + c) + (b + d) i (2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5) i = 6 + 8i

Subtraction: subtract in the same way as binomials with like terms General example Numerical example

(a + bi) − (c + di) = (a − c) + (b − d) i (2 + 3i) − (4 + 5i) = (2 − 4) + (3 − 5) i = − 2 − 2i

Multiplication: distribute in the same way as binomials, evaluate any powers of i, then combine any like terms General example Numerical example

(a + bi) (c + di) = ac + adi + bci + bdi2 = ac + (ad + bc) i + bdi2 = (ac − bd) + (ad + bc) i (2 + 3i) (4 + 5i) = 8 + 10i + 12i + 15i2 = 8 + (10 + 12) i + 15i2 = (8 − 15) + (10 + 12) i = − 7 + 22i

The conjugate of the complex number a + bi is a − bi. A complex number multiplied by its conjugate is a non-negative, real number. General example Numerical example

122

(a + bi) (a − bi) = a2 − abi + abi − b2 i2 = a2 + b2 i2 = a2 − b2 (5 + 2i) (5 − 2i) = 52 − 10i + 10i − 4i2 = 25 − 4i2 = 25 + 4 = 29

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Example 3 The following complex numbers are written in the form a + bi. State the values a and b. a −8 + i

Apply the idea

Reﬂect and check

a = −8

Both terms together are considered one complex number.

b=1

Create a strategy This is a real number which means the imaginary part is missing, so we can write the imaginary part as 0i. That means can be written as

+ 0i.

Apply the idea a=

Reﬂect and check

and b = 0

can be classiﬁed as complex, real, and rational.

Create a strategy We need to rewrite the expression in terms of i, simplify the radical, then determine the values of a and b.

Apply the idea Factor out − 1

Product of radicals

Deﬁnition of i

Factor 40

Product of radicals

Evaluate the square root

There is only one term in this expression, and it contains i. This means the real part is missing which makes this a pure imaginary number.

Reﬂect and check This number can be classiﬁed as complex, imaginary, and pure imaginary.

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Example 4 Simplify each of the following expressions. Justify each step using the commutative, associative, and distributive properties. a (−3 + 5i) + (7 − 2i)

Create a strategy When adding or subtracting complex numbers, we can combine like terms by adding the real parts together and the imaginary parts together.

Apply the idea (−3 + 5i) + (7 − 2i) = (−3 + 7) + (5i − 2i) = 4 + 3i

Commutative and associative properties Evaluate the addition

b (−6 − i) − (8 − 5i)

Apply the idea (−6 − i) − (8 − 5i) = (−6 − i) + (−8 + 5i)

Distributive property

= (−6 + −8) + (−i + 5i)

Commutative and associative properties

= − 14 + 4i

Evaluate the addition

Reﬂect and check It is important to distribute the negative sign to the second complex number ﬁrst because the commutative and associative properties only hold for addition and multiplication, not subtraction.

c (2 − 4i) (−4 + 2i)

Create a strategy When multiplying complex numbers, we can use the distributive property: ( A + B) (C + D) = AC + AD + BC + BD

Apply the idea (2 − 4i) (−4 + 2i) = 2 (−4) + 2 (2i) − 4i (−4) − 4i (2i) 2

124

Distributive property

= − 8 + 4i + 16i − 8i

Evaluate the multiplication

= − 8 + 4i + 16i − 8(−1)

Substitute i2 = − 1

= − 8 + 4i + 16i + 8

Evaluate the multiplication

= 20i

Evaluate the addition

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Example 5 What is the result when each of the following is multiplied by its conjugate? a 4 + 3i

Create a strategy First, we need to identify the conjugate of the expression. The conjugate of 4 + 3i is 4 − 3i. When multiplying complex numbers, we can use the distributive property: ( A + B) (C + D) = AC + AD + BC + BD

Apply the idea (4 + 3i) (4 − 3i) = 4 (4) + 4 (−3i) + 3i (4) + 3i (−3i) 2

Distributive property

= 16 − 12i + 12i − 9i

Evaluate the multiplication

= 16 − 12i + 12i − 9(−1)

Substitute i2 = − 1

= 16 − 12i + 12i + 9

Evaluate the multiplication

= 25

Evaluate the addition

Reﬂect and check Recall that the product of complex conjugates always results in a real number. We can use this fact to check the reasonableness our solution. If the solution is imaginary, then there must be a mistake in the calculations.

Create a strategy First, we need to rewrite the expression as a complex number. Then, we need to identify the conjugate of the expression.

Apply the idea As a complex number, the expression is: Rewrite as a product with − 1

Product of radicals

Deﬁnition of i

Evaluate the square root

This is a pure imaginary number because the real part is 0, so we can also write this as 0 − 2i. This helps us see that the conjugate of − 2i is 2i. (−2i) (2i) = − 2 (2) i (i) = − 4i

Distributive property Evaluate the multiplication

= − 4(−1)

Substitute i2 = − 1

Evaluate the multiplication

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Create a strategy First, let’s write the expression in the form a + bi.

Next, we need to identify the conjugate of the expression.

The conjugate of

Apply the idea Multiply the fractions

Distributive property

Evaluate the multiplication

Substitute i2 = − 1

Evaluate the multiplication

Evaluate the addition

Reﬂect and check The only difference between a complex number and its conjugate is the sign of the imaginary parts. The signs of the real parts of complex conjugates are the same, but the signs of the imaginary parts are opposite. This helps us see that the conjugate of different.

is not

because the signs of the real parts of the expression are

Idea summary All numbers are complex numbers. They are in the form a + bi where a is the real part and bi is the imaginary part. Both parts together make one complex number. We simplify complex numbers by combining the real parts with the imaginary parts. Sometimes, we have to evaluate powers of i and continue simplifying the expression. The conjugate of the complex number a + bi is a − bi.

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Practice What do you remember? 1

Give an example of each of the following: a

A complex number

A pure imaginary number

A real number

A pair of complex conjugates

Fill in the blanks: a ± bi is a ⬚ number, where ⬚ and ⬚ are real numbers and ⬚ = real (⬚) and imaginary (⬚i) parts, but either part can be 0.

Which diagram best represents the relationship between the number types? A

All numbers

Pure Real Complex numbers numbers imaginary numbers

Complex numbers

Pure imaginary numbers

Real numbers

Pure imaginary numbers

Determine whether each number can be classiﬁed as the following: i

A complex number

A real number

iii

A pure imaginary number

An irrational number

e 5

Pure Real Complex numbers numbers imaginary numbers

Real numbers

Complex numbers

. A complex number is made up of both

Match each radical with its simplﬁed form. a

iii

− 7i

−7

Express the following in terms of i: a

State the complex conjugate of each of the following complex numbers: a

3 + 4i

8 − 5i

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127

Let’s practice 8

Explain why − 8i2 = 8.

Consider expressions of the form in. a

Copy and complete the given table. i Radical form Complex form

−1

⬚

−1

⬚

i16

i14

iii

i19

i22

i29

i17

i11

−4i8

6i9

3i3 − 5i4 − 6i5

(3i)4

Simplify each expression. Give your answer in the form a + bi. a

(−5 + 7i) − 6

(−1 − 4i) − 7i

(9.6 + 6.6i) + (−5.1 + 9.9i)

(−8 + i) + (3 + 4i) + (−5 − 7i)

(4 + 3i) − (2 − 9i)

(−5 + i) − (−2 − 9i) − (8 + 6i)

(2 + 7i) − (−8 + 5i) + (−6 + i)

−21 − 3i

Which number is equivalent to (−8 + i) − 3i − (13 − 5i)? −21 + 3i

21 + 3i

21 − 3i

Simplify each expression. Give your answer in the form a + bi. a

−8 (6 + 5i)

(6 − 8i)2

7i(3 − 2i)

(8 + 9i) (1 + 6i)

(2 + 9i) (9i − 2)

5i(3 − i)2

−2i(5 − 4i)2

For each of the following solutions, ﬁll in missing steps and reasons: 15i − (6 − 4i) + 2 = 15i − 6 + 4i + 2

⬚

= 2 − 6 + 15i + 4i

⬚

=⬚

Combine like terms

(9 − 5i) (6 − 4i) = 9 (6 − 4i) − 5i (6 − 4i) = 54 − 36i − 30i + 20i

⬚

Deﬁnition of i2

=⬚

= 54 − 36i − 30i – 20

⬚

= 54 − 20 − 36i − 30i

⬚

= 34 − 66i

⬚

For each of the following, ﬁnd the missing values that make each equation true: a c e

128

⬚

Select all options that are equivalent to (i).

⬚

Simplify each of the following expressions. Give your answer in the form a + bi.

A 13

Simplify each of the following complex numbers:

A i43

SOL

Check to see if the pattern continues by evaluating the next four powers of i.

(5 + ⬚) + (⬚ − 2i) = 13 + 5i

(⬚ − i) (5 + 2i) = ⬚ + i

i⬚ = − 1

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b d

(⬚ − 9i) − (14 − ⬚) = 20 + 11i

(⬚ − 5) (2 + 3i) = 3i − 37

Multiply each of the following expressions by its conjugate. Fully simplify your answer. a

−2 − 4i

Sophia wants to evaluate

−4i

and writes out the following solution: 1 2 3

Identify the error she made.

Correct Sophia’s work.

Simplify the expression

Simplify each of the following expressions. Give your answer in the form a + bi.

. Show your work/thinking.

a 20

Simplify each expression. Give your answer in the form a + bi. a

Let’s extend our thinking 22

Prove that the product of a complex number and its conjugate is always a real number.

Consider any two complex numbers. Determine whether each of the following statements is always, sometimes or never true. Justify your answer. a

The sum of two complex numbers is a real number.

The product of two complex numbers is a real number.

The sum of two complex numbers is a pure imaginary number.

The product of two complex numbers is a pure imaginary number.

The sum of two complex numbers is a complex number.

The product of two complex numbers is a complex number.

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129

Electrical circuits have a power source and can include resistors which slow down the ﬂow of the current. a

This ﬁgure shows a circuit with two resistors in series. The total resistance is given by the formula: RTotal = R1 + R2 If R1 = 1 + 2i ohms and R2 = 3 + i ohms, ﬁnd the total resistance in the circuit. R1 = 1 + 2i R2 = 3 + i

For a circuit with two resistors in parallel, the formula for total resistance can be given as:

R2 R1

Give an example of two complex numbers resistances that would result in a total resistance that is a real number. 25

130

Not all properties that are true for real numbers are true for complex numbers. Below is a proof which results in a false statement. 1

Using that

Using that 12 = 1

Using that 1 = − 1 ⋅ − 1

Using that

Deﬁnition of i

Deﬁnition of squaring

Deﬁnition of i2

Identify which line the uses a property which is not valid for complex numbers.

Test a property of equality to see if it is valid over the complex numbers.

If a property applies to the complex number system, does it apply to the rational number systems? Justify your answer.

Consider how we can use complex numbers to factor a sum of squares. a

Factor a2 + b2 using complex numbers.

Using the identity found in part (a) or otherwise, factor 9x2 + 25 into two complex linear expressions.

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2.05 Quadratic equations with complex solutions After this lesson, you will be able to... • write quadratic equations to represent real-world situations. • solve quadratic equations. • verify solutions to quadratic equations algebraically, graphically, and with technology. • justify the reasonableness of solutions. • explain the steps for solving a quadratic equation. • interpret solutions in context. • identify the number and type of solutions to a quadratic equation.

Quadratic equations with complex solutions Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 2.05 to answer these questions. 1.

What do you think the solutions to the equation are based on the red parabola?

What do you think the solutions to the equation are based on the blue parabola?

How do you think you could find those solutions algebraically?

Graphing to solve quadratic equations with complex solutions is not practical because the graph would need to have 3 dimensions: the horizontal plane, the vertical plane, and the imaginary plane, which is perpendicular to the real x-plane. This means we will need to use algebraic methods for solving quadratic equations with complex solutions. We have seen that quadratic equations of the form ax2 + bx + c = 0 can have 2 real solutions, 1 real solution, or no real solutions. We used the discriminant to determine the nature of the solutions. If there are no real solutions, then the solutions are complex roots (of a quadratic). Recall the quadratic formula which contains the discriminant:

b2 − 4ac discriminant

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131

y x x

b2 − 4ac > 0

b2 − 4ac = 0

b2 − 4ac < 0

The equation has two real solutions.

The equation has one real solution.

The equation has two complex solutions.

When solving quadratic equations with real coefficients that have non-real roots, we can now find the solutions by expressing them as complex numbers, with the roots being complex conjugates. Conjugate (of complex number) A number created by changing the sign of the imaginary part of a complex number. Example: The conjugate of a + bi is a − bi. In Algebra 1, we discussed several methods for solving quadratic equations which could also be used to find the complex solutions of a quadratic equation: • Factoring is best to use when the coefficients are relatively small numbers and when the value of the discriminant is a perfect square. • The square root property is best to use when the equation is in the form x2 = k or in vertex form a (x − h)2 + k = 0. • Completing the square can be used to solve any quadratic equation, but it is easiest when a = 1 and b is even. • The quadratic formula is the best method for all other types of quadratic equations, especially ones where the coefficients are large numbers.

Example 1 Determine the number and nature of the solutions to the following equations: a 5x2 + 2x + 2 = 0

Create a strategy We can calculate the value of the discriminant to ﬁnd the types of solutions to the equation. The discriminant is equal to b2 − 4ac, and for this equation we have a = 5, b = 2, c = 2.

Apply the idea

Reﬂect and check 2

The discriminant is (2) − 4(5) (2) = − 36, so there are two complex solutions.

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If the discriminant is negative, we would need to ﬁnd the square root of a negative number in the quadratic formula, which results in non-real solutions. We can see that if 4ac > b2 the discriminant will always be negative.

b 16x2 − 24x + 9 = 0

Create a strategy The discriminant is equal to b2 − 4ac, and for this equation we have a = 16, b = − 24, c = 9.

Apply the idea

Reﬂect and check 2

The discriminant is (−24) − 4(16) (9) = 0. This tells us there If the discriminant is zero, there is only one real solution will be one real solution. because the root is repeated.

It is not possible to have one real solution and one complex solution for a quadratic equation with real coeﬃcients.

Example 2 Solve the following equations, stating your solutions in the form a ± bi: a 2x2 − 6x + 19 = 0

Create a strategy A quadratic equation in standard form ax2 + bx + c = 0 has the solutions have a = 2, b = − 6, c = 19.

, and for this equation we

Apply the idea

State the quadratic formula

Substitute a = 2, b = − 6, c = 19

Evaluate the square and products

Evaluate the difference in the radicand

Rewrite the radicand

Evaluate the radical

Rewrite as two fractions

Simplify the quotients

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133

Reﬂect and check We can see that the two complex solutions,

and

, are complex conjugates.

That is, they are of the form a + bi and a − bi. To write this solution in set notation, we simply list both solutions, separated by a comma, within set brackets:

b 4x2 + 9 = 0

Create a strategy We can rewrite this to be in the form of x2 = , then use the square root property to solve.

Apply the idea Given equation

Subtract 9 from both sides

Divide both sides by 4

Square root property

Division property of radicals

Evaluate the radical

Reﬂect and check We can check our solutions by subtituting them into the original equation. Le’ts check each solution one at a time. First, check x =

: Given equation in for x

Substitute

Evaluate the square

Evaluate the multiplication

Replace i2 with its value of − 1

Evaluate the multiplication

Add − 9 and 9

This shows x =

134

makes the equation true, so it is a valid solution.

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Now, let’s check x =

: Given equation in for x

Substitute

Evaluate the square

Evaluate the multiplication

Replace i2 with its value of − 1

Evaluate the multiplication

Add − 9 and 9

Since substituting our solution into the equation resulted in a true statement, we have conﬁrmed both x = x=

are solutions to this quadratic equation. This solution can also be represented as

and

Example 3 Consider the quadratic function p (x) = x2 − 6x + 16. Find the roots of the equation p (x) = 0.

Create a strategy The roots of the equation p (x) = 0 are the same as the solutions to the equation x2 − 6x + 16 = 0. Since a = 1 and b is even, it would be easy to complete the square to ﬁnd the roots.

Apply the idea

Given equation

Subtract the constant from both sides Complete the square

Factor the left side, evaluate the right side

Square root property Add 3 to both sides Rewrite the radicand Substitute The roots are x = 3 +

and x = 3 −

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135

Reflect and check Another method used for finding roots is the quadratic formula. Let’s confirm our solutions using this method.

State the quadratic formula

Substitute a = 1, b = − 6, c = 16

Evaluate the square and products

Evaluate the difference in the radicand

Rewrite the radicand

Evaluate the radical

Rewrite as two fractions

Simplify the quotients , which is the same answer we got when solving by completing the square.

This shows the solution set is

Example 4 Consider the equation x2 +

= − 2.

a Determine the nature and number of solutions.

Create a strategy To determine the nature and number, we need to ﬁnd the discriminant. But before calculating the discriminant, we need to move all terms to the same side.

For this equation, a = 1, b = , c = 2.

Apply the idea

Substitute coeﬃcients into the discriminant

Evaluate the multiplication

Evaluate the subtraction

Since the discriminant is negative, the equation has 2 complex solutions.

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Reﬂect and check Recall that the set of complex numbers contains both real numbers and imaginary numbers. Although real numbers belong to the set of complex numbers, it is not common to refer to the real solutions as “complex” in the context of quadratic equations. Generally, only solutions of the form a ± bi are referred to as complex solutions.

b Find the roots of the equation.

Create a strategy Since we already have the value of the discriminant, we can use the quadratic formula and substitute the discriminant into the radicand.

Apply the idea 1

State the quadratic formula

2 Substitute known values

3 Rewrite the radicand

4 Evalute the radical 5 Evalute the division

Reﬂect and check To get from step 4 to 5, we divided the numerator of the complex fraction by the denominator:

This solution set can also be represented as

Idea summary The discriminant, b2 − 4ac, can help us determine the nature and number of solutions to a quadratic equation without needing to fully solve the equation. • • •

b2 − 4ac > 0 two real solutions b2 − 4ac = 0 one real solution b2 − 4ac < 0 two complex solutions

We can use the square root property, completing the square, or the quadratic formula to solve quadratic equations with complex solutions. We cannot ﬁnd complex solutions by graphing. Complex solutions always come in pairs called complex conjugates.

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137

Practice What do you remember? 1

For the quadratic equation ax2 + bx + c = 0, state the discriminant.

Describe the values of the discriminant for the quadratic equation ax2 + bx + c = 0, where a, b, and c are rational numbers, given that there are:

Two distinct real solutions

Two distinct rational roots

Exactly one real root

No real solutions

Two distinct imaginary solutions

Exactly one imaginary root

Determine the number and nature of the solutions.

For each of the following equations: i

Calculate the value of the discriminant.

x2 + 8x + 16 = 0

5x2 − 4x +

4x2 + 4x + 2 = 0

x2 + 22x + 121 = 0

9x2 + 12x − 7 = 0

9x2 − 12x + 4 = 0

0.1x2 + 0.5x + 1 = 0

2x2 − 2x = x − 1

Which of the following describes the root(s) of the equation 3x2 + 5 = 6x? A

Exactly one real root

Two distinct real roots

Exactly one imaginary root

Two distinct imaginary roots

Solve each equation using an appropriate strategy. Give your answer as an exact value. a

138

5 (x − 2)2 − 100 = 0

5x2 − 20x + 5 = 0

0 = 79 + 4x − 4x2

Simplify each of the following expressions to a complex numbers in the form a + bi. a

0 = 4 (x − 6)2 − 40

Simplify each of the following expressions to a complex numbers in the form a + bi. a

Determine if each statement is true or false. a

The quadratic formula will always result in two distinct solutions.

The quadratic formula will always give a real solution.

The quadratic formula can be used to solve any quadratic equation with real coeﬃcients.

The quadratic formula will result in complex conjugates if the radicand is negative.

The quadratic formula will always simplify to give rational coeﬃcients for real and imaginary parts.

The result after using the quadratic formula to solve an equation is x = solutions this quadratic equation has.

Mathspace Virginia SOL Algebra 2 mathspace.co

. Determine the number of real

Fill in the justiﬁcation or work for each step. Given equation

⬚ ⬚ ⬚

Evaluate the division

Explain which strategy you would use and why.

Solve the equation.

x2 = − 25

+5=7

−(x + 3)2 = 49

−4(x − 1)2 = 0

x2 + 18x + 81 = 0

x2 + 5x + 7 = 0

−3x2 + 5x =

(x − 3) (x + 5) = 6

Consider the following quadratic functions: i

Find the roots of the equation p (x) = 0.

Determine the nature of these roots.

p (x) = 9x2 − 3x − 20

p (x) = x2 − x − 20

p (x) = − 0.2x2 − x − 0.8

p (x) = 0.8x2 − 0.8x + 1

p (x) = x + 6x + 11

A solution to a quadratic equation is 10 + A

Let a = ⬚, b = ⬚, c = ⬚

For each equation:

⬚

Evaluate the square root and use

⬚

10 +

10 −

. Which of these must also be a solution to this equation? C

−10 +

140

−10 −

After some market testing, Hisham created a revenue model selling Kofta out of his home kitchen. He found that at $3 per pound he could sell 153 pounds per month. For every additional dollar per pound, he will sell 9 fewer pounds. a

Explain why the revenue model will be R(x) = (3 + x) (153 − 9x), where x represents the cost increase above $3/lb.

Determine the selling price(s) that would result in a monthly revenue of $576.

Determine the selling price(s) that would result in a monthly revenue of $900.

Determine the selling price(s) that would result in a monthly revenue of $1000.

Let’s extend our thinking 21

Sketch the graph of a quadratic function f (x) = ax2 + bx + c, where: a

a > 0 and b2 − 4ac > 0

a > 0 and b2 − 4ac < 0

a < 0 and b2 − 4ac < 0

a < 0 and b − 4ac = 0

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Heiko sells sports customized clothing. For sweaters, his profits this quarter can be modeled with the quadratic function, where p (x) is his profit and x is the number of sweaters he sells: p (x) = − 0.7x2 + 77x − 700 In order to afford a new embroidery machine, he needs to earn a profit of $1500.00. Explain whether or not it is possible him to afford a new embroidery machine.

Consider the quadratic equation x2 + 2x + 10 = 0. a

Solve the quadratic equation by completing the square and using square roots, showing all your work.

Solve the quadratic equation using the quadratic formula, showing all your work.

State the similarities and differences between the strategies.

For a quadratic function in standard form, explain when completing the square is more eﬃcient and when the quadratic formula is more eﬃcient.

Find the values of n for which x2 − 8nx + 1296 = 0 has one solution.

Find the value of k so that 16x2 + 8x + k = 0 has equal roots.

Consider the equation 4x2 + 8x + k = 0.

Find the value of the discriminant in terms of k.

Find the value of k for when the equation has one unique solution.

Find the values of k for when the equation has real solutions.

Find the values of k for when the equation has no real solutions.

Find the values of k for when the equation has two real and distinct solutions.

Consider the equation x2 + 6x + k + 8 = 0. a

Find the values of k for which the equation has no real solutions.

If the equation has no real solutions, determine the smallest integer value that k can have.

Show that if 7 + 4i is a root of x2 + bx + c = 0, then 7 − 4i is also a root.

Determine the equation of a quadratic function with: a

• Vertex at (5, −4) • One of the zeros at x = 5 − 2i

b • A constant term of 8 • One of the zeros at x = 1 +

A hammock is strung between two trees. f (x), is used to approximately model the hammock, where one of the trees is modeled by the y-axis and the ground is modeled by the x-axis. One of the zeros of f (x) is x = 4 −

. Find the equation of f (x). f (x) (4, 2) 0

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2.06 Quadratic inequalities After this lesson, you will be able to... • write one-variable quadratic inequalities to represent real-world situations. • solve quadratic inequalities and represent the solution on a number line. • verify solutions to quadratic inequalities algebraically, graphically, and with technology. • justify the reasonableness of solutions. • explain the steps for solving a quadratic inequality. • interpret solutions in context.

Quadratic inequalities A quadratic inequality is a polynomial inequality with a degree (highest exponent) of 2.

Exploration Consider the graph of the quadratic function y = − x2 + 4 and the solutions of the corresponding inequalities on the number lines below. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5 2

Solution to −x2 + 4 ≥ 0

Solution to −x + 4 ≤ 0 1.

What relationships do you notice between the graph of the function and the solutions of the corresponding inequalities?

How can we use the graph to ﬁnd the solutions −x2 + 4 < c where c is any real number?

The solution set of a quadratic inequality are the values that make the inequality true. Similar to quadratic equations, we can visualize where these solution sets come from by considering the graph of the corresponding quadratic function.

x2 > 4 y = x2 Corresponding function

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In this inequality, we are looking for the x-values that make the y-values greater than 4, as shown by the green sections of the graph.

y 9 8 7

The blue sections of the graph are less than 4, so those values do not satisfy the inequality.

6 5

We can show the solution set on a number line, as shown below to only indicate the values of x that satisfy the inequality.

4 3 2 1

−4 −3 −2 −1

−5

−4

−3

−2

−1

Notice that the end points at −2 and 2 are unﬁlled. This is because those values do not satisfy the inequality: (−2)2 ≯ 4 and (2)2 ≯ 4. 4

Another method to solve x2 > 4 is by making one side of this inequality zero and then solving the equivalent inequality that results:

3 2

x2 − 4 > 0

1 −4 −3 −2 −1 −1

x 1

By making one side of the inequality zero, we can now use the x-intercepts of y = x2 − 4 as the boundary points and the solution set is where the graph lies above the x-axis, that is, the y-values are above zero.

−2 −3 −4

If we change the inequality sign, the solution set will change as well. Consider the inequality x2 − 4 ≤ 0. Now, the blue region of the graph above will be the solution set because those are the values that are below zero. This time, the end points are ﬁlled because the solutions do satisfy the inequality.

(−2)2 − 4 ≤ 0 and (2) 2 − 4 ≤ 0

4 − 4 ≤ 0 and 4 − 4 ≤ 0

0 ≤ 0 and 0 ≤ 0 −5

−4

−3

−2

−1

Example 1 Solve the inequality x2 − 2x − 8 < 7 and graph the solution set on a number line.

Create a strategy To graph the inequality on a number line, we ﬁrst want to subtract 7 from both sides. Then, we can solve the related equation for x to ﬁnd the zeros.

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Apply the idea Subtracting 7 from both sides gives us x2 − 2x − 15 < 0. Next, we want to solve the related equation x2 − 2x − 15 = 0 to ﬁnd the zeros. x2 − 2x − 15 = 0 (x − 5) (x + 3) = 0

Factor the quadratic

This gives us zeros of x = 5, x = − 3. To solve the inequality (x − 5) (x + 3) < 0 we now want to identify the regions where this is true. Instead of drawing the entire graph, we can use a number line as the x-axis, plot the zeros, and imagine a parabola passing through those zeros. Then, we will use test points to determine which range of values makes the inequality true. Plotting the zeros on the number line shows us that there are three regions we need to test: 1. The left region when x < −3 2. The middle region when −3 < x < 5 3. The right region when x > 5

−5 −4 −3−2−1 0 1 2 3 4 5 6 7 8

We can use test points to determine which region of values would satisfy the inequality. For example, we can use x = − 4 as the test point from the left region, x = 1 as the test point from the middle region, and x = 7 as the test point from the right region. We substitute these test points into the inequality (x − 5) (x + 3) < 0. x = −4 x=1 x=7

(x − 5) (x + 3) (−4 − 5) (−4 + 3) = 9 (1 − 5) (1 + 3) = − 16 (7 − 5) (7 + 3) = 20

Less than 0? No Yes No

Since the inequality is less than zero, we need to look for when the answer is negative. The only negative answer comes from the test point x = 1 which lies in the middle region. This tells us (x − 5) (x + 3) < 0 when −3 < x < 5. We can now graph the solution on a number line: −5

−4

−3

−2

−1

Reﬂect and check Notice that we cannot solve this in the same way we would solve the equivalent quadratic function. That is, we cannot set both factors to be less than zero and solve them independently. This method would give us the incorrect solution of x < 5, x < −3. Using the graph to check, we see that the graph is less than zero in between the intercepts.

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y 2 −4 −3 −2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18

x 1 2 3 4 5 6

Example 2 Graph the corresponding quadratic function and solve each of the inequalities. a x2 < 9

Create a strategy The corresponding function is y = x2.

Apply the idea The graph of y = x2 is: 9

8 7 6 5 4 3 2 1 −4 −3 −2 −1

x 1

2 3 4

The solution set to the inequality x2 < 9 is equivalent to when y < 9 on the graph. 9

We can see that the graph is less than 9 between the x-values of −3 and 3. These endpoints will not be included because they do not satisfy the inequality.

8 7 6 5 4 3 2 1 −4 −3 −2 −1

x 1

2 3 4

The solution set for x2 < 9 is: −3 < x < 3

Reﬂect and check We could have created an equivalent inequality by subtracting 9 from both sides to get x2 − 9 < 0. This would have translated the graph of the corresponding function down 9 units. Because one side of the inequality is 0, we can look at the x-axis rather than where the y-values are at 9.

y −4 −3 −2 −1 −1

x 1

2 3

−2 −3 −4 −5 −6 −7

This graph is below the x-axis when −3 < x < 3, so the solution is the same.

−8

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b −x2 + 16 ≤ 0

Create a strategy The corresponding quadratic function to this inequality is y = − x2 + 16. This is the graph of y = x2 after it has been reﬂected across the x-axis and translated up 16 units.

Apply the idea The graph of y = − x2 + 16 is: y 16 12 8 4 x −4 −3 −2 −1

−4

The solution set to the inequality −x2 + 16 ≤ 0 is equivalent to when y ≤ 0 on the graph, which occurs on and below the x-axis. 16

From the graph, we can see that the end points of the solution set for the inequality will be

x = − 4 and x = 4

8 4 x −4 −3 −2 −1

−4

So the solution set for −x2 + 16 ≤ 0 is: x ≤ −4 or x ≥ 4

Reﬂect and check It is a good idea to check our answer by substituting in an x-value that is inside the solution set to make sure that it satisﬁes the inequality. For example, x = 5 is inside the solution set, so we can check that: −x2 + 16 ≤ 0 −(5)2 + 16 ≤ 0 −25 + 16 ≤ 0 −9 ≤ 0 It does satisfy the inequality, so we have chosen the correct interval for our solution set.

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c 0 ≤ (x − 4) (x − 1)

Create a strategy The corresponding quadratic function to this inequality is y = (x − 4) (x − 1). The parabola will be facing upward with x-intercepts at (1, 0) and (4, 0).

Apply the idea The graph of y = (x − 4) (x − 1) is: 5

4 3 2 1 −1

−1

x 1

−2 −3

The solution set to the inequality 0 ≤ (x − 4) (x − 1) is equivalent to when 0 ≤ y on the graph. Another way to look at that is y ≥ 0. This will be the part of the graph on or above the x-axis. 5

From the graph, we can see that the end points of the solution set for the inequality will be

x = 1 and x = 4

3 2 1 −1

−1

x 1

−2 −3

The solution set to 0 ≤ (x − 4) (x − 1) is: x ≤ 1 or x ≥ 4

Reﬂect and check Using the test point method, we could create a table to see when the solutions would be greater than zero or less than zero. We can use the inequality (x − 4) (x − 1) ≥ 0 which is the same as the one we graphed above after applying the symmetric property. x≤1 1≤x≤4 x≥4

Test point x=0 x=2 x=5

(x − 4) (x − 1) (0 − 4) (0 − 1) = 4 (2 − 4) (2 + 1) = − 6 (5 − 4) (5 + 1) = 6

greater than or equal to 0? Yes No Yes

Since the inequality says the values need to be greater than zero, we look for where the values would be positive. We can see from the table this occurs when x ≤ 1 or when x ≥ 4.

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Example 3 Write a corresponding quadratic inequality for the given solution set on the number line. −5

−4

−3

−2

−1

Create a strategy The end points of the solution set are the solutions to the related equation, so we can use them to write the related equation in factored form. Since the end points are included in the solution set, we know that the inequality symbol will be either ≤ or ≥. We can then substitute a value from the solution set into the equation and replace the equal sign with the inequality symbol that would make the inequality true.

Apply the idea Since the end points of the solution set are −3 and 1, the related equation will be: (x + 3) (x − 1) = 0 In the equation we created, the parabola is facing upward. We can imagine an upward facing parabola with x-intercepts at (−3, 0) and (1, 0). Between those points, the graph would be below the x-axis. So, the shaded region on the number line, with the end points included, represents when y ≤ 0. Therefore, a corresponding quadratic inequality for the solution set is: (x + 3) (x − 1) ≤ 0

Reﬂect and check Since the inequality has 0 on the right-hand side, we could multiply the quadratic expression by any positive scale factor and still get the same result. However, multiplying by a negative scale factor would require us to reverse the direction of the inequality symbol in order to make the solution set true. So if we had created a downward facing parabola, the inequality would have been −(x + 3) (x − 1) ≥ 0.

Example 4 A company that produces children’s toys makes a total proﬁt, P in hundreds of dollars, given by the function P (x) = − 5x2 + 80x − 315, where x is the number of toys produced in hundreds. a Write an inequality whose solution represents when the company will make a proﬁt.

Create a strategy In order for the company to make a proﬁt, the proﬁt function must be positive. That is, P (x) > 0. We can create the inequality using the given quadratic function.

Apply the idea 2

−5x + 80x − 315 > 0

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Reﬂect and check Notice that we used > instead of ≥ because the company has not made a proﬁt if they have $0.

b Use the table to determine when the company would make a proﬁt. x P (x)

5 −40

6 −15

7 0

8 5

9 0

10 −15

11 −40

Create a strategy We need to look for the x-values that make P (x) > 0. Keep in mind that the values are in hundreds.

Apply the idea 7<x<9 The company will make a proﬁt if they make more than 700 toys but less than 900 toys.

Reﬂect and check If the company makes less than 700 toys or more than 900 toys, they will lose money. This is likely due to the cost of the materials for the toys and the cost of producing the toys. We can see that the company makes a maximum proﬁt of $500 when they make 800 toys, so this should be the number of toys that they aim to produce.

Idea summary The solutions to a quadratic inequality are any values that make the inequality true. When using a graph to solve a quadratic inequality with a number on one side, we look for where the y-values are equal to that number. If the inequality symbol is < or ≤, the solution is where the graph is below that y-value. If the inequality is > or ≥, the solution is where the graph is above that y-value.

Practice What do you remember? 1

Determine if each statement is true or false. a

x2 > 9 is a quadratic inequality.

2x − 5 ≥ 16 is a quadratic inequality.

Quadratic inequalities always have a single value as the solution.

Inequalities generally have solution sets which are a range of values.

If we subsitute a value into an inequality and the inequality is satisﬁed, then that value is in the solution set.

For each inequality, determine whether or not each x-value is part of the solution set. i

x=3

x=0

iii

x = −2

x2 > 1

x2 ≥ 3x

x2 − x − 2 ≤ 0

2x2 + 5x − 8 < 0

For each solution set provided, plot the solution on the number line. Be sure to clearly show if an endpoint is ﬁlled or unﬁlled. a

x ≤ −2

{x : x ≤ −3 and x ≥ 4}

{x : x ≥ 3}

[−4, 2)

(−∞, −3] ∪ [5, ∞)

{x : −5 < x < −1}

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Fill in the blanks. a

5x − 3 ≤ 12

Given inequality

5x ≤ ⬚

Add ⬚ to both sides of the inequality

x⬚3

Divide both sides of the inequality by ⬚

−3x > 6

Given inequality

x⬚−2

Divide both sides of the inequality by ⬚

−2x2 + 3x − 9 ≥ −5

Given inequality

2x2 − 3x + 9 ⬚ 5

⬚ both sides of the inequality by −1

2x − 3x + 4 ⬚ 0

Subtract ⬚ from both sides of the inequality

6x2 − x − 12 = 0

Given equation

(2x − ⬚) (3x + ⬚) = 0

(2x − 3)

Rewrite in factored form

0 and (3x + 4) = 0 ⬚ and x = ⬚

Use the ⬚ property

Solve using inverse operations

Consider the graph of the function:

y 6

y = 6 − x − x2

Determine the solution set for the inequality.

6−x−x >0

3 2 1

−4 −3 −2 −1 −1

−2

The graph of y = x2 − 2x − 8 is given.

Determine the solution set for each inequality.

150

x − 2x − 8 > 0

x − 2x − 8 > −8

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

x − 2x − 8 < −5

x − 2x − 8 ≤ −9

x − 2x ≤ 8

Determine the solution set for each inequality: a

x2 ≥ 4

x ≤ 25

x2 < 9

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x <4

x2 > 16

x2 ≥ 1

x2 > 9

x ≤1

y x 1 2 3 4 5

Piper wants to solve the quadratic inequality (x − 3) (x − 10) > 0. She has found that x = 3 and x = 10 are the solutions to (x − 3) (x − 10) = 0. a

Determine the possible intervals for the solution set.

Piper chooses one x-value in each interval to test if it makes the inequality true. She does this by evaluating (x − 3) (x − 10) for these values and the end points of the intervals. Copy and complete the table. x

(x − 3) (x − 10)

Determine the solution set for the quadratic inequality.

Sketch the solution set for the quadratic inequality on a number line.

Adélie Penguins can hold their breath for up to six minutes. When they are swimming, they porpoise and jump out of the water to breathe. Height (ft) 2.5 2

(1, 1.5)

1.5 1 0.5 0

(1.75,0) 0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

Time (s) 2

The given diagram shows the height in feet after x seconds from when the penguin left the water. Using the diagram, write an inequality that represents the times when the penguin is more than 1 foot above the water. Do not solve it.

Let’s practice 10

For each quadratic inequality: i

Find the solution set.

Sketch the solution set on a number line.

(x − 1) (x − 4) < 0

(x + 2) (1 − x) ≥ 0

(x + 3) (x − 2) > 0

x(x − 3) ≤ 0

For each quadratic inequality: i

Determine the related equation.

Find the solutions to the related equation.

iii

Sketch the solution set for the inequality on a number line, if possible.

x2 − 6x + 8 < 0

x2 + x ≥ 0

x2 + x > 6

x2 ≤ 5x − 4

2x2 − 8 > 0

−3x2 + 12 ≤ 0

−4x2 − 16x − 16 > −8x

x2 + 3 < 1

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Giuseppe solves the inequality (2x + 1) (x + 2) < 14. He shows these three steps for his solution. Explain what strategy Giuseppe is using and what he is doing at each step. Step 1:

(2x + 1) (x + 2) < 14

Given inequality

2x2 + 5x + 2 < 14

Distribute the parentheses

2x + 5x − 12 < 0

Subtract 12 from both sides

(2x − 3) (x + 4) < 0

Rewrite in factored form

(2x − 3) (x + 4) = 0

Write the related equation

and x = − 4

Step 2: x

−5

−4

(2x − 3) (x + 4)

−12

2 0

Step 3: Solution set is −6 −5 −4 −3 −2 −1 0 1 2 3 4

Fatimah solves the inequality x2 − 8x − 9 ≥ −16. a

Fatimah shows these three steps for her solution. Explain what strategy Fatimah is using and what she is doing at each step. Step 1 4 2 −1−2 −4 −6 −8 −10 −12 −14 −16 −18 −20 −22 −24

Step 2

4 2

−1 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20 −22 −24

1 2 3 4 5 6 7 8 9

y = x2 − 8x − 9

y x 1 2 3 4 5 6 7 8 9

y = x2 − 8x − 9

Step 3: Solution set is (−∞, 1] ∪ [7, ∞) −2 −1 0 1 2 3 4 5 6 7 8 9 10

Verify Fatimah’s solution. Explain your process.

Given an inequality like x2 − 8x − 9 ≥ −16 or (2x + 1) (x + 2) < 14, which strategy would you use to solve it? Explain your choice.

Adrian wants to solve the quadratic inequality x2 < 5 by taking the square root of both sides. His work is shown below:

x2 < 5

x<±

Square root both sides of the inequality

,x<

Write the solution set as two linear inequalities

Remove x <

Identify where Adrian has made an error and explain what it is.

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since it is included in x < 5

For each quadratic inequality: i

Sketch the corresponding quadratic function on the coordinate plane.

Determine the solution set for the inequality in interval notation.

x2 − 4 < 0

e 17

x2 + 1 > 0

Solve using any strategy.

(x − 2)2 − 4 < 0

(x + 5)2 > 9

−x2 + 12x − 12 ≥ 20

x2 − 1 ≤ 3

(x − 3)2 ≤ 0

(x − 1) + 2 < 0

Describe the strategy you will use.

iii

Verify your solution using technology or otherwise.

(x − 3) (x + 2) ≤ 0

(x − 1) (x + 3) > 5

x2 − 5x + 6 ≤ 0

48x2 − 86x + 14 > 4

Graph the solution set to the inequalities on a number line. −x2 + 5x − 3 < −3

x2 + 2x > −1

Lafayette is starting a bakery that specializes in French macarons. Through market research, he creates a monthly proﬁt function P (x), where x is the price of one dozen macarons and graphs it.

−400 −800 −1200 −1600 −2000 −2400 −2800

(32, 2000) P(x) x 8

(0, −3120)

Write an inequality whose solutions represent when the proﬁt is positive.

Find and interpret when the proﬁt will be positive.

Find and interpret the solution set to each inequality in terms of the context. i

2000 1600 1200 800 400

x(x + 3) ≥ 0

For each inequality:

a 19

(x − 1) (x − 3) ≥ 0

P (x) < 0

P (x) ≥ 720

Abbey is building a rectangular garden with a length of x feet, and a width which is 10 feet longer than the length. a

If Abbey wants her garden to take up no more than 24 square feet, create an inequality representing the possible area of the garden.

Find solution set to the inequality.

Find and interpret a solution set that is reasonable based on the context.

Violet is bouncing a tennis ball in her yard. The height of the ball over time forms a quadratic relationship. Violet estimates that the ball reaches its highest point of 12 feet in the air about 2 seconds after she bounces it. Omar, who lives next door, sees the ball when it is above the fence. a

If the fence between them is 6 feet high, create an inequality representing the times when Omar can see the ball.

Estimate the solution set to the inequality created in the previous part.

Explain whether or not the solution set is reasonable.

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Let’s extend our thinking 22

For each number line, state a possible quadratic inequality that has the solution set shown. a −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9 10

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9 10

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

2 3 4 5 6 7 8 9 10

The solution set for x2 + x − 6 ≤ 0 is shown. −5

−4

−3

−2

−1

For each of the following related inequalities. Draw their solution set on the number line and compare to x2 + x − 6 ≤ 0. a 24

x2 + x − 6 > 0

−x2 − x + 6 ≤ 0

5x2 + 5x − 30 ≤ 0

5x2 + x − 6 ≤ 0

Clifford and Trenton want to ﬁnd an inequality that has a solution set matching the number line: −5

−4

−3

−2

−1

Clifford comes up with the inequality 2 (x + 3) (x − 2) > 0 and Trenton comes up with the inequality −5 (x + 3) (x − 2) < 0. Determine who is correct. Justify your reasoning. 25

Consider the statement: “The solution set for a quadratic inequality is symmetric about the same x-value as its corresponding quadratic equation.” Explain whether you think the statement is true or false.

Without solving algebraically, explain why the inequality x2 + 1 < 0 has no solutions.

Explain why (x − 2) (x + 4) > 0 and ∣x + 1∣ − 3 > 0 have the same solution set. Show this in at least two different ways.

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The line and parabola have no points of intersection, so the system has no solution.

The line and parabola have one point of intersection, so the system has one solution.

The line and parabola have two points of intersection, so the system has two solutions. y

The parabolas have one point of intersection, so the system has one solution.

1 −4 −3 −2 −1 −1

x 1

The parabolas have two points of intersection, so the system has two solutions.

The parabolas do not have any points of intersection, so the system has no solution.

When the two quadratic equations in a system are equivalent, the parabolas will lie on top of one another when graphed. For example, the graphed system is

−2 −3 −4 −5

This means all points on the parabolas will make both equations true. Since there are inﬁnitely many points that lie on the parabolas, there are an inﬁnite number of solutions.

−6

The solution to a system of equations in a given context is viable if the solution makes sense in the context and is non-viable if it does not make sense. 156

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Example 1 Consider the following systems of equations:

a Graph the equations on the same coordinate plane.

Create a strategy The solution(s) to a system of equations can be represented graphically as their point(s) of intersection. We can use technology to graph the two equations, or, if drawing them by hand, it will be useful to ﬁrst ﬁll out a table of values for both equations. We can use what we know about function types to pick the best range of table values. For example, we know the vertex of the quadratic equation will be at x = of x = 1. −3 12

x 2

y = x − 2x – 3 −3 6

x y = −x + 3

−2 5 −2 5

−1 0 −1 4

0 −3 0 3

1 −4 1 2

so we will want to choose x-cordinates on either side

2 −3 2 1

Apply the idea

3 0 3 0

4 5 4 −1

Reﬂect and check 5 4 3 2 1

Notice that from the table of values both functions have the points (−2, 5) and (3, 0). We want these points of intersection to be visible on our graph. We also want the vertex of the parabola, (1, −4), to be visible.

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

We want to ensure that the x-values on our graph cover at least the interval −3 ≤ x ≤ 4 and the y-values on our graph cover at least the interval −5 ≤ y ≤ 6.

b Identify the coordinates of the solution(s) to the system of equations.

Apply the idea The points of intersection occur at (−2, 5) and (3, 0). When creating the table, we saw that these values made both equations true, so these coordinate pairs are the solutions to the system of equations.

Reﬂect and check Alternatively, we could have solved the system of equations algebraically, by equating both equations and solving for x. y = x2 − 2x – 3

First equation

−x + 3 = x − 2x – 3

Substitute y = − x + 3

Add x to both sides

0=x −x–6

Subtract 3 from both sides

0 = (x + 2) (x − 3)

Factor the quadratic

3=x −x–3

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Using the zero product property, we can see the two solutions to this new quadratic are x = − 2 and x = 3. We can now substitute these into one of the given equations to ﬁnd the corresponding y-values. y = −x + 3

Equation 2

= − (−2) + 3

Substitute x = − 2

=2+3

Evaluate the multiplication of the signs

Evaluate the addition

This corresponds with the solution (−2, 5). y = − (3) + 3

Substitute x = 3 into Equation 1

Evaluate the addition

This corresponds with the solution (3, 0).

Example 2 Consider the quadratic-quadratic system of equations:

a Determine the number of solutions to the system of equations.

Create a strategy To determine the number of solutions to the system of equations, we can graph the system on a coordinate plane and look for points of intersection.

Apply the idea 4

From the graph, we can see that the parabolas intersect in two places.

Therefore, there are 2 solutions to the system of equations.

2 1 −2

−1

x 1

−1 −2

b Find the solution(s) to the system of equations.

Create a strategy From the graph, we see that one of the solutions is (1, 2) but the other solution does not have integer coordinates. For this reason, we must use technology to estimate the coordinates of the other point or solve algebraically. When solving algebraically, using the substitution method is usually easiest, but the elimination method is also valid.

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Apply the idea Let’s begin by numbering the equations to make them easier to work with. 1

y = − 2x2 + 4

y = x2 − 4x + 5

Since both equations already have y isolated, we can start by substituting equation 1 into equation 2 to eliminate y from the equation. y = x2 − 4x + 5 2

Equation 2

−2x + 4 = x − 4x + 5

Substitute y = − 2x2 + 4

4 = 3x2 − 4x + 5

Add 2x2 to both sides

0 = 3x2 − 4x + 1

Subtract 4 from both sides

0 = (3x − 1) (x − 1)

Factor the quadratic

Now, we can solve for x using the zero product property. Set the ﬁrst factor equal to 0

Add 1 to both sides

Divide both sides by 3

Set the second factor equal to 0

Add 1 to both sides

Finally, we will substitute the x-values back into one of the equations to solve for y. Equation 1

Substitute x =

Evaluate the multiplication

Rewrite with a common denominator

Evaluate the addition Substitute x = 1

Evaluate the multiplication

Evaluate the addition

The solutions to the system are

and (1, 2).

Reﬂect and check We can verify these solutions with technology or by substituting the values back into the equations to see if the solutions make both equations true. Since we solved the system algebraically, we will check the answer by graphing with technology and using the tracing tool.

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When the coordinates of the point are not integers or are rational numbers with more than 3 decimal place values, the calculator will give an approximation (a rounded estimate) of the values.

y 4

(0.333, 3.778)

Note that

(1, 2)

2 1 −2

−1

which shows that our solutions are correct.

−2

Example 3 Find the solution(s) for the following linear-quadratic system of equations:

Create a strategy We can approach this graphically or algebraically. y 25 20 15 10 5 −4 −3 −2 −1 −5

x 1 2 3 4 5 6 7 8 9

But we can see from the graph that the points of intersection are not clearly identiﬁable. In cases like this, an algebraic approach is preferable. As both equations are already in terms of y, we can use the substitution method to solve.

Apply the idea y = x2 − 5x

Substitute y = 3x + 1

1 = x − 8x

Subtract 3x from both sides

Subtract 1 from both sides

3x + 1 = x − 5x 0 = x − 8x – 1

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Second equation

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This equation is not easily factorable, so we will use the quadratic formula to ﬁnd the solutions.

Quadratic formula

Substitute a = 1, b = − 8, c = − 1

Evaluate the square and products

Evaluate the sum in the radicand

Simplify the radical

Rewrite as two fractions

Simplify the quotients

We have found the x-coordinates of the points of intersection. We can substitute these into either equation to ﬁnd the corresponding y-coordinate.

So one solution is

, and using the same method, we ﬁnd the other solution is

Example 4 A base jumper jumps from the bridge of the Petronas towers, 560 ft high, immediately deploys his parachute, and then descends at a constant rate of 25 ft/s. At the same time, a ball is thrown from the observation deck of the tower, 1214 ft feet high. It follows a path in the form a(x − h)2 + k where a is the force of gravity which is −16 ft/s2 and reaches its maximum of 1250 ft. after 1.5 seconds. a Write a system of equations to model this problem.

Create a strategy Notice that the base jumper’s trajectory follows a linear path in the form y = mx + b. The ball’s trajectory follows a quadratic path in the form y = a(x − h)2 + k.

Apply the idea For the base jumper, their initial height is 560 ft. and they are falling at a rate of −25 ft/s. Using a linear model, the height, y, after x seconds, is given by: y = −25x + 560 . We know the ball follows a path in the form a(x − h)2 + k where a is the force of gravity which is −16 ft/s2. Since it reaches its maximum of 1250 ft. after 1.5 seconds, we know h = 1.5 and k = 1250. Therefore, the ball’s height can be modeled by the function: y = −16(x − 1.5)2 + 1250

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b Graph the height of the base jumper and ball on the same coordinate plane.

Create a strategy We can use technology to obtain a graph of the two functions, using the context and graphs to identify an appropriate domain and range for the graph. Alternatively, we can graph by hand using key features of the graph such as: • The equation for the base jumper’s height is a linear function, so we can identify the y-intercept, ﬁnd another point on the line using the slope, and then graph the line through those two points. • The equation for the ball is in vertex form, so we can see the direction of opening from the coeﬃcient, plot the vertex, then plot the y-intercept of (0, 1214), and substitute another value for x to get the shape.

Apply the idea Height over time y (feet) 1200 1000 800 600 400 200

x (seconds) 5

Reﬂect and check Consider what domain is appropriate for each graph in the context. The graphs should not extend beyond their valid domain. c Determine the time the base jumper and ball are at the same height.

Create a strategy Let’s use technology to solve this problem. We can start by inputting the two equations for the base jumper and ball into a graphing calculator. We already have a sketch, so know we should end up with a parabola and a line.

We can estimate the coordinates by eye, but we should determine the coordinates more precisely using an intersection tool.

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Apply the idea From the graph, we can see that the graphs intersect at a single point in the domain appropriate to the context. To ﬁnd the coordinates of this point, we can use the intersection tool. For the built-in GeoGebra graphing calculator, we need to click on the point of intersection.

The point where the graphs intersect within the restricted domain of the context is: (9.07, 333.26). So, the ball and base jumper will be at the same height of approximately 333.26 ft at 9.07 seconds after the throw/jump.

Reﬂect and check We can conﬁrm this algebraically as well: y = − 25x + 560 2

First equation

−16 (x − 1.5) + 1250 = − 25x + 560

Substitute y = − 16 (x − 1.5)2 + 1250

−16x2 + 48x + 1214 = − 25x + 560

Square the binomial and combine like terms

−16x + 73x + 654 = 0

Add 25x and subtract 560 from both sides

The solutions of this quadratic equation would give us the x-coordinates for any points of intersection between the two graphs. The discriminant of this equation is (73)2 − 4 ⋅ (−16) ⋅ 654 = 47 185 > 0 which means there are two real solutions. We could then use the quadratic formula to get the x-values of the points of intersection, ignoring negative solutions for this context.

Idea summary A linear-quadratic or quadratic-quadratic system can be solved using the graphing method or substitution method. Using the graph is best when the intersection point(s) are clearly visible. Otherwise, solving the system algebraically or using an intersection tool with technology will lead to a more precise solution.

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Practice What do you remember? 1

Consider the equation 2x2 − 2x + 2 = 2x + 3. a

Rewrite the equation into a system of equations.

Find the solution(s) to the system of equations.

Are the following statements true or false? a

The coordinates of the point of intersection of two functions, (x, y), satisfy the equations of both functions.

There is always a point of intersection between two functions.

The point of intersection might only satisfy one of the equations.

A parabola and a line can have two points of intersection with the same x-values.

A system of a quadratic and linear function can have one, two, or no solutions.

If we solve a system algebraically, we can verify the solution graphically.

Irrational solutions cannot be identiﬁed graphically without using technology.

Identify the number of points of intersection for each pair of functions: a

y 8 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2

8 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2

x 1 2 3 4 5

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8 7 6 5 4 3 2 1

1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2

1 2 3 4 5

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2

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8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

1 2 3 4 5

x 1 2 3 4 5

Sketch a system of equations that matches the criteria. a

A linear function and a quadratic function with 2 points of intersection.

A linear function with non-zero slope and a quadratic function with 1 point of intersection.

Two quadratic functions with different directions of opening and 2 points of intersection.

Two quadratic functions with the same direction of opening and 1 point of intersection.

Consider the graphs of y = x2 and y = 2x: a

Identify the number of points of intersection that there are between the two graphs.

One point of intersection occurs at the point (0, 0). Write the coordinates of the other point of intersection.

8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

x 1 2 3 4 5

Identify the coordinates of the solutions, if any, for each of the following systems of equations:

y 7 6 5 4 3 2 1

−1

−1 −2 −3

y 3 2 1

−5 −4 −3 −2 −1 −1 2

−2

x 1

−3

−4 −5

y 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1−1

−2 −3 −4

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y 6

8 7

1 −4 −3 −2 −1 −1

x 1

−2

−8 −7 −6 −5 −4 −3 −2 −1

x 1

Consider the quadratic function y = x2 − 3 and the linear function x + y = 3. a

Graph the functions on the same coordinate plane.

State the number of solutions there to the equation x2 − 3 = − x + 3.

Identify the coordinates of the solutions of y = x2 − 3 and x + y = 3.

Consider the quadratic functions y = 2 (x − 3)2 and the linear function y = − 2(x − 1)2 + 8. a

Graph the functions on the same coordinate plane.

State the number of solutions there to the equation 2 (x − 3)2 = − 2(x − 1)2 + 8.

Identify the coordinates of the solutions of y = x2 − 3 and x + y = 3.

Consider the following systems of equations: i

Graph the equations on the same coordinate plane.

Identify the coordinates of the solutions to the system of equations. b

One point of intersection of the equations y = x2 + 5x − 3 and y = 4x + 3 occurs at x = 2. Determine the coordinates of that point of intersection.

A bike and a car both start at the top of a hill at the same time. A car is traveling at a constant speed of 45 ft/s. The bike starts at 9 ft/s, but begins to accelerate down the hill according to the displacement formula , where v0 is initial velocity and a = 28 ft/s2, when other forces are accounted for. Write a system of equations which could be used to ﬁnd when the bike and car are the same distance down the hill. Do not solve.

Let’s practice 12

Determine whether or not each point is a solution to the each system of equations. i a

166

(0, 3)

ii b

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(3, 0)

iii c

(−3, 6)

iv d

(1, 6)

For each system of equations: i

Solve each system algebraically. Show your work.

Graph the equations on the same coordinate plane to verify the solution(s).

a 14

For each system of equations: i

Solve each system algebraically. Show your work.

Graph the equations on the same coordinate plane to verify the solution(s).

a 15

Find the solution(s) for each of the following linear-quadratic systems of equations. a

Find the solution(s) for each of the following quadratic-quadratic systems of equations. a

Consider the parabola y = x2 − 5. Fiona discovered that the line with equation y = parabola at exactly one point.

intersects the

Determine the equation of another straight line that has exactly one point of intersection with the parabola, and that has the same y-intercept as y = 18

Francesco attempted to solve the system of equations given by:

He mistakenly concluded that the points of intersection are (2, 4) and (−1, 1). Identify and correct the errors in his solution. Find the y-values:

Find the x-values: 1 2 3 4

x+2=x −4 0 = x2 + x − 2 0 = (x + 2) (x − 1) x = 2 and x = − 1

For x = 2, we get y = 2 + 2 = 4 For x = − 1, we get y = (−1) + 2 = 1 State the solutions. Solutions are: (2, 4) and (−1, 1)

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Two projectiles are launched simultaneously from different heights. Their paths follow parabolic trajectories, and their height h in meters after t seconds is modeled by quadratic equations. • Projectile A is launched from a height of 20 meters, reaches its maximum height of 80 meters after 3 seconds, and lands after 6 seconds. • Projectile B is launched from a height of 10 meters, reaches its maximum height of 50 meters after 2 seconds, and lands after 5 seconds. Which of the following pairs of quadratic equations correctly models the heights of Projectile A and Projectile B as a function of time t? A

hA(t) = −10(t − 3)2 + 80

hA(t) = −12(t − 3)2 + 80

hA(t) = −5(t − 3)2 + 80

hB(t) = −8(t − 2)2 + 50 hB(t) = −6(t − 2)2 + 50 C

hA(t) = −8(t − 3)2 + 80

hB(t) = −7(t − 2) + 50 hB(t) = −5(t − 2)2 + 50 20

The plans for a proposed tunnel through a mountain side are shown. The height of the mountain above sea level is given by the equation y = − y=

+4, and the height of the tunnel above sea level is given by the equation

+ 2, where y represents the height above sea level in miles. y

Entrance to Tunnel

nel

ed Tun

Propos

Sea Level

Solve the system

Interpret the solutions to the system.

The ﬁnancial team at The Gamgee Cooperative wants to calculate the proﬁt, P (x), generated by producing x units of wetsuits. The revenue produced by the product is given by the equation is R (x) =

+ 40x.

The cost of production is $410 plus $5 per unit produced.

Find an expression for P(x).

Determine how many wetsuits the company must sell for their revenue to be greater than the cost of production. Explain how you got your answer.

Justify the reasonableness of the answers in part (a).

In business, supply and demand functions are ways of expressing the supply or demand of an item as a function of its unit price. Market equilibrium is the term that describes when the supply is equal to the demand. Consider the supply and demand functions for a new book bag, where x is unit price in dollars and y is in thousands of units supplied or demanded. Supply function: y = 2x2 Demand function: y = − 4x + 3x2 We want to ﬁnd the equilibrium quantity and the corresponding price by solving the system of equations.

168

Determine the values of x at market equilibrium.

Interpret the meaning of the solutions to the system in this context.

Verify the solution and determine its reasonableness.

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Amina is animating the background for scene in the mountatins. She has drawn one mountain and its proﬁle is shown on the graph where (x) represents the horizontal distance from the left edge of the screen and (h) indicates the height above the bottom of the screen, both in hundreds of pixels. h(hundreds of pixels) 50 40 30 20 10

x(hundreds of pixels) 10

100

110

Create an equation to ﬁnd where the hill is 2500 pixels from the bottom of the screen. Do not solve it.

A new mountain will be drawn that can be represented by the equation: h = − 0.1(x − 70)2 + 30

Determine the horizontal distance from the left side where the two mountains have the same height. Explain your answer. 24

Two amusement park rides, “Sky Coaster” and “Mountain Flyer,” follow parabolic paths. The “Sky Coaster” starts at a height of 50 feet, reaches its maximum height of 150 feet after 5 seconds, and returns to 50 feet after 10 seconds. The “Mountain Flyer” starts at a height of 40 feet, reaches a maximum height of 120 feet after 4 seconds, and returns to 40 feet after 8 seconds. a

Create a system of two quadratic equations that represents the height of each ride as a function of time.

At what time will both coasters be at the same height?

Let’s extend our thinking 25

The parabola y = x2 + 1 has been graphed on the coordinate plane. 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2

x 1 2 3 4 5

Determine the values of k such that the line y = k has at least one point of intersection with the parabola. 26

One point of intersection of the curves y = x2 − 3x + 15 and y = kx − 1 is at x = 4. Find k.

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The parabolas y = c − ax2 and y = x2 + 2x + c intersect at the point (1, 0). a

Find the value of c.

Find the value of a.

Graph the parabolas on the same coordinate plane.

Consider the system of equations.

Determine the value of c that will result in one unique solution to the system of equations.

Determine the values of c that will result in two real solutions to the system of equations.

Reuben used technology to graph the system of equations y = 0.5x2 + x + 8 y = − 0.5x + 7 His calculator displayed the graphs as shown. y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Reuben thinks the solution is (−2, 8), but his classmate said there should be two solutions. How can Reuben ﬁnd the second solution?

Sketch a graph of the equations that clearly shows both solutions. Explain how you chose the values for your axes.

Identify the second solution to the system of equations.

An aerial camera is hovering around a baseball game ﬁlming at 30 feet above the ground. In celebration a player throws the ball straight up in the air. The height can be modeled using the equation h = h0 + v0t − 16t2 where h0 is the initial height in feet, v0 is the initial velocity, and t is time in seconds. a

The ball was thrown from 6 feet above the ground and leaves their hand at 40 ft/s.

Construct a system of equations to model the heights of the aerial camera and the baseball. Use the model to determine if there is any risk of the ball hitting the aerial camera. Justify your conclusion. b

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Use the model created in part (a) to construct another model that represents the situation. Determine how the second model would help determine the risk and compare the models.

The range of the signal from a radio station has boundaries within a circular radius and is modeled by the equation x2 + y2 = 4000. A car driving along a highway through the broadcast range enters at (−60, 20) and exits at (36, 52). a

Construct a system of equations to model the car’s position in the broadcast range.

Revise the model in part (a) for a car that will never be in the radio station’s broadcast range and explain your choice.

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For each of the following statements, state whether it is always, sometimes, or never true. Justify your answer. a

A vertical line will have exactly one point of intersection with a quadratic function.

A quadratic function and a linear function have exactly one point of intersection.

Two quadratic functions have four points of intersection.

Two quadratic functions which intersect twice have leading coeﬃcients with opposite signs.

Write a system of equations with a linear and quadratic function that have: a

No points of intersection

Exactly one point of intersection

Two points of intersection with the same y-coordinate.

Two points of intersection with different y-coordinates.

Prove that two different quadratic functions with the same leading coeﬃcient have a point of intersection only if the coeﬃcient of the linear terms are not equal.

Using substitution or otherwise, solve the system:

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Polynomial Expressions, 3 Equations, & Functions Big ideas • The properties of real numbers can be applied to many types of expressions. • Changing the form of an expression or equation can reveal information that was previously unknown. • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models.

Chapter outline 3.01 3.02 3.03 3.04 3.05 3.06

Operations with polynomials Polynomial identities Divide polynomials Characteristics of polynomial functions Zeros and factors Solve polynomial equations

174 184 193 201 212 226

3.01 Operations with polynomials After this lesson, you will be able to... • add, subtract, and multiply polynomial expressions in one and two variables. • represent polynomial expressions in different forms and show their equivalence (including with algebra tiles and area models).

Operations with polynomials In Algebra 1, we learned the following deﬁnitions related to polynomials: Polynomial

Degree (of a polynomial)

The sum or difference of terms which have variables raised to non- negative integer powers and coeﬃcients that are constant

The largest exponent or the largest sum of exponents of a term within a polynomial

Leading coefficient The coefficient of the first term of a polynomial written in descending order of exponents The term which has a fixed value and no variables is called the constant term. The term with highest exponent on the variable is called the leading term, and the exponent of the leading term is the degree of the polynomial.

leading term

The standard form of a polynomial is anxn + an − 1 xn − 1 + … + a1 x + a0, where n is a non-negative integer and each ai is a coeﬃcient.

leading coefficient degree

quadratic term

coefficient n

n−1

p(x) = an x + an − 1 x

constant term 2

+ ... + a2 x + a1 x + a0

term

linear term

Polynomials can also have names speciﬁc to the number of terms they have. A monomial is a polynomial with one term. A binomial is a polynomial with two terms. A trinomial is a polynomial with three terms. They can also be named based on their degree. A degree 0 polynomial is constant. A degree 1 polynomial is linear. A degree 2 polynomial is quadratic. 3 is cubic, 4 is quartic and so on.

Exploration In the deﬁnition of polynomials, the coeﬃcients are multiplied to the variables, the variables are raised to nonnegative integer powers, and the terms are added and subtracted together. This allows the function to be one, smooth curve with no breaks, holes, sharp turns, or stopping points. Use technology to graph the following functions: • y = x2 + 2x − 1 5

• 3

• y = − 3x − 4x + 2x + x − 0.2x

• y = ∣x∣

• y = 4x +

174

Determine if the function is a polynomial and explain your answer.

For the functions that are not polynomials, rewrite the function to show that a variable is not raised to a non-negative integer or that the terms are not added and subtracted together.

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One way to determine if an expression or equation is a polynomial is to examine its graph. The graph of a polynomial is a function with one smooth curve over a continuous domain. A polynomial will not have a negative exponent on a variable, a rational exponent on a variable, or have a variable in absolute value bars. Polynomials are closed under addition, subtraction, and multiplication. This means that the sum, product, or difference of polynomials will also be a polynomial. When adding and subtracting polynomials, we use the method of combining like terms. Addition: if we assume m < n, General example Numerical example

(anxn + … + a0) + (bmxm + … + b0) = anxn + … + (am + bm) xm + … + (a0 + b0) (2.7x5 − 1.8x3 + 0.9x − 2) + (3.8x4 + 2x3 − x + 5.1) = 2.7x5 + 3.8x4 + 0.2x3 − 0.1x + 3.1

By deﬁnition, m and n will be non-negative integers, and the coeﬃcients will remain constant. Therefore, the result is another polynomial. Subtraction will work the same way as addition. Assuming m < n, General example

(anxn + … + a0) − (bmxm + … + b0) = anxn + … + (am − bm) xm + … + (a0 − b0)

Numerical example The coeﬃcients will remain constant, and the exponents will be non-negative integers. The result is another polynomial. To multiply polynomials, we apply the distributive property which will require us to use the product of powers law of exponents when multiplying variables. Product of powers law When multiplying two exponential expressions with the same base, add the exponents. Example: am ⋅ an = am + n a 3 ⋅ a 5 = a3 + 5 = a8 General example Numerical example

(anxn + … + a0) (bmxm + … + b0) = (anbm) xn + m + … + (anb0) xn + … + (a0bm) xm + … + (a0 b0) (5x2 + 2x + 1) (x2 − 3x + 5) = 5x4 − 13x3 + 20x2 + 7x + 5

Because n and m were non-negative integers, n + m will also be non-negative. The exponents will still be constants, so the result is another polynomial.

Example 1 Determine whether each of the following can be classiﬁed as a polynomial. a y = 5x2y +

− 2y

Create a strategy To be a polynomial, all coeﬃcients must be constant, and all exponents must be non-negative integers.

Apply the idea

Reﬂect and check

The coeﬃcients 5, , and −2 are all real number constants. The exponents on the variables are all non-

There are 3 terms in this polynomial, so it is a trinomial. The leading coeﬃcient is 5, and the degree of the polynomial is 3.

negative. Therefore, this is a polynomial.

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175

Create a strategy We can use the negative exponent property to rewrite this expression.

Apply the idea

Reﬂect and check Negative exponent property

Because there is a variable with a negative exponent, this is not a polynomial.

We will eventually learn that this is a rational expression. Rational functions have asymptotes that separate the graph into multiple pieces. Therefore, it is not continuous. In general, whenever there is a variable in the denominator, it is not a polynomial expression or equation.

Create a strategy Before looking at the coeﬃcients and exponents, we need to make sure the expression is fully simpliﬁed. For this equation, the constant can be rewritten as a fraction.

Apply the idea Given function The coeﬃcients ,

, 0.5, and

Negative exponent property

are all real number constants. The exponents on the variables are all non-negative,

so this is a polynomial.

Example 2 Fully simplify each polynomial expression. a (−5x3 + 7x2 − 4) + (3x3 − 9x + 2)

Create a strategy To simplify the given expression, we’ll start by rearranging the terms in descending order of their exponents, which makes it easier to identify like terms (terms that have the same variable raised to the same power). Then, we’ll combine the coeﬃcients of like terms.

Apply the idea The expression (−5x3 + 7x2 − 4) + (3x3 − 9x + 2) is equivalent to the expression (−5x3 + 7x2 − 4) + 1 (3x3 − 9x + 2). We can distribute the positive 1 to the second expression, and the terms will remain the same. This means we can remove all parentheses. (−5x3 + 7x2 − 4) + (3x3 − 9x + 2) = − 5x3 + 7x2 − 4 + 3x3 − 9x + 2 3

= − 5x + 3x + 7x − 9x − 4 + 2 = − 2x + 7x − 9x – 2 3

Therefore, the simpliﬁed expression is −2x + 7x − 9x − 2.

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Distribute +1 Reorder the terms Combine like terms

Reflect and check Since the highest power of x in any term is 3, we can conclude that our simplified expression is a third-degree polynomial. The expression has been arranged in descending order by the power of x, starting from x3 and proceeding to the constant term, indicating that it is in the standard form for polynomials. Expressing polynomials in standard not only organizes the terms efficiently but also ensures that all like terms have been appropriately combined.

b (−7x3 + 5.5x2 − 2.1x) − (4.3x3 − 1.7x2 + 3x − 0.5)

Create a strategy We can simplify this expression by combining like terms. To do so, we must be careful to apply the subtraction to each term in the second polynomial. This is the same as multiplying each term in the second expression by −1. Also, remember that like terms have the same variables with the same exponents.

Apply the idea First, we will distribute the subtraction to the second polynomial.

(−7x3 + 5.5x2 − 2.1x) − (4.3x3 − 1.7x2 + 3x − 0.5) = − 7x3 + 5.5x2 − 2.1x − 4.3x3 + 1.7x2 − 3x + 0.5

Next, we will group like terms together. = − 7x3 − 4.3x3 + 5.5x2 + 1.7x2 − 2.1x − 3x + 0.5 Finally, we can combine like terms. = − 11.3x3 + 7.2x2 − 5.1x + 0.5

Reflect and check The different forms of the expressions highlight different things. The original form clearly identifies the two polynomials in the expression. However, the simplified form helps us easily identify the leading coefficient, degree, and constant term of the entire polynomial expression.

c (−2a2 + 5b − 3) (3a − 4b)

Create a strategy To simplify this expression, we will apply the distributive property, which involves multiplying each term in the ﬁrst polynomial by each term in the second polynomial.

Apply the idea (−2a2 + 5b − 3) (3a − 4b) = − 2a2 (3a) − 2a2 (−4b) + 5b (3a) + 5b (−4b) − 3 (3a) − 3 (−4b)

= − 6a3 + 8a2b + 15ab − 20b2 − 9a + 12b

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Reﬂect and check An area model visually represents the distributive property used in multiplication. Here, we used the product of powers law of exponents. This law says that when we multiply monomials, we add their exponents. −2a2

+5b

−3

−6a3

15ab

−9a

−4b

8a2b

−20b2

+12b

The area of each rectangle in the model represents the product of the side lengths, akin to multiplying each term of the first polynomial by each term of the second polynomial. This model confirms our simplified expression.

d (3y2 + 2x − 4) + (4x2 − x + 3y) − (2y − 5x + 3)

Create a strategy To simplify, we ﬁrst distribute the subtraction to the last polynomial, then we can reorder and combine like terms.

Apply the idea (3y2 + 2x − 4) + (4x2 − x + 3y) − (2y − 5x + 3) 2

= 3y + 2x − 4 + 4x − x + 3y − 2y + 5x − 3 2

= 3y + 4x + 2x − x + 5x + 3y − 2y − 4 − 3 = 3y + 4x + 6x + y − 7 2

Start with the given expression Distribute the subtraction Group like terms Combine like terms

The simpliﬁed expression is 3y + 4x + 6x + y − 7.

Reﬂect and check It is important that we accurately distribute signs and combine like terms. To organize our work, we can underline like terms with the same color and underline the other terms with different colors.

Distribute the minus sign Reorder like terms Simplify and combine like terms

This helps us visually represent the polynomial expression and conﬁrm that we have simpliﬁed correctly.

e 4y (2x2 − x + 5) + 6x (3y2 − 2y + 4) − 5y (4x − y)

Create a strategy To simplify this expression, we will distribute each monomial across its respective polynomial, carefully combining like terms afterward.

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Apply the idea In the given expression, 4y (2x2 − x + 5) + 6x (3y2 − 2y + 4) − 5y (4x − y), notice that 4y, 6x, and −5y are coeﬃcients that need to be multiplied to the expressions in the parentheses adjacent to them. We can multiply these expressions individually: • 4y (2x2 − x + 5) = 8x2 y − 4xy + 20y • 6x (3y2 − 2y + 4) = 18xy2 − 12xy + 24x • −5y (4x − y) = − 20xy + 5y2 Now, we can combine the results into a single polynomial and continue to simplify the expression. 8x2 y − 4xy + 20y + 18xy2 − 12xy + 24x − 20xy + 5y2 2

= 8x y + 18xy + 5y − 4xy − 12xy − 20xy + 24x + 20y

Rewrite as a single polynomial Reorder the terms

Combine like terms = 8x y + 18xy + 5y − 36xy + 24x + 20y The simpliﬁed expression is 8x2 y + 18xy2 + 5y2 − 36xy + 24x + 20y.

Example 3 Form a fully simpliﬁed polynomial expression for the perimeter of the rectangle shown.

Create a strategy The perimeter of a shape is the sum of its side lengths. In this case, the shape is a rectangle, so we can add the two labeled side lengths and then double the result.

Apply the idea Formula for perimeter

Substitute expressions for the length and width

Rewrite the fractions as decimals

Combine like terms

Distributive property

Reflect and check There are benefits to having the rectangle’s perimeter in both its original and simplified forms. Although these forms mean the same thing, using the simplified form makes it easier to substitute values for x and y. On the other hand, the original form helps us easily see the dimensions of the rectangle.

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Example 4 A rectangular swimming pool is 16 yds long and 6 yds wide. It is surrounded by a pebble path of uniform width x yds.

x yd x yd 16 yd

6 yd

x yd

Find an expression for the area of the path in terms of x. Fully simplify your answer.

Create a strategy The area of the path, Apath, will be the area of the larger rectangle minus the area of the pool. The area of pool, Apool, is represented by the inner green rectangle which has a length of 16 yds and a width of 6 yds. The area of the large rectangle, Alarge rectangle, is a combination of the area of the path and the area of the pool. We can see that the large rectangle has a length of 16 yds plus an x yds on either side of the pool. This gives us a length of x + 16 + x = (16 + 2x) yds We can apply the same logic to the width of the large rectangle to ﬁnd a width of (6 + 2x) yds.

Apply the idea Apath = Alarge rectangle – Apool = (16 + 2x) (6 + 2x) − (16) (6) 2

= 96 + 32x + 12x + 4x – 96 2

= 4x + 44x 2

Equation for area of the path Substitute expressions for each area Distributive property Combine like terms

Apath = (4x + 44x) yd

Reflect and check There are good things about knowing the rectangular path’s area in both its original unsimplified form, and simplified forms. The original form of the rectangular path’s area, Apath = (16 + 2x) (6 + 2x) − (16) (6), reveals several important details. It shows the dimensions of both the path and the pool, the combined area of the pool and the path, and the area of the pool alone. In contrast, the simplified version is specifically useful for calculating the area of the path alone.

Idea summary A polynomial is the sum or difference of terms which have variables raised to non- negative integer powers and coeﬃcients that are constant. A polynomial written in different forms can make some things easier or harder to see. To determine if two polynomials written in different forms are equivalent, we can use visual aids like area models or algebra tiles.

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Practice What do you remember? 1

Match the following description with its expression a

Trinomial; degree 5

7x4 − 1

Not a polynomial

−x + 6x2 − x3 + 5

Binomial; constant term of −1

iii

Polynomial; leading coeﬃcient of −1

Add or subtract each of the following polynomial expressions. Give your answer as a fully simpliﬁed polynomial. a

(−8x2 − 3) + (−4x2 − 8x)

(3x3 − 9x2 + 8x − 7) + (−7x3 − 9x)

(−6x3 + 4.27x2 + 0.4x) − (3.8x3 + 2x + 1)

(−5x2 + 6x − 4) − (−7x2 + 7x + 9)

e f 3

Simplify the following expressions: a

6u7 (9u7 + 9u6)

2a (5a2 + 2a + 3)

(5m + 8) (m − 3)

(7w + 5)2

Multiply each of the following polynomial expressions. Give your answer as a fully simpliﬁed polynomial. a e

(−0.8x6 + 0.9x4 − 1.1x3 − 1.3) + (x6 + 0.5x5 − 0.3x3 − 0.2x2)

8u(10u + v) − 5 u(u + 2) − 9 (u + 8)

x(x + 7) + 3 (x + 10)

x(x + 6) + 4 (5x + 3)

9y( y + 6) + 8y − 2

(2n + 5) (5n + 2) − 4

6 (−5x + 3) (x + 2) − 2

5 + 8y( y + 6) − 4y

Kennedy wanted to use the area model to multiply (2x − 1) (2x − 3). Her area model is shown. She simpliﬁes her area model as

−3

4x2

−6x

−3

4x2 − 6x + 2x − 3 4x2 − 4x − 3 Explain the error in her area model and ﬁx the mistake. 6

Find a simpliﬁed polynomial that represents the perimeter of the following ﬁgures: a

x+4

4x3 + 2x2 + 5

4x2 + x 2x + 3

8x2 + 9x + 5

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(x + 4) m

Consider the rectangular prism shown with its dimensions labelled. Form a fully simpliﬁed polynomial expression for the volume of the given prism.

(x + 8) m

12 m

Let’s practice 8

Add or subtract each of the following polynomial expressions. Give your answer as a fully simpliﬁed polynomial. a

(2x2 y − 3xy2 + 4y2 − 7) − (3x2 y + 5xy − 8y2)

(8a2 b2 − 7a2 b + 8ab − 11) + (−3a2 b2 + 6a2 b + 5ab)

Simplify the following expressions: a

−3.05ab(4.6a2 − 1.5a − 0.9b2)

c e

(5c6 − 2) (4c5 + 2c3 − 3c)

g 10

9 (0.2y + 0.8) ( y + 2.1)

(5a + 0.5) (−0.5a − 4 − 5a2)

(−3a2 − 2b + 2) (4a + 3b)

(x2 y − 2x + 4y) (3x + 6y)

Find the value of b.

Consider A (x) = 5x2 + 2, B (x) = − 3x + 3 and C (x). If A (x) − B (x) − C (x) = 7x2 − 4x − 1, ﬁnd C (x).

Consider P (x) = ax2 + (b − 5) x − 1 and Q (x) = x2 − 5x + 2. The result of adding P (x) and Q (x) is R (x) = 2x2 + 2x + 1. a

Find the value of a.

A company’s revenue can be modeled by R (x) = 10.2x4 + 3.06x2 + 7.14x − 6.85. If their proﬁt can be modeled by P (x) = 10x4 + 2.7x2 + x + 1.92, ﬁnd their expenses.

Simplify the expression (3x + 5)3.

Alisa has a piece of rectangular cardboard measuring 10 cm by 6 cm. It is to be converted into a box with no lid by cutting out square corners measuring x cm in length and folding up the sides.

Is she correct? Justify your answer by showing whether or not the two are equivalent. b

182

Find the volume of the box as a simpliﬁed polynomial expression.

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x Fold

10 cm

Fold

She wants to ﬁnd the surface area of the cardboard with the corners removed. She thinks that the area of the original piece with the four corners taken away is the same as adding the individual rectangles she will be folding.

Fold

Fold x

x x

6 cm

Taran is building the new reﬂecting pool in the city’s downtown area, and needs to consider the landscaping surrounding the pool. His design is shown, where the shaded region represents a grassy area for citizens to walk and enjoy the reﬂecting pool. a

Determine the area that will be covered by grass, writing your answer as a simpliﬁed polynomial.

A fence will surround both the pool and the surrounding grassy area. Find an expression to represent the total length of the fencing needed.

5x − 4y

Elizabeth wants to know the difference in cross-sectional area of the door of her tent and the tent itself. The proportions of the tent are exactly the same as the door, as seen in the picture. The triangular door has a length and height of w and h respectively. The top of the tent is 9 cm higher than the door, and its ends are 4 cm longer than the ends of the door. a

Express the height of the tent in terms of h.

Express the base length of the tent in terms of w.

Now, ﬁnd a simpliﬁed expression for the difference between the crosssectional area of the tent and the tent door.

x + 3y

x − 2y

Let’s extend our thinking 17

Explain how you can ﬁnd the coeﬃcient of x in the given expression without fully distributing the multiplication. (x2 − 3x − 5) (5x − 4)

Consider f (x) = 2x2 + 5x − 1 and g (x) = − x2 − 8x + 2. a

Determine a simpliﬁed expression for ( f + g) (x), the sum of f (x) and g (x).

Evaluate each of the following: i

f (2)

g (2)

iii

( f + g) (2)

Based on part (b) draw a conjecture about the addition of polynomials.

Create a similar conjecture that for multiplication. Test your conjecture with an example.

Han is trying to determine the dimensions of a rectangular box that will result in the largest volume. The height of the box is 4 in, length of x in, and the perimeter of the base is 12 in. a

Find a fully simpliﬁed expression for the volume of the box.

Determine the domain restrictions on the value of x.

Han believes that the maximum volume is V = 40 in3. Determine whether or not Han is correct, justify your answer.

Jiang’s mom needs to purchase 41 more chickens for their farm. Each chicken costs $39. Explain how you can use the polynomial identity (a + b) (a − b) = a2 − b2 and mental math to ﬁnd the total cost of the new chickens.

Judith makes four claims about polynomials. Determine if each claim is true or false and use evidence to justify your answer. a

The sum, difference, or product of two polynomials is another polynomial.

The sum of two binomials is always a binomial.

The product of two binomials is always a trinomial.

The product of two polynomials with degree greater than 1 always has a higher degree than both of the original polynomials.

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Sum of cubes identity

Difference of cubes identity

Two perfect cube expressions being added to each other

Two perfect cube expressions being subtracted from each other

a3 + b3 = (a + b) (a2 − ab + b2)

a3 − b3 = (a − b) (a2 + ab + b2)

In the proof of an identity, it is our job to prove both sides are equal, so we must work with one side of the equation and show that algebraic manipulation leads to the other side. We cannot manipulate both sides of the equation at once because changing both sides assumes that both sides are already equal.

Example 1 The perfect square trinomial identity is (a + b)2 = a2 + 2ab + b2. a Prove the identity.

Create a strategy We can prove the identity using a geometric diagram.

Apply the idea We will begin by drawing a square with side lengths of a + b. The area of this square is the left side of the equation, (a + b)2.

Next, we can ﬁnd the areas of the squares and rectangles within the larger square.

The area of the larger square is equal to the sum of the areas of the squares and rectangles within it. Therefore, (a + b)2 = a2 + 2ab + b2.

Reﬂect and check Alternatively, we could prove the identity through algebraic manipulation. We are trying to prove (a + b)2 = a2 + 2ab + b2 We can only manipulate one side of the equation, so we will expand the left hand side using the distributive property. (a + b)2 = (a + b) (a + b)

Expand the power

Distributive property

Commutative property of multiplication

= a + ab + ba + b = a + ab + ab + b 2

= a + 2ab + b

Add like terms

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b Use the identity to evaluate 982.

Create a strategy Because 98 is squared, we can represent it with the left side of the equation. (a + b)2 = 982

Apply the idea When choosing values for a and b, we want to choose values that would be easy to calculate with. One option is to use a = 100 and b = − 2. (100 + −2)2 = 982 Now, we can use the right side of the identity to easily calculate the value of 982. (a + b)2 = a2 + 2ab + b2 2

State the identity 2

(100 + −2) = (100) + 2 (100) (−2) + (−2)

Substitute a = 100 and b = − 2

Evaluate the multiplication and powers

Evaluate the addition and subtraction

98 = 10 000 − 400 + 4 98 = 9604

Reﬂect and check You can use your calculator to conﬁrm this answer, but this is a helpful strategy to use when calculator use is not permitted.

Example 2 Use an identity to expand the expression (x − 6) (x2 + 6x + 36). State which identity was used.

Create a strategy When we multiply the ﬁrst terms in both sets of parentheses, we get an x3 term. This tells us that the expression must use one of the cube identities, sum of cubes or difference of cubes.

Apply the idea The given expression (x − 6) (x2 + 6x + 36) represents the difference of cubes identity, as it follows the pattern a3 − b3 = (a − b) (a2 + ab + b2). In this case, a = x and b = 6, so: (x − 6) (x2 + 6x + 36) = x3 − 63 = x3 − 216

Reﬂect and check We can check our result by expanding the expression using the distributive property.

Notice that this method requires more steps, so it is more eﬃcient to expand by using a polynomial identity.

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Example 3 Factor the following expressions and identify which identity was used. a 25r2 − 60rs + 36s2

Create a strategy Look for patterns that match known polynomial identities to factor the expression. Notice that the ﬁrst and last terms, 25r2 and 36s2, are perfect squares. They can be rewritten as (5r)2 and (6s)2. Also, observe that the middle term, −60rs, is twice the product of the square roots of the ﬁrst and last terms. −2 (5r) (6s) = − 60rs This suggests we can factor the expression using the perfect square trinomial identity (a − b)2 = a2 − 2ab + b2.

Apply the idea The expression 25r2 − 60rs + 36s2 can be factored using the identity a2 − 2ab + b2 = (a − b)2 where a = 5r and b = 6s. 25r2 − 60rs + 36s2 = (5r − 6s)2

Reﬂect and check One way to check our result is by substituting a speciﬁc value for r and s into both the original and factored expressions and verifying that they yield the same result. Let’s check for r = 1 and s = 2: 25 (1)2 − 60 (1) (2) + 36 (2)2 = (5 (1) − 6 (2))2 2

25 − 120 + 144 = (5 − 12) 49 = 49



Substitute r = 1 and s = 2 into both expressions Evaluate the multiplication Simplify both sides

b 64m6 − 81n4

Create a strategy This expression has two terms that are subtracted. The only identity that has the same structure is the difference of squares identity. Although it may not be obvious at ﬁrst, both terms are in fact perfect squares, so we can use the difference of squares identity to factor the expression.

Apply the idea

Reﬂect and check

The expression is an example of the difference of squares, following the pattern a2 − b2 = (a + b) (a − b).

Although this expression was the difference of two square terms, not every binomial with a subtraction sign represents the difference of two squares. For example, 50x2 − 9 is not a difference of two perfect squares.

We can rewrite both terms as perfect squares: • 64m6 = (8m3)2 • 81n4 = (9n2)2 3

This shows a = 8m and b = 9n . 64m6 − 81n4 = (8m3)2 − (9n2)2

We have to be careful that we only apply the identity when the terms are in fact perfect squares because the identity cannot be used on non-square terms.

= (8m3 + 9n2) (8m3 − 9n2)

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c 125g3 + 64h3

Create a strategy The given expression is a sum of two terms, both of which are perfect cubes. This means we can use the sum of cubes identity to factor the expression.

Apply the idea The expression is an example of the difference of cubes identity, a3 + b3 = (a + b) (a2 − ab + b2). When can rewrite both terms as perfect cubes: • 125g3 = (5g)3 • 64h3 = (4h)3 This shows a = 5g and b = 4h. 125g3 + 64h3 = (5g + 4h) ((5g)2 − (5g) (4h) + (4h)2) = (5g + 4h) (25g2 − 20gh + 16h2)

Reflect and check The factorization for the sum of cubes and difference of cubes are very similar. The only difference between them is the signs of the terms. a3 + b3 = (a + b) (a2 − ab + b2) a3 − b3 = (a − b) (a2 + ab + b2) To remember the difference between the signs of each identity, we can use the acronym SOAP: • S (Same): The first sign (in the binomial factor) is the SAME as the sign in the original expression. This reflects the “S” in SOAP. • (Opposite): The second sign (first sign in the trinomial factor) is OPPOSITE to the original expression’s sign, aligning with the “O” in SOAP. • AP ( Always Positive): The last sign (second sign in the trinomial factor) is ALWAYS POSITIVE, regardless of the original expression’s sign. This corresponds to the “AP” in SOAP.

Example 4 Use the algebra tiles given to verify that x2 − 6x + 9 = (x − 3)2.

−

− 1

Create a strategy The ﬁrst algebra tile has an area of x2, representing the ﬁrst term in the trinomial. The next group of 6 tiles each have an area of x, but their sign is negative. This represents the second term, −6x. The remaining group of 9 square tiles each have an area of 1, representing the constant term 9. To verify this identity, we can rearrange the algebra tiles to form a square with a side length of x − 3.

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Apply the idea x−3

− − −

+ + + + + + + + +

x−3

This veriﬁes that x2 − 6x + 9 = (x − 3)2.

Reﬂect and check Let’s expand the expression (x − 3)2 using the distributive property to check our answer algebraically. (x − 3)2 = (x − 3) (x − 3) = x (x − 3) − 3 (x − 3) = x2 − 3x − 3x + 9 = x2 − 6x + 9

Idea summary Important identities we use often are: • • • •

Perfect square trinomials: a2 + 2ab + b2 = (a + b)2 or a2 − 2ab + b2 = (a − b)2 Difference of squares: a2 − b2 = (a + b) (a − b) Sum of cubes: a3 + b3 = (a + b) (a2 − ab + b2) Difference of cubes: a3 − b3 = (a − b) (a2 + ab + b2)

We can verify identities mathematically through algebraic manipulation or using geometric diagrams. These identities can be used to describe numerical relationships.

Practice What do you remember? 1

Match each name to its polynomial identity. a

Difference of squares

a3 − b3 = (a − b) (a2 + ab + b2)

Perfect square trinomial

a2 − b2 = (a + b) (a − b)

Sum of cubes

iii

a2 − 2ab + b2 = (a − b)2 or a2 + 2ab + b2 = (a + b)2

Difference of cubes

a3 + b3 = (a + b) (a2 − ab + b2)

For each of the following expressions: i

Name the identity represented by the polynomial expression.

Use the identity to expand the polynomial expression.

(5x − 3) (5x + 3)

(x − 9) (x + 9x + 81)

(−4x − 7y3)2

(3a + 5b2) (9a2 − 15ab2 + 25b4)

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For each of the following polynomials: i

Name the identity represented by the polynomial expression.

Use the identity to factor the polynomial expression.

16a2 − 40ab + 25b2

36x2 − 25y2

125x3 + 27

64b3 − 125c3

49x2 −

x3 −

Let’s practice 4

Khai-ro decides to use the area model to complete a polynomial multiplication question, (5x + 8) (5x + 8). He is working from the worked example of (3x + 7) (3x − 7). Fill in the area model to ﬁnd the product (5x + 8) (5x + 8). Find the product using the model or another method. Give your answer as a fully simpliﬁed polynomial. 5x

9x2

21x

−7

−21x

−49

(3x + 7) (3x − 7) = 9x2 + 21x − 21x − 49 = 9x2 − 49 5

Consider the expression (2x − 9y3)2. a

Explain two different ways to rewrite the polynomial expression.

State which strategy you prefer and use it to fully expand the polynomial expression.

Consider the sum of two cubes: a3 + b3 a

Prove a3 + b3 = (a + b)3 − 3ab(a + b).

Prove x3 − y3 = (x − y)3 + 3xy (x − y).

Consider the difference of two cubes: x3 − y3 a

Prove a3 + b3 = (a + b) (a2 − ab + b2). Prove x3 − y3 = (x − y) (x2 + xy + y2).

Factor each polynomial expression fully using appropriate techniques: a

27a3 − 8b3

−8t2 + 18

3y − 36y + 108

x3 + 8x2 y + 16xy2

3x2 + 24x + 48

x6 − 64

432 − 2a3

8p ( p2 − 100) − 5 ( p2 − 100)

1 − n4

xy + 8x

Use the identity (u + v)2 = u2 + 2uv + v2 with v = 2 to ﬁnd the value of 10022.

Follow the steps to show that (a2 + b2) (c2 + d2) = (ac − bd)2 + (bc + ad)2.

190

Expand the left-hand side of the equation.

Expand the right-hand side of the equation.

Use the identity to ﬁnd the value of (272 + 112) (422 + 942).

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Op Art, short for Optical Art and is style of visual art that deals with optical illusion. Rosaria is studying art and was inspired by Victor Vasarely to create the artwork shown. From different perspectives, it can look like the smaller cube is cut out of the larger cube, outside the larger cube, or other interpretations.

This art reminded her of the difference of cubes. We can use a large cube with a smaller cube cut out of it to give a geometric interpretation to a3 − b3. 2 b

3 a

Write an expression for the volume of each of the following solids: i

b c

Solid 1, V1

Solid 2, V2

iii

Solid 3, V3

Explain why V1 + V2 + V3 = a3 − b3.

Using parts (a) and (b), show why a3 − b3 = (a − b) (a2 + ab + b2) geometrically.

Let’s extend our thinking 12

Answer the following: a

4080 is the product of three consecutive integers. Find the three integers using trial and error.

The product of any three consecutive integers in which x is the ﬁrst integer is given by x (x + 1) (x + 2). Which number(s) will the product always be divisible by? Select all the correct options. A 5

Consider the following: a

Expand and simplify (m2 − n2)2 + (2mn)2.

Use part (a) to prove that (m2 − n2)2 + (2mn)2 = (m2 + n2)2

This identity can be used to generate Pythagorean triples. Find the side lengths of a right triangle using x = 8 and y = 6.

Use the identity to ﬁnd the other two side lengths of a right triangle in which one leg measures 30 units.

Consider the following pattern: 42 = 32 + 3 + 4 52 = 42 + 4 + 5 62 = 52 + 5 + 6 We wish to show that this pattern will continue for any square number. Let x2 be the square number. a

According to the pattern, create an equation for x2.

Prove the identity from part (a).

Use the identity to evaluate 1012.

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Prove the following: (a + bi)3 = (a3 − 3ab2) + (3a2 b − b3)i

Answer the following: a

Complete the missing value: 2 × 3 × 4 × 5 + 1 = (⬚)2

We want to show that 1 more than the product of four consecutive integers is always a perfect square. Let x be the ﬁrst of any four consecutive integers. Complete the gaps to show that x (x + 1) (x + 2) (x + 3) + 1 can be expressed as the square of a value. x(x + 1) (x + 2) (x + 3) + 1 = (x2 + x) (x2 + ⬚x + ⬚) + 1

= x4 + ⬚x3 + ⬚x2 + ⬚x + ⬚ = (x2 + ⬚x + 1) (x2 + ⬚x + 1)

= (x2 + ⬚x + 1)⬚

Consider a two-digit number where m is the tens digit and n is the unit digit. a

Write an expression for the value of the two-digit number in terms of m and n.

The digits are switched to form a new two-digit number. Express the value of the new two digit number in terms of m and n.

Show that the difference between the original two-digit number and the new two-digit number is always a multiple of 9.

Consider the following: a

Expand (2x2 + x + 2) (x2 + 2x + 2).

Use part (a) to evaluate 212 × 122.

If the identity in part (a) is to be used to evaluate the products of three digit numbers, determine what x represents. Explain your process.

The standard equation of a circle with radius r and center (h, k) is (x − h)2 + ( y − k)2 = r2 a

Use your knowledge of identities to create an equivalent equation that could be used to represent the equation of a circle.

Verify that your equation from part (a) is equivalent to the standard equation of a circle using the circle shown.

12 11 10 9 8 7 6 5 4 3 2 1 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2

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x 1 2 3 4

3.03 Divide polynomials After this lesson, you will be able to... • divide polynomials in one and two variables (this addresses simplifying rational expressions in preparation for working with rational functions). • represent polynomial division in different forms and show their equivalence (including with algebra tiles and area models).

Divide polynomials Recall from our work with rational numbers, that when we divide a sum by a real number, we can use the fact that:

for any real numbers, a, b or c. We can extend this concept to the division of polynomial expressions. For any polynomial expressions A, B or C :

The simplest form of division of polynomials is when the divisor is a monomial. The process involves dividing each term of the polynomial by the monomial, then simplifying each individual fraction using the properties of exponents. When dividing by a monomial that contains a variable, we use the quotient rule of exponents, to simplify each term. Quotient rule When dividing two exponential expressions with the same base, subtract the exponents. Example:

Dividing polynomials involves a process known as algebraic manipulation. We can view the dividend as the numerator of a fraction and the divisor as the denominator. By factoring, we can simplify and divide out any factors that are common between the numerator and denominator. 1. Completely factor both the numerator and denominator. 2. Identify all common factors that are present in both the numerator and the denominator. These could be monomial or binomial factors. 3. Divide out all common factors from the numerator and denominator. 4. Simplify the resulting expression.

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Example 1 The rectangle has an area of 4x4 − 12x square units, and its width is 4x units. Find the length of the rectangle.

Create a strategy We can begin by identifying the known values and rearranging the formula for the area of the rectangle to highlight the unit we need to solve for.

Formula for the Area of a Rectangle Solve the formula for length

Area = 4x4 − 12x and Width = 4x

Apply the idea Substitute expressions for area and width

Divide each term by 4x

Simplify the expression

Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is x3 − 3. The length of the rectangle is x3 − 3 units.

Reﬂect and check To check the answer we found for the length, we can write the dividend as the product of the quotient and divisor. Length ⋅ Width = Area 3

Formula for the Area of a Rectangle

(x − 3) (4x) = 4x − 12x

Distributive property

The product of the quotient and the divisor does result in the dividend, so our answer is correct.

Example 2 Find the quotient of (15s4 t5 − 5s3 t3 + 20s2 t) ÷ 40st2.

Create a strategy Since the divisor is a monomial, divide each term in the dividend by 40st2.

Apply the idea Divide each term by 40st2 The quotient is

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Simplify the expression

Reﬂect and check Notice the quotient rule was used when dividing each term by 40st2.

Example 3 Factor the numerators and denominators, then divide the polynomial expressions. a

Create a strategy Since the denominator is a binomial, we ﬁrst want to factor the numerator and denominator. Then, we can divide common factors.

Apply the idea We can begin factoring a GCF. In this case, the numerator has a common factor of x and the denominator has a common factor of 3. Factoring these out gives:

We can now see the common factor of 2x + 7 which can be simpliﬁed from the numerator and denominator by the multiplicative inverse property of equality, since

The quotient is .

Create a strategy We ﬁrst want to factor the numerator and denominator, then divide common factors.

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Apply the idea First, we will factor the numerator. In this case, the numerator is a quadratic. To factor it by grouping, we need to ﬁnd two numbers that have a sum of b = 5 and a product of ac = − 24. The two numbers are 8 and −3, so Rewrite the numerator

Factor out each GCF

Factor out the common factor

Next, we will factor the denominator. The denominator is a difference of two cubes, since 27 = 33, so we can factor it using that identity.

Finally, we can divide by the common factor of y − 3.

The quotient is

Reflect and check In this case, the simplified form appears to have about the same “complexity” as the original expression, since it has the same number of terms between the numerator and denominator as the original expression did. The reason that this form is simpler is that the exponents involved are smaller. The numerator and denominator of the original expression had degrees of 2 and 3 respectively, while the simplified expression has degrees of 1 and 2 respectively.

Create a strategy Find and factor out the greatest common factor in the numerator and the denominator. Then, divide by any common factors between the numerator and denominator.

Apply the idea We can begin factoring a GCF. In this case, the numerator has a common factor of 3cd and the denominator has a common factor of 3cd. Factor out each GCF

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Divide by the common factor

Example 4 The rectangle shown has an area of 15n3 + 13n2 + 33n.

a Find a polynomial expression for its height.

Create a strategy The area of a rectangle is given by the formula Area = length ⋅ height. The length is n units, so we can ﬁnd the height by dividing the area by the length.

Apply the idea To ﬁnd the height, we divide the area by the length:

Since the divisor is a monomial, we can divide each term by n and simplify using the quotient rule.

The height of the rectangle is 15n2 + 13n + 33 units.

b Write two polynomials that could represent ﬁnding the area of a triangle with the same dimensions.

Create a strategy To ﬁnd the area of a triangle with the same base and height as the rectangle, we use the formula Area =

⋅ base ⋅ height. We will create two expressions to represent this area.

Apply the idea One way to express the area of such a triangle is directly applying the formula with our known base, n, and height, 15n2 + 13n + 33. The area of the triangle can be expressed as

n (15n2 + 13n + 33) or

(15n2 + 13n + 33).

Another expression can be found by distributing the coeﬃcient: 7.5n3 + 6.5n2 + 16.5n

Reﬂect and check These polynomial expressions for the triangle’s area capture the same geometric relationship in different forms, offering ﬂexibility in how we approach and solve problems involving areas of shapes with similar dimensions.

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Idea summary For dividing polynomials where the divisor is a monomial, we can use the fact that

To divide polynomials with non-monomial divisors: 1. Completely factor the numerator and denominator 2. Divide out all common factors between the numerator and denominator 3. Simplify the resulting expression (if necessary)

Practice What do you remember? 1

Divide each expression. a e

(2xy) ÷ (40y)

Divide the expressions: a c

(30a3b2 − 18a2b) ÷ (3ab)

198

(28c5d5 − 20c3d2 + 12cd) ÷ (4c3d2)

Factor the numerator and denominator.

Identify the greatest common factor between the numerator and denominator.

iii

Divide the polynomial expressions. b

Simplify by factoring ﬁrst:

Let’s practice 5

For each expression:

a 4

(9k) ÷ (12k)

Fill in the missing values to make the statement true:

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Factor and divide: a

(8 − 2x) ÷ (3x2 − 12x)

(4a2 − 9a + 5) ÷ (3a2 − 5a + 2)

j l

(x3 + 3x2 + 2x + 6) ÷ (21 + 4x − x2)

Divide the polynomial expressions: a e

(3x − 12) ÷ (9x + 15)

i k

(x3 − y3) ÷ (x2 − y2)

(2x − 2) ÷ (4x3 − 4x)

You divide a polynomial by a monomial to result in 8x6 − 5x4 + 6x2. a

This result was after the original polynomial was divided by 2x2. Find the original polynomial.

If the terms are rearranged, does this change the original polynomial? Justify your answer.

A rectangle has a width of 4x + 5 and an area of 5x3 + 7x2 − 18x − 8. Find a polynomial expression for its length.

For the space station, an engineer has designed a new rectangular solar panel that has an area of (24x3 − 24x2 + 10x − 2) ft2. The length of the solar panel is (6x2 − 3x + 1) ft. Find the width of the solar panel.

The area of a parallelogram is given by (x4 − 7x2 + 4x − 6) cm2, and its base measures (x + 3) cm in length. Find the perpendicular height of the parallelogram.

The triangle shown has an area of 13n3 + 11n2 + 29n. a

Find a polynomial expression for its height.

Write two polynomials that could represent ﬁnding the area of a rectangle with the same dimensions.

It costs (x5 − 3x4 − 33x3 − 32x2 + 26x − 4) dollars to replace the lawn in the backyard. If the new lawn costs (x + 2) dollars per square foot: a

Find the area of the lawn in square feet.

If the lawn is rectangular and the width of the lawn is (x2 − 8x + 2), draw a diagram of the lawn with all sides labeled.

Determine which two of these quotients are equivalent. Justify your answer. A

Luciana has been asked to ﬁnd the quotient of 3m2 − 18m − 48 and 2m3 − 24m2 + 64m. Her work is shown below, but contains errors. 1

Factor out the GCF

Factor completely

3 Divide out (m − 8) and 2m from the numerator and denominator. Identify where Luciana has made an error and explain what it is. 3.03 Divide polynomials mathspace.co

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Qin has attempted to divide the polynomial expressions and showed his work: 1

Factor the numerator and denominator

Divide out (x + 3) from the numerator and denominator

Additive property of fractions

Simplify the fractions

Identify where Qin has made an error and explain what it is.

Let’s extend our thinking 17

The result of a

is x2 + 5x − 7.

Explain how you can ﬁnd p (x).

Find p (x).

A polygon has an area represented by A = 4x2 + 8x + 4. The ﬁgure has at least one dimension equal to 2x + 2. Draw the ﬁgure and label its dimensions

Explain which values must be restricted from the domain when dividing polynomials and why those restrictions are needed.

For each expression, ﬁnd the value(s) of k that would result in a quotient that is a linear expression: a

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3.04 Characteristics of polynomial functions After this lesson, you will be able to... • identify domain, range, and intercepts of a polynomial function. • compare and contrast characteristics of polynomial functions with other functions. • identify increasing, decreasing, and constant intervals. • identify relative and absolute maxima and minima. • find the value of f (x) given x, and vice versa. • describe the end behavior.

Characteristics of polynomial functions A polynomial function is a function that involves variables raised to non-negative integer powers. The standard form of a polynomial function is given by f (x) = an xn + an − 1 xn − 1 + an − 2 xn − 2 + … + a2 x2 + a1 x + a0 where n is a positive integer and an, an − 1, an − 2, …, a2, a1, a0 are constant coeﬃcients. The domain of every polynomial function is (−∞, ∞). The degree, n, of a polynomial function can be found by identifying the highest exponent on the independent variable. The degree tells us information about the key features, such as the range, end behavior, number of x-intercepts, and turning points. A polynomial of degree n can have at most n x-intercepts, with those of odd degree having at least one. Absolute ( global) maximum

Relative (local) maximum

The largest value over the domain of a function

A point where a function changes from increasing to decreasing

Absolute ( global) minimum The smallest value over the domain of a function

Relative (local) minimum A point where a function changes from decreasing to increasing

3 2 1 −4 −3 −2 −1 −1

x 1

Even degree polynomial • Range depends on the minimum or maximum and leading coeﬃcient • Move in the same direction at the ends • Has an absolute minimum or maximum

−2 −3 −4

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Odd degree polynomial • Range: (−∞, ∞) • Move in opposite directions at the ends • No absolute minimum or maximum

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

The rate of change of a polynomial function is variable, meaning it changes over the course of the domain. 4

3 2 1

−4 −3 −2 −1 1 2 3 4 −1 Inflection point −2 −3 −4

In some polynomials, the function increases (or decreases) at a fast rate, then the rate of change slows around a point called an inflection point, or turning point. In other words, the function continues increasing (or decreasing), but the rate is slower around the point of inflection. This is one type of a point of inflection, sometimes referred to as a horizontal point of inflection, not all points of inflection will see the rate slow around the point. A polynomial of degree n can have up to n − 1 turning points, with those of even degree having at least one.

Example 1 y

60 50 40 30 20 10

Given this polynomial function:

−4 −3 −2 −1 −10

−20 −30 −40 −50 −60

a Determine the end behavior of the function.

Create a strategy We want to determine what happens to the y-values when the x-values are very small and very large.

End behavior as x gets very small

60 50 40 30 20 10

−4 −3 −2 −1 −10 −20 −30 −40 −50 −60

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x 1

4 End behavior as x gets very large

Apply the idea As x gets very small, y gets very large. So, as x → −∞, y → ∞. As x gets very large, y gets very small. So, as x → ∞, y → −∞.

b State the domain and range of the function.

Create a strategy

Apply the idea

The domain represents the x-values of the function, and the range represents the y-values of the function. Recall that the graph of a polynomial function is one smooth curve over a continuous domain.

As a polynomial is defined for any real number input x, the domain is (−∞, ∞). This can be seen in the graph where the function is a smooth continuous curve and continues to be defined to the left, as the x-values get small, and the right, as the x-values get large. Since the y-values continue to get increasingly small and increasingly large indefinitely, the range is also (−∞, ∞).

c Determine if the degree of the function is even or odd.

Create a strategy The function moves in opposite directions at the extremities, and does not have a global maximum or minimum.

Apply the idea

Reﬂect and check

The function has an odd degree.

We can use other key features about the graph of the polynomial to help us determine its equation and other information as needed.

Example 2 This graph shows f (x) = x4 + 2x3 − 7x2 − 8x + 12. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

a Determine f (2).

Create a strategy We want to ﬁnd the value of the function f (x) when x = 2, which corresponds on the graph to the ordered pair (2, y).

Apply the idea When x = 2, the function crosses the x-axis, which represents the point (2, 0). Therefore, f (2) = 0.

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Reflect and check The x-intercepts of a graph can be used to find the roots and binomial factors of the original function. This function has x-intercepts at x = − 3, −2, 1, and 2, so we know that (x + 3) (x + 2) (x − 1) (x − 2) are part of the polynomial, which simplifies to x4 + 2x3 − 7x2 − 8x + 12. Some polynomials have a vertical stretch or compression in the form of a numerical greatest common factor, which does not apply in this case.

b Name the type of extrema seen on the graph.

Create a strategy Since the function has an even degree, the function will have a global minimum or maximum. While the global minimum or maximum may occur for multiple x-values, the y-value of these points will be the same.

Apply the idea The graph of the function opens up, so the function has a global (absolute) minimum at y = − 4.

Reflect and check The leading coefficient of the polynomial also tells us which direction the graph of the function will open. x4 + 2x3 − 7x2 − 8x + 12 has a leading coefficient of 1. A polynomial with a positive leading coefficient opens up, and will therefore have a global minimum. While there is only one global minimum there are two x-values that have this minimum y-value. There is also a local maximum, but we cannot see it on the part of the graph we are shown.

c Determine the range, writing your answer in set notation.

Create a strategy The range of an even function is determined by its extrema. In the case of an even degree polynomial that opens up, all of the y-values of the function will be increasing from the y-value of the global minimum.

Apply the idea The global minimum has a y-value of −4. In set-builder notation, this will be written as {y∣y ≥ −4}.

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Example 3 The graphs of two functions are shown: y

f (x)

g(x)

x −2

−1

x −2

−1

a Identify and compare the intervals where each function is increasing, decreasing, or constant.

Create a strategy Identify the intervals by observing the graph. Note where it rises (indicating increasing behavior), falls (indicating decreasing behavior), or remains ﬂat (indicating constant behavior). Then compare where the two graphs have a similar pattern.

Apply the idea For f (x), we can observe the following: • f (x) is increasing on (−∞, 0) and (2, ∞) • f (x) is decreasing on (0, 2) For f ( g), we can observe the following: • g (x) is increasing on (−∞, 0) and (2, 3) • g (x) is decreasing on (0, 2) • g (x) is constant on (3, ∞) The two functions are both increasing on (−∞, 0) and (2, 3), but g (x) is constant on (3, ∞) while f (x) increases steadily after x = 3. They are also both decreasing on (0, 2).

b Identify and compare the x and y-intercepts of each function.

Create a strategy The x-intercepts are where the graph crosses the x-axis, and the y-intercept is where the graph intersects the y-axis.

Apply the idea For f (x): • x-intercept: (−1, 0), (2, 0) • y-intercept: (0, 4) For g (x): • x-intercept: (−2, 0), (2, 0) • y-intercept: (0, 4) The two functions have the same y-intercept of (0, 4) and x-intercept of (2, 0). However, they each have a second x-intercept, and these are at different points.

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Idea summary The degree, n, of a polynomial function is the same as the highest exponent on the variable. It tells us information about the key features, such as the: • • • •

range number of x-intercepts end behavior number of turning points

The domain of every polynomial function is (−∞, ∞).

Practice What do you remember? 1

Is each function a polynomial or not? a

f (x) = 2x3 − 4x5 + 3

f (x) =

+ 2x5 + 2

f (x) =

Range

f (x) = 3x3 +

For each graph, use interval notation to identify the i

Domain

1 −4 −3 −2 −1 −1

−4 −3 −2 −1 −1

−2

−3

−4

1 −4 −3 −2 −1 −1

−2

206

+ 7x

x 1

1 −4 −3 −2 −1 −1

−2

−3

−4

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−1

The graph of a linear function, g(x), quadratic function, h(x), and an exponential function, f (x), are shown. 4

3 2 1 f (x)

−4 −3 −2 −1 1 −1 g(x) −2

−3 −4

h(x)

Which graph(s) represent(s) polynomials?

Which graph has the greatest y-intercept?

What end behavior do all three graphs have in common?

Which graph has an absolute minimum/maximum?

For each graph, select all true statements: A

y → ∞ as x → ∞

y → −∞ as x → ∞

y → −∞ as x → −∞

y → ∞ as x → −∞

Has relative maxima/minima

Has absolute maxima/minima

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

−4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

4 3 2 1

5 4 3 2 1

4 x 1

3 2 1 −4 −3 −2 −1 −1

−2 −3 −4

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For each function, f (x), Use the graph to ﬁnd: f (x) = 4

f (0)

iii

f (2)

1 −4 −3 −2 −1 −1

−2

−3

−4

4 3 2 1

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6

−4 −3 −2 −1 −1

−2

x 1

f (x) = 0

3 2

x 1

−4 −3 −2 −1 −1

−2 −3 −4

Let’s practice 6 7

The domain of every polynomial function is ⬚ For each graph of the function y = f (x):

Identify the x- and y-intercepts.

Identify any relative or absolute extrema.

iii

Find the intervals where the function is increasing and where it is decreasing.

Describe the end behavior of the function.

y 10 8 6 4 2

−5 −4 −3 −2 −1 −2 −4 −6 −8 −10

208

x 1 2 3 4 5

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10 8 6 4 2 −4 −2 −2 −4 −6 −8 −10

x 2

5 4 3 2 1

−5 −4 −3 −2 −1 −2 −4 −6 −8 −10

10 8 6 4 2

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5

For each function: i

Decide if it has an even or odd degree.

iii

Determine the x-intercepts.

iv Find the interval(s) where the function is increasing.

f (x) = x2 + 2x − 1

f (x) = (x − 1)2 (x + 2)

Write the range using inequalities.

y = x − 2x − 11x + 12x + 36

y = (x2 − 2x + 1) (x + 4)

f (x) = − 4 + x3 − x2

f (x) = − x4 + 4x2

The graph of g(x) = (x + 1)2 (x − 3)2 − 4 is shown. Use the graph or the polynomial function to ﬁnd a

g(2)

g(x) = 12

g(x) = − 4

g(2)

g(x) = 21

g(x) = 8

g(x) = (x + 1)2 (x − 3)2 − 4 14 12 10 8 6 4 2 −2

−2 −4 −6

g(x) x 1

Consider the function y = x3 − 4x. a

Complete the table of values. x y

−1

−3

−2

−1

Sketch a graph of y = x3 − 4x.

Describe the end behavior of y = x3 − 4x.

State the intervals where the function values are positive.

The given graph shows a model for the revenue earned over the ﬁrst year of a small business launching.

40 000

Revenue in $( y)

35 000

Identify when the revenue is increasing.

Identify when the revenue is negative.

30 000

Predict the revenue after one year.

25 000

Predict when the revenue reaches $25 000.

20 000 15 000 10 000 5000

Time in months (x) 1 2 3 4 5 6 7 8 9 10 11 12

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The given graph shows the population of crows at a local park over 15 weeks. a

Estimate when the number of crows is decreasing.

Predict the number of crows after 10 weeks.

Predict when the number of crows will reach 70.

Describe what happens as the number of weeks increases indeﬁnitely, as x → ∞. Explain if this is reasonable.

Number of crows ( y)

90 80 70 60 50 40 30 20 10

Time in weeks (x) 2

Consider the function f (x) = (x − 2)3 + 1 and the function g (x) whose graph is shown. a

Compare and contrast the intervals over which the functions are increasing or decreasing.

Determine the domain and range of each function.

9 8 7 6 5 4 3 2 1

Consider the functions: 2

y = (x + 4x + 4) (x + 3x − 4)

• Function B: y 9 8 7 6 5 4 3 2 1

−3−2 −1 −1 −2 −3

True or False? Both Function A and B have the same number of x-intercepts. Explain your reasoning.

Name the type of minima and maxima represented by each function.

True or False? Both functions show the same end behavior. Explain your reasoning.

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x 1 2 3 4 5 6 7 8 9

x 1

−2 −3

• Function A:

g(x)

−5 −4 −3 −2 −1−1

Let’s extend our thinking 15

Explain how the end behavior of a function is determined by the leading term.

Florin is using the following graph to describe information about the equation of the polynomial function. Since the graph is not symmetric about the origin or the y-axis, Florin states that there is not enough information to determine anything about the equation of the function. Determine whether Florin is correct and explain your reasoning.

y 8 6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

A marine biologist is observing the path of a dolphin leaping in and out of the water. Letting y represents the height above sea level and x represents the distance in meters after where the dolphin ﬁrst exits the water, they modeled the dolphin’s path with the polynomial y= a

x(x − 1) (x − 4) (x − 6) (x − 9) (x − 10)

Complete the table of values, rounding to two decimal places. x y

x 2 4 6 8

9.5

Graph the polynomial modeling the path of the dolphin.

Explain whether or not you think the polynomial is a good model for the dolphin’s path.

Write an equation for each polynomial described: a

An even degree polynomial with an absolute minimum of (0, 0)

An odd degree polynomial with at least three terms that opens down with three x-intercepts.

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3.05 Zeros and factors After this lesson, you will be able to... • identify the zeros of a polynomial function. • explain how the zeros of a polynomial function relate to its factors. • write the factored form equation of a polynomial function given its zeros and multiplicity.

Zeros and factors The zeros of a function are the input values which make the function equal to zero. This means a is a zero of f (x) if f (a) = 0. We also refer to these solutions as roots of the equation f (x) = 0. Fundamental theorem of algebra A polynomial of degree n where n ≥ 1 has a total of n roots in the set of complex numbers. The fundamental theorem of algebra says the number of roots, including complex and repeated solutions, of any polynomial is equal to the degree of the polynomial. Remember that complex roots include the real and imaginary roots. The real zeros of a function will be the x-intercepts of its graph. We can use the factor theorem to make connections between the roots and factors of a polynomial function. Factor theorem If x = a is a root of the polynomial equation f (x) = 0, then (x − a) is a factor of f (x). The multiplicity of a zero is the number of times that its corresponding factor appears in the function. The multiplicities of the zeros in the function will sum to the degree of the polynomial by the fundamental theorem of algebra. Zeros with different multiplicities look different graphically. y

Zeros of multiplicity 1

Zeros of multiplicity 2

y x = −3 x = −1 Mult. 1 Mult. 2

x x=2 Mult. 3

Zeros of multiplicity 3

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Graph of y = (x + 3) (x + 1)2 (x − 2)3

A root of multiplicity 1 crosses through the x-axis with no point of inﬂection (turn). A root with an odd multiplicity greater than 1 crosses through the x-axis with a point of inﬂection (turn). Roots with even multiplicity are tangent to the axis which means they touch the x-axis, then change direction and do not cross the x-axis.

Exploration Match each graph to its equation. • y=

(x + 3) (x + 1) (x − 2)

• y=

(x + 3) (x + 1) (x − 2)2

• y = − (x − 2) (x + 1)2 • y = (x − 2)2 (x + 1)2 y

y 8

−4 −3 −2 −1 −2

−4

−6

−8

5 4 3 2 1

3 2 1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −2

−2 −3 −4

−4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

Explain how you found the correct equation for each graph.

What are the similarities and differences between each of the graphs?

What are the similarities and differences between each of the equations?

If the roots are known, what other information would we need to know to ﬁnd the equation for a speciﬁc function?

There are many polynomial functions that have the same roots. To find the equation of a specific function, we need to know: • the roots • the multiplicity of each root • the degree of the function • another point on the graph (to find the leading coefficient) The leading coefficient, the real roots, the imaginary roots, and their multiplicities are what determines the exact equation of a function.

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When the coeﬃcients of a polynomial meet certain criteria, complex roots and irrational roots will come in conjugate pairs. Complex conjugate roots theorem

Irrational conjugate roots theorem

If P (x) is a polynomial with real coeﬃcients and a + bi is a root of P (x), then its complex conjugate a − bi is also a root.

If P (x) is a polynomial with rational coeﬃcients and a +

is an irrational root of P (x), then its

conjugate a −

is also a root.

Example 1 Given the function f (x) = − 2(x − 4) (3x + 2)2 a State the degree of the polynomial.

Create a strategy The degree of a polynomial is the highest exponent on the independent variable when written in standard form.

Apply the idea Since the polynomial is given in factored form, it will need to be rewritten in standard form to determine its degree. f (x) = (−2x + 8) (3x + 2)2

Distribute −2 to ﬁrst binomial

Square (3x + 2) using the sum of squares identity

Multiply polynomial expressions

= (−2x + 8) (9x + 12x + 4) 3

= − 18x + 48x + 88x + 32

Since the highest exponent in f (x) is 3, the degree of the function is 3.

Reflect and check Since the degree is only related to the independent variable, we can ignore any coefficients or constant terms when calculating the degree from a factored polynomial. −2(x − 4) (3x + 2)2 → (x) (x)2 = x3 The degree of the polynomial provides information about the number of x-intercepts, end behavior, range, and turning points of the graph. Prior to graphing, we would expect f (x) to have at most 3 x-intercepts, up to 2 inflection points, a range of (−∞, ∞), and no absolute extrema.

b Determine the zeros of the polynomial and their multiplicities

Create a strategy To determine the zeros of a function, each unique factor is set equal to zero and solved. The multiplicity of a zero is the number of times that the zero’s factor appears in a function.

Apply the idea To solve for the zeros, each unique factor of f (x) will be set equal to zero and solved for x. Set factor equal to zero

Solve for x

Set factor equal to zero

Solve for x

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In the factored form of f (x), we see that (x − 4) appears once and (3x + 2) appears twice. So, f (x) has a zero of 4 with a multiplicity of 1, and a zero of

with a multiplicity of 2.

Reﬂect and check The multiplicities of the zeros give additional information about the graph of f (x). A zero with an even multiplicity will have the graph change directions without passing through the x-axis, while a zero with an odd multiplicity will pass through the x-axis. The function f (x) will have x-intercepts at

and (4, 0). At

, the graph touches the x-axis, then changes

directions. At (4, 0), the graph crosses through the x-axis. Also, since the fundamental theorem of algebra says that the number of real or complex solutions is equal to the degree of the polynomial, and our number of real zeros including repeating solutions equals the degree, we know there are no complex (non-real) zeros to f (x).

Example 2 Consider the graph of a cubic function shown. Determine the equation of the function.

f (x)

30 20 10 −5 −4 −3 −2 −1 −10

x 1

−20 (−2, −27) −30 −40

Create a strategy To determine the equation, we need to determine the zeros and their multiplicities first. We can use that information with the factor theorem to write the factors of the function. The zeros and factors are not enough to determine the exact equation of this specific function though, so we will use the given point (−2, −27) to find the leading coefficient.

Apply the idea From the graph, we can determine the zeros by ﬁnding the x-intercepts. These are at x = − 3 and x =

. The graph

crosses through the x-axis at x = − 3, so it must have a multiplicity of 1. The graph is tangent at x = of the function is 3, so this zero has a multiplicity of 2.

, and the degree

Using the factor theorem with the zeros and their multiplicities, we have the expression

Alternatively, we can express a factor with a zero of

as (2x + 1): (x + 3) (2x + 1)2

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Next, we need to use the given point to ﬁnd the leading coeﬃcient. f (x) = a (x + 3) (2x + 1)2

Equation of the function 2

−27 = a (−2 + 3) (2 ⋅ − 2 + 1)

Substitute values from the given point

−27 = a (1) (−3)

Evaluate the multiplication and addition

−27 = 9a

Evaluate the multiplication

−3 = a

Divide both sides by 9

Therefore, f (x) = − 3 (x + 3) (2x + 1)2 is the cubic function.

Reﬂect and check To verify this answer, we can use the equation to ﬁnd another point and check that it lies on the graph shown. The y-intercept is an easy point to verify, so we will substitute x = 0 into the equation. f (x) = − 3 (x + 3) (2x + 1)2

Equation of the function 2

f (0) = − 3 (0 + 3) (2 ⋅ 0 + 1) 2

Substitute x = 0

= − 3 (3) (1)

Evaluate the multiplication and addition

= −9

Evaluate the multiplication

Looking at the graph, we see that the y-intercept is at (0, −9), so the equation we found is correct.

Example 3 A polynomial function f (x) has the following characteristics: • Degree of 3 • Zeros include x = 3 and x = 2i • Real coeﬃcients • Has a y-intercept at (0, −12) Determine the equation of the function.

Create a strategy The zeros of the function that were given are x = 3 and x = 2i. The latter is an imaginary root, so we need to determine if we can use the complex conjugates theorem by looking at the information given about the coeﬃcients. The coeﬃcients must be real in order to use the theorem.

Apply the idea Since x = 2i is a zero and the polynomial has real coefficients, then x = − 2i must also be a zero by the complex conjugates theorem. From these, we get the expression (x − 3) (x − 2i) (x + 2i) The degree of the product of the factors is 3. Next, we need to use the y-intercept to find the leading coefficient.

216

f (x) = a (x − 3) (x − 2i) (x + 2i)

Equation of the function

−12 = a (0 − 3) (0 − 2i) (0 + 2i)

Substitute values from the y-intercept

−12 = a (−3) (−2i) (2i)

Evaluate the addition

−12 = − 12a

Evaluate the multiplication

1=a

Divide both sides by −12

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Therefore, f (x) = (x − 3) (x − 2i) (x + 2i) is the polynomial function, which can also be written as f (x) = (x − 3) (x2 + 4), or f (x) = x3 − 3x2 + 4x − 12.

Reflect and check If the information did not tell us that the coefficients were real, then we could not have used the complex conjugates theorem. This theorem can only be used if the coefficients are real since imaginary solutions must come in pairs and the imaginary components would disappear when factors are expanded.

Example 4 For each given function, determine the zeros using appropriate methods and state the multiplicity of each zero. a g(x) = 3x3 − 15x2 + 24x − 12

Create a strategy The factor theorem connects the roots of a polynomial to its factors, so factoring the polynomial (if possible) will allow us to determine the zeros. How often each factor appears in the function will also tell us the multiplicity of each zero.

Apply the idea Replace g(x) with 0 and factor the polynomial. 0 = 3x3 − 15x2 + 24x – 12 3

0 = 3(x − 5x + 8x − 4)

Set g(x) = 0 Take out a greatest common factor

From here, we need to ﬁnd a way to break up one of the terms so we can factor by grouping. For this one, we can rewrite 8x into 4x + 4x which will allow us to ﬁnd another factor. 0 = 3(x3 − 5x2 + 4x + 4x − 4)

Rewrite since 8x = 4x + 4x

0 = 3[x(x − 5x + 4) + 4 (x − 1)]

Factor by grouping

0 = 3[x(x − 4) (x − 1) + 4 (x − 1)]

Factor the quadratic

0 = 3[(x − 1) (x (x − 4) + 4)]

Factor GCF of (x − 1)

0 = 3(x − 1) (x − 4x + 4)

Simplify the factors

0 = 3(x − 1) (x − 2)

Factor completely

Once completely factored, set each factor equal to zero and solve. 0=x–2

0=x−1

2=x

1=x

We see that the factor (x − 2) appears twice and the factor (x − 1) appears once. The zeros are 2 with a multiplicity of 2, and 1 with a multiplicity of 1.

Reﬂect and check Since the degree of the polynomial is 3 and the multiplicities of the zeros also add to 3, we know that there will not be any complex (non-real) solutions to g(x).

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b f (x) = − 3x3 + 12x2 − 3x

Create a strategy After setting the function equal to zero, we will determine if the polynomial can be factored, taking out a greatest common factor if possible. The quadratic formula will be used where needed if a prime quadratic appears after factoring. The number of each zero will determine the multiplicity.

Apply the idea Replace f (x) with 0 and determine if the polynomial can be factored. 0 = − 3x3 + 12x2 − 3x 2

0 = − 3x(x − 4x + 1)

Set f (x) = 0 Take out a greatest common factor

Since x2 − 4x + 1 can not be factored, we will use the quadratic formula, x = b = − 4, and c = 1.

, using values of a = 1,

Replace values of a, b, c in quadratic formula Evaluate b2 − 4ac and 2a

Simplify radicals and write as separate solutions

Simplify Since −3x is a factor of f (x), the remaining zero will be the solution to −3x = 0, or x = 0. Each zero appears once, so the zeros 0, 2 − , and 2 + all have multiplicities of 1.

Reﬂect and check Since f (x) had a greatest common factor with a variable, we knew that there was at least one real solution. The value of the discriminant, b2 − 4ac, lets us know whether the remaining solutions will be both real, both complex, or one of each. The value of the discriminant of x2 − 4x + 1 is positive, which means that both remaining solutions of f (x) are real. Since the degree of the polynomial is 3, that means that there would be no complex zeros to the polynomial f (x).

Example 5 Show that the fundamental theorem of algebra is true for quadratic functions.

Create a strategy We can use the quadratic formula and the discriminant to explain the different types of solutions there are to a quadratic equation, then show that each case satisﬁes the fundamental theorem of algebra.

Apply the idea All quadratic functions have a degree of 2, so they all have 2 zeros according to the fundamental theorem of algebra. The zeros of any quadratic function can be found by the quadratic formula

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According to the discriminant, there are only 3 types of solutions that a quadratic equation can have. • When b2 − 4ac > 0, there are 2 real solutions. • When b2 − 4ac = 0, there is 1 real solution. • When b2 − 4ac < 0, there are 2 imaginary solutions. The ﬁrst case says there are 2 real solutions.

Graphically, we can see that each zero has a multiplicity of 1 because the graph crosses through the x-axis. x

Therefore, the quadratic has 2 solutions in the set of complex numbers.

The second case says there is 1 real solution.

Graphically, we can see that this zero has a multiplicity of 2 because it is tangent to the x-axis. This means the quadratic is in the form y = k (x − α )2 which can also be written as y = k (x − α ) (x − α ), where k and α are real numbers. Therefore, the quadratic has 2 solutions in the set of complex numbers. x

The ﬁnal case says there are 2 imaginary solutions.

Graphically, this means there are no x-intercepts. However, if we extend to a complex plane, rotate the graph 90° and then reﬂect the graph, it will intersect the complex x-plane twice. Recall the set of complex numbers contains both real and imaginary solutions. Therefore, the quadratic has 2 solutions in the set of complex numbers.

This shows that the fundamental theorem of algebra is true for quadratic functions.

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Idea summary The fundamental theorem of algebra says that a polynomial of degree n has n complex solutions, roots, or zeros. If x = a is a solution or root of a polynomial equation f (x) = 0, then: • • • •

f (a) = 0 x = a is a zero of the polynomial f (x) f (x) has an x-intercept at (a, 0) (x − a) is a factor of f (x)

The multiplicity of a zero is the number of times a zero is repeated. This can be found by the exponent of its corresponding factor in the function. When graphed, the multiplicities appear as follows: • • •

Multiplicity of 1 crosses through the x-axis Even multiplicity is tangent to the x-axis Odd multiplicity greater than 1 crosses through the x-axis with a point of inﬂection

Imaginary roots come in complex conjugate pairs like 7i and −7i.

Practice What do you remember? 1

Select all that apply. The zero of a function can also be called a

root

extreme

x-intercept

y-intercept

y = (x + 3) (x + 2) (x − 2) 2

y = (x − 2) (x + 5)

solution

y = − (x + 4) (x + 2) (x − 1)

y = x (3x − 1) (2x + 5)

For each graph: i

State the x-values of the x-intercepts.

Write the equation of the quadratic function in factored form.

y 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

220

asymptote

Write the ordered pairs representing the x-intercepts for each function represented. a

b 4

3 x 1 2 3 4 5

Mathspace Virginia SOL Algebra 2 mathspace.co

2 1 −4 −3 −2 −1 −1 −2 −3 −4

x 1

3 2 1

10 8 6

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

4 2 −2 −1 −2

x 1

−4 −6

x 1

For each polynomial, state the multiplicity of the root x = − 5. a c

f (x) = (x − 2) (x + 5)3 (x − 5) 4

h(x) = (x + 4) (x + 5)

p(x) = (x − 5)2 (x − 2)2 (x + 5)

g(x) = 5(x + 5)2 (x − 3)

For each of the following functions: i

State the degree of the function.

Identify the zeros of the function and their multiplicities.

f (x) = (x − 7) (x − 1) (x + 6)

f (x) = 5(x + 1)2 (x − 6)

f (x) = 2(x2 − 2x + 1) (x + 3)

f (x) = x(x2 − 9)

Let’s practice 6

Determine whether or not each of the following is a factor of f (x) = x3 + 3x2 − 3x − 1: a

(x − 3)

(x − 1)

(x + 2)

Determine the remaining zero of a polynomial with rational coeﬃcients and with: A degree of 3 a • b • A degree of 2 • A constant term of 8 • A coeﬃcient of −10 for x • x-intercepts at x = 2, x = 1 + • Solution at x = 5 −

Given that the following functions have only integer x-intercepts, determine the zeros of each function. a

f (x) = x3 − 2x2 − x + 2

f (x) = 2x3 − 14x + 12

For each of the following functions: i

Sketch the graph of the function.

Use the graph to approximate the roots

f (x) = − x4 + x3 + 2x2

g(x) = x3 − x2 − 4x + 4

f (x) = (x2 + 2x + 1) (x2 − 4x + 3)

h(x) = − 2x3 + 18x

Write a function in factored form given the following zeros and their multiplicities: a

x = − 5 is a root with multiplicity 2; x = 1 is a root with multiplicity 1

x = 10 is a solution with multiplicity 3; x = − 6 is a solution with multiplicity 2; x = 2 is a solution with multiplicity 2

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For each zero of the graph shown, determine if its multiplicity is even or odd, and defend your reasoning.

y 8 6 4 2 −10 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Juliet says the equation x2 − 16x + 64 = 0 has one solution. Jerome says Juliet is wrong and that the fundamental theorem of algebra states that a quadratic equation must have two solutions. Determine who is correct and why.

The function h (x) is shown on the graph and has a leading coeﬃcient of −1. 30

25 20 15 10 5 −4

−3

−2

−1

x 1

−5

−10 −15

State the coordinates of the y-intercept of h (x).

Write h (x) in factored form.

The function d (x) is shown on the graph, has a leading coeﬃcient of 1 and a degree of 3. 25 20 15 10 5 −5

222

State the roots of h (x).

−4

−3

−2

−1

x 1

−5 −10 −15 −20 −25

State the x-intercepts of d (x).

Determine the multiplicity of each zero.

Write d (x) in factored form.

Write the equation for d (x) in standard form.

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Dean graphed the function y = − 3(x − 2) (x + 3)2 as given: y

−3

−18

Identify the two errors that Dean made and graph the function correctly. 16

Consider the following graph of a function f (x): y

x −6

−4

−3

−2

−1

State all the linear factors of f (x).

Write an expression for a polynomial h (x) with a leading coeﬃcient of 1, as a product of linear factors that has exactly the same roots as f (x).

Determine whether the following has roots which include all the roots of f (x): 2 (x + 4) (x − 5) (x + 3) (x − 4) (x − 2)

(x + 4) (x + 3) (x − 2)2 (x − 5)

iii 4 (x − 4) (x − 3) (x + 2) (x + 4) (x + 5)

(x + 4) (x + 3) (x − 2) (x − 4) (x − 5) (x + 7) (x − 11)

d 17

−5

Apart from f (x), state how many other polynomials there are with the same roots and the same graph as f (x).

Find the equation of the following cubic functions in standard form: Has real coeﬃcients a • b • Has real coeﬃcients • Zeros at x = 0, x = 2, and x = 5 • x-intercepts at x = 5, x = − 3, and x = 1 • When x = − 1, y = 72 • When x = − 2, y = − 84 c

• Has real coeﬃcients • Roots at x = 0, x = 6, and x = − 6 • When x = − 4, y = 20

d • Has real coeﬃcients • x-intercepts at x = − 5, x = − 1, and x = 4 • When x = 2, y = − 126

• Has real coeﬃcients • Roots at x = 4, x = 1, and x = 5 • When x = 2, y = 18

Has real coeﬃcients g • • Zeros at x = − 3, x = − 1, and x = − 4 • When x = − 2, y = − 10

• Has real coeﬃcients • Zeros at x = − 1, x = − 4, and x = 3 • When x = − 5, y = 64

h • Has real coeﬃcients • Solutions at x = 4, x = 1, and x = − 2 • When x = 3, y = 50

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The dimensions of a box are shown: x+5

4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

−2 −3 −4 −5 −6 −7 −8

Use the graph to ﬁnd the missing factor representing the height, h.

If the shortest length of the box is 2 units, ﬁnd the dimensions of the box.

The concentration, in parts per million, of a medicine in a patient’s bloodstream after x hours can be modeled using the function C(x) = 0.0003x(x − 50)2. a

State the zeros of the function and their multiplicities.

State what each root represents in context.

ppm 5 4 3 2 1 hours 10

−1

Let’s extend our thinking 20

The overall proﬁt of a company during its ﬁrst 10 years in business is modeled by a polynomial, as shown on the graph: a

Identify the times when the company was breaking even.

Given that the company took out a $36 000 loan when it ﬁrst started, determine the equation of the polynomial modeling the company’s proﬁt.

Explain whether or not the company should expect this model to be accurate for the future.

70 000 60 000 50 000 40 000 30 000 20 000 10 000 −10 000 −20 000 −30 000 −40 000

The graphed function has a degree of 9. Determine the factored form of the polynomial.

Profit (dollars)

Years 1 2 3 4 5 6 7 8 9 10 11

y 4 3 2 1 −1

x 1

−2 −3 −4

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(5, −3.2)

Write a polynomial function p of least degree with rational coeﬃcients which has a solution of x = 1 + i and whose graph passes through the origin.

Write a polynomial function q with rational coeﬃcients, a leading coeﬃcient of 1, and zeros of x = 2i and x=1+ .

Explain why an integer polynomial must always have radical zeros in conjugate pairs.

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3.06 Solve polynomial equations After this lesson, you will be able to... • determine the number and type of solutions of a polynomial equation. • solve polynomial equations. • verify solutions to polynomial equations algebraically, graphically, and with technology. • justify reasonableness of solutions. • explain the steps to solving a polynomial equation. • interpret solutions in context.

Solving polynomial equations A polynomial equation is an equation with polynomial expressions on both sides of the equation. The standard form of a polynomial equation is given by anxn + an − 1 xn − 1 + an − 2 xn − 2 + … + a2 x2 + a1 x + a0 = 0 where n is a positive integer and an, an − 1, an − 2, …, a2, a1, a0 are constant coeﬃcients. There are various methods for solving polynomial equations, including: • factoring • quadratic formula • graphing Let’s compare these methods by ﬁnding the solutions of −12x3 + 72x = 30x2 Regardless of method, the polynomial should be rewritten in standard form: 0 = 12x3 + 30x2 − 72x Solve by factoring • To solve a polynomial in factored form, we can set each factor equal to 0 and solve. 0 = 6x(2x2 + 5x − 12)

Take out a greatest common factor

0 = 6x(2x − 3) (x + 4)

Factor the quadratic completely 6x = 0 2x − 3 = 0 x + 4 = 0

Not all polynomials are factorable, so an alternative method may be necessary. Solve using the quadratic formula, • A polynomial equation containing a quadratic expression can be solved using the quadratic formula. After factoring out the 6x that gave x = 0 as a solution, we will use the quadratic formula for 2x2 + 5x − 12 with a = 2, b = 5, and c = − 12.

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The solutions from both the quadratic formula and the greatest common factor give the same set of zeros, . This formula works for all quadratics, both factorable and non-factorable. However, this method cannot be used with higher-degree polynomials unless they can be factored down to a quadratic ﬁrst. Solve by graphing • The graph of a polynomial equation set equal to 0 will show the real solutions as x-intercepts. To graph the corresponding polynomial function of the given example, we will substitute f (x) or y for 0. f (x) = − 12x3 − 30x2 + 72x 20 −4 −3 −2 −1 −20 −40 −60 −80 −100 −120 −140 −160 − −180

y x 1

We see integer roots at −4 and 0, and can estimate a solution at 1.5, or . This method can be used to ﬁnd integer solutions or to estimate non-integer solutions, but it is not the best method for ﬁnding exact non-integer roots or imaginary solutions.

Example 1 Given the polynomial equation (4x2 − 81) (x2 − 3x − 10) = 0: a Determine the number and type of solutions.

Create a strategy Since the equation is already factored as two quadratics, the sign of each quadratic’s discriminant, b2 − 4ac, will tell us the number and type of solutions: • Positive: 2 real solutions • 0 : 1 real solution • Negative: 2 complex, non-real solutions

Apply the idea For the quadratic expression 4x2 − 81, we will use values of a = 4, b = 0, and c = − 81 to ﬁnd the discriminant. b2 − 4ac = (0)2 − 4(4) (−81)

Substitute values of a, b, and c

= 0 + 1296 = 1296

Simplify 2

The discriminant is positive, so 4x − 81 has 2 real solutions. For the quadratic expression x2 − 3x − 10, we will use values of a = 1, b = − 3, and c = − 10 to ﬁnd the discriminant. b2 − 4ac = (−3)2 − 4(1) (−10)

Substitute values of a, b, and c

= 9 + 40 = 49

Simplify

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The discriminant is positive, so x2 − 3x − 10 has 2 real solutions. Combined, the number of real solutions to the polynomial is 4.

Reﬂect and check We can check that the polynomial has 4 solutions using the fundamental theorem of algebra. This theorem says that the degree of the polynomial is equal to the number of its complex solutions. If we were to multiply the given factors together, the degree with the highest term would come from multiplying 4x2 in the ﬁrst expression by x2 in the second expression. 4x2 ⋅ x2 = 4x4 This shows us that the degree of the polynomial is 4, so it has 4 complex solutions. We found 4 real solutions, and real numbers belong to the set of complex numbers, which means there are no imaginary or non-real solutions.

b Solve the equation.

Create a strategy First, we determine whether the expression on the left-hand side of the equation can be expressed as a product of linear factors. If so, we can apply some factoring techniques.

Apply the idea The equation contains a factored expression on the left-hand side and 0 on the right-hand side. The factors are both of degree 2. We need to express these quadratic factors as linear factors, if possible. Observe that 4x2 − 81 = (2x)2 − (9)2. This means that 4x2 − 81 is a difference of two squares, so we have 4x2 − 81 = (2x − 9) (2x + 9). (4x2 − 81) (x2 − 3x − 10) = 0 2

Original equation

(2x − 9) (2x + 9) (x − 3x − 10) = 0

Difference of two squares identity

(2x − 9) (2x + 9) (x − 5) (x + 2) = 0

Factor the quadratic

The zero product property states that if A ⋅ B ⋅ C ⋅ D = 0, then A = 0, B = 0, C = 0 or D = 0. So, we get the following equations: 2x − 9 = 0 2x + 9 = 0 x−5=0 x+2=0 Solving each equation for x, we obtain the following solutions: x= ,x=

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, x = 5, x = − 2

Reﬂect and check We can check the answer by using technology to graph y = (4x2 − 81) (x2 − 3x − 10), then looking for the x-intercepts.

900

The function is equal to zero at

750

600

, x = − 2, x = , x = 5 2

450

300

which conﬁrms these are the solutions of (4x − 81) (x − 3x − 10) = 0.

150 −5 −4 −3 −2 −1 −150

x 1 2 3 4 5

−300

Example 2 Given the equation 4x5 − 9x3 = 72 − 32x2: a Determine the number and type of solutions.

Create a strategy The degree of a polynomial equation corresponds to the number of complex solutions, including repeated solutions. After writing the equation in standard form, we can graph the corresponding polynomial function, identify the x-intercepts, then compare the intercepts and their multiplicities with the degree of the equation. This will help us determine the number of real and imaginary solutions.

Apply the idea We can see that the equation 4x5 − 9x3 = 72 − 32x2 is not in standard form because it is not equal to 0. We can convert the equation to standard form by moving the terms 72 and −32x2 to the left-hand side of the equation using inverse operations. 4x5 − 9x3 + 32x2 − 72 = 0 Now, we can graph the corresponding polynomial function, f (x) = 4x5 − 9x3 + 32x − 72 30 20 10 −4 −3 −2 −1 −10 −20 −30 −40 −50 −60 −70

x 1

The graph shows 3 distinct x-intercepts, and the graph crosses the x-axis at each intercept. This means each intercept has a multiplicity of 1. This tells us the polynomial equation has 3 real solutions. Since the degree of the polynomial equation is 5, the polynomial must also have 2 complex, non-real solutions.

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b Find the roots of the polynomial equation and use the roots to conﬁrm part (a).

Create a strategy Once in standard form, a polynomial equation can be solved in various ways, including factoring. Once factored, each factor will be set equal to zero and solved.

Apply the idea In the previous part, we wrote the equation in standard form, giving 4x5 − 9x3 + 32x2 − 72 = 0. Factoring and solving will give the exact real roots and allow us to solve for the imaginary solutions. By factoring, 4x5 − 9x3 + 32x2 − 72 = 0

Standard form

Group terms based on common factors

(4x + 32x ) + (−9x − 72) = 0 2

4x (x + 8) − 9(x + 8) = 0

Factor out the GCF: 4x2 and −9

(4x2 − 9) (x3 + 8) = 0

Factor out the common binomial

(2x − 3) (2x + 3) (x + 8) = 0

Difference of squares identity

(2x − 3) (2x + 3) (x + 2) (x − 2x + 4) = 0

Sum of cubes identity

Using the zero product property for the linear factors, we solve the following equations:

Since the expression x2 − 2x + 4 is not factorable, we will use the quadratic formula to solve:

Quadratic formula

Substitute a = 1, b = − 2 and c = 4

Evaluate the operations

Rewrite the radical

Simplify

Therefore, the solutions are:

These solutions match the predicted number of each type of solution in part (a) since there are 3 real solutions and 2 complex, non-real solutions.

Reﬂect and check We can check the solutions by substituting the values and conﬁrming equality. First checking the real solutions: Substitute x =

Evaluate the multiplication



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Substitute x =

Evaluate the multiplication



4(−2)5 − 9(−2)3 = 72 − 32(−2)2 −128 + 72 = 72 − 128 −56 = − 56

Substitute x = − 2 Evaluate the multiplication 

Checking the imaginary solutions: Substitute x = 1 +

Evaluate the multiplication

 Substitute x = 1 −

Evaluate the multiplication



All ﬁve values are valid solutions for the polynomial since each value balanced the two sides of the equation.

Example 3 Gabby is designing a marble basin that will house a set of planters in her garden. The basin’s sides and bottom should be 1 foot thick. Its outer length should be twice the outer width and outer height. She wants the basin to hold 36 cubic feet of water. a Gabby has determined that 36 = (2x − 2) (x − 2) (x − 1) represents the dimensions of her basin. Determine all possible solutions for f (x).

Create a strategy Since we do not have zero on one side of the equation, we will expand the factors and subtract the 36 to the other side. After the polynomial is expanded and the 36 is subtracted, the polynomial can be factored and its factors set equal to zero.

Apply the idea 36 = (2x − 2) (x − 2) (x − 1) 2

36 = (2x − 6x + 4) (x − 1)

Original equation Multiply the ﬁrst two binomials

Multiply the trinomial and binomial

Subtract 36

Factor the GCF

36 = 2x − 8x + 10x – 4 0 = 2x − 8x + 10x – 40 0 = 2 (x − 4x + 5x − 20) 2

Factor by grouping

Factor the GCF

0 = 2[x (x − 4) + 5 (x − 4)] 0 = 2(x + 5) (x − 4)

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Once factored, each factor will be set equal to zero and solved. 0 = x2 + 5 −5 = x

Write the ﬁrst factor

Subtract 5 from both sides

Square root both sides

0=x–4

Write the second factor

4=x

Add 4 to both sides

The solutions of f (x) are x =

, 4.

Reﬂect and check We know that (x − 4) has a real solution since it is a linear expression. Since (x2 + 5) is a quadratic, we could use the value of the discriminant, b2 − 4ac, to determine what types of solutions it has. With a = 1, b = 0, and c = 5, the discriminant is (0)2 − 4(1) (5) −20 Since the discriminant of x2 + 5 is negative, we know that f (x) has two complex, non-real solutions.

b Determine which solution(s) would be best for Gabby’s design, and describe your choice using the dimensions of the basin.

Create a strategy The value of x represents the thickness of the planter, so only real solutions should be considered. If there are multiple real solutions, they can be plugged in for x to see which values are the most realistic for the context.

Apply the idea From the previous part, the solutions of f (x) are x =

, 4. Only x = 4 is a real solution.

Since x = 4 is the only real solution and x represents the thickness, it is the only choice for Gabby’s design. By substituting x = 4 for each factor of (2x − 2) (x − 2) (x − 1), we have dimensions of 2(4) − 2 = 6 4 − 2 = 2 4 − 1 = 3 The dimensions of the basin are 6 feet by 2 feet by 3 feet.

Reﬂect and check If our polynomial had multiple real solutions, we would evaluate the different real values in the context of the problem and see which gives the most realistic result.

Idea summary The roots of a polynomial equation, p (x) = 0, can be found using multiple methods after the equation has been written in standard form. Methods include •

Factoring: If p(x) is factorable, then the zero product property can be used to ﬁnd the solutions.

•

, Quadratic formula: If the equation contains a factor ax2 + bx + c, the quadratic formula, can be used to ﬁnd its solutions. Graphing: The x-intercepts represent the real zeros of a polynomial. Graphing is best used to ﬁnd integer solutions.

•

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Practice What do you remember? 1

Factor each polynomial completely a

6x2 − 36x + 48

−31 − 4 (x − 9) = 2 (x + 2) + 1

(x − 3)2 = 4

−x2 = 4x + 4

(5x − 4) (x + 3) (x − 2) = 0

(x + 3)2 (x − 6) (5x − 2) = 0

x3 − 125 = 0

x4 = 16x2

3x3 = 5x2

64x − 81x = 0

m3 − 9m − 5m2 + 45

Use the graphs to determine the real zeros of each polynomial. y 10 8 6 4 2

−5 −4 −3 −2 −1 −2 −4 −6 −8 −10

140 120 100 80 60 40 20

x 1 2 3 4 5

5 4 3 2 1

−10−8 −6 −4 −2 −20 −40 −60

−3 −2 −1 −1 −2 −3 −4 −5

n3 − 121n

Solve for the real roots of the following equations:

e 4

x3 − 37x + 84

Determine the number of solutions to each equation. a

2 4 6 8 10

y 5 4 3

1 2 3 4 5 6 7

1 −3 −2

−1

x 1

−1

Consider the equation 2x5 + 36x4 − 80x3 = 0. a

Rewrite the equation by factoring out a greatest common factor.

Solve the equation.

Consider the equation 4x3 − 12x2 − 5x + 15 = 0. a

Factor by grouping to rewrite the equation in factored form.

Solve the equation.

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Let’s practice

(x2 − 16) (x2 + 12x + 36) = 0

10x4 + 60x2 = 70x3

x3 + 4x2 − 4x − 16 = 0

x3 + 40 = 3x2 + 18x

x3 = 7x − 6

2x3 + 5x2 − 22x + 15 = 0

x3 − 0.18x2 − 0.136x − 0.012 = 0

Solve each polynomial equation. a

9m5 = 27m3

11x5 = x6 + 30x4

2x5 = 2x4 + 40x3

3y4 + y2 = y5 + 5y3 − 6y + 4

Solve each equation, stating any complex solutions in the form a + bi. a

x3 + 4x = 3x2 + 12

y2 + 5 = 2y

x2 + 2x = − 3

10x2 + 26 = 2x3 + 34x

For each equation: i

Determine the number and type of solutions

Solve the following equations, stating any complex solutions in the form a + bi:

x4 = 81

27 = x3

7x3 + x = 0

4x3 + 8x = 5x2 + 10

x4 + 5x2 = 36

15x2 = 6x3 − 3x4

48t2 = 3t6

15x3 + x5 = 16x

A box is formed by cutting squares of length x cm from the corners of a piece of cardboard 10 cm by 30 cm. a

Find an expression for the volume of the box in terms of x.

Find the length of the square cutout if the volume is 176 cm3.

x 30 cm

Fold Fold

Solve the following equations by using an appropriate method:

Fold

Fold x

x x

A retail outlet sells x units of a product each year. The annual cost for this product is given by C = 9x + 3x2 and the annual revenue is given by G = 9 − 2x − 4x2 + x3 where C and G are both in dollars. a

Form an expression for P, the net proﬁt after producing and selling x units.

Find the number of units that must be sold in order to make a proﬁt of $40 672.

A two-sided skate ramp is to have dimensions as shown in the image. One of the sloped sections is 50% longer (horizontally) than the other. a b

10 cm

If the total volume of the two sloped sections is 240 ft3, construct an equation in terms of x to represent the situation. Determine the dimensions of the skate ramp.

3x 2x

2x + 6

23x + 10

Desmond is predicting the projected price of a new stock in the market over time, which can be modeled by the polynomial P (x) = (x3 + 2x2 − 11x − 12) (x2 + 4) He wants to use different key features of the polynomial to make predictions about the stock’s price.

234

Determine the real roots of the polynomial, representing the time when the stock price reaches zero.

Which roots would be reasonable for Desmond to use? Justify your answer.

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Luca claims that f (x) = x4 − 11x3 + 36x2 − 16x − 64 and g(x) = 2x3 − 4x2 − 14x − 8 both have the same set of zeros at −1 and 4. Naia was able to show that 4 is a solution to both, but does not believe that −1 is a zero to both. Who is correct? Show your answer both algebraically and graphically.

Ely attempted to solve the equation 4x2 + 9x = x3 + 36 and realized that he made a mistake. After checking his solutions against his graph and verifying algebraically, he realized that he made an error with his factoring. His work is shown below.

1 2

0 = x3 + 4x2 − 9x + 36 2

0 = x (x + 4) − 9(x + 4)

Moved terms to one side to equal 0 Factor by grouping

0 = (x − 9) (x + 4)

0 = (x − 3) (x + 3) (x + 4)

Difference of squares

x = − 4, −3, 3

Set each factor equal to 0 and solve

Which solution is incorrect? Show your answer algebraically.

How did the graph conﬁrm this solution was incorrect? Include the graph in your explanation.

Fix Ely’s error and name the correct solutions to the equation. Verify the corrected solution algebraically.

Let’s extend our thinking 17

Consider the equation x8 = 1. a

Use the fundamental theorem of algebra to state how many roots (real or complex) the equation has.

Fully factor the polynomial p (x) = x8 − 1 using real coeﬃcients.

Use the fact that i2 = − 1 to further factor the polynomial.

By considering that

= i, solve the equation x8 = 1 for its roots.

Keagan is solving the polynomial equation 2x4 − 11x3 + 21x2 − 36x = 0. He determines that x = 4 is a solution and factors the polynomial into the form (x − 4) (2x2 − 3x + 9) = 0. He then uses the quadratic formula and obtains

as complex solutions.

Determine where Keagan has gone wrong, and help ﬁx his answers. 19

Consider the polynomial equation (3x2 + 1)2 − 12x2 + k = 4, where k is a constant. Determine which values of k result in the polynomial having only real solutions.

Explain why a real-valued polynomial of odd degree always has at least one real zero.

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4 Radical Functions Big ideas • Changing the form of an expression or equation can reveal information that was previously unknown. • Expressions are the building blocks of algebra. They can be used to represent and interpret realworld situations. • The properties of real numbers can be applied to many types of expressions. • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • A solution set is the collection of all values that make an equation or inequality true. • Many function types share similar characteristics.

Chapter outline 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08

Simplify radicals Rational exponents Add and subtract radical expressions Multiply and divide radical expressions Radical functions Solve radical equations Function composition Inverse functions

238 244 253 258 268 288 301 314

4.01 Simplify radicals After this lesson, you will be able to... • simplify numeric and algebraic radicals.

Simplifying radical expressions Radical expressions have many parts as shown in the diagram: Parts of a Radical Radical symbol

Index

Radicand

Radical

index

perfect square

The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. If an index does not appear on the root, it is implied to be 2

A number that is the result of multiplying two of the same integer perfect cube A number that is the result of multiplying three of the same integer together

radical A mathematical expression that uses a root, such as a square root , or nth root radicand The value or expression inside the radical symbol

To simplify a radical expression, the radicand must be broken down into factors using the index as a guide for simplifying. We use properties to help us simplify, such as

a, b n

are positive integers or variables is a positive integer th

We can also use the property of taking the n root of an nth power

a is a positive integer or variable n is a positive integer The index of a radical indicates the number of identical prime factors to look for to create a perfect nth. Square root expressions are written in simplest form if the radicand has no perfect square factors other than one. For cube roots, this means there are no remaining factors of the radicand that are perfect cubes.

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The expression

can be simpliﬁed through prime factorization.

Recall that the square root of a negative number does not result in a real number. When simplifying square roots with to get an imaginary number. a negative radicand, we can take out a factor of

Example 1 Simplify

. Assume all variables are positive.

Create a strategy Write out the prime factorization and use the properties to take square roots of perfect squares and simplify.

Apply the idea Find prime factorization

Rewrite using perfect squares

Product of radicals property

Square root of a perfect square

Simplify

Reﬂect and check The phrase “assume all variables are positive” allows us to evaluate an expression such as

Because the variable is positive,

. For example:

. However, if the variable was negative, then

• If y = 1, the simpliﬁed expression would be • If y = − 1, the simpliﬁed expression would be These two results are not the same since

If the instructions for the problem do not speciﬁc whether the variables are positive or negative, we would need to use absolute value bars to account for both positive and negative results. In this case, the absolute value notation of the simpliﬁed expression would be

Because the square of any real number is always non-negative, x stays positive whether x is positive or negative, so the absolute value bars around x are not necessary.

And by the deﬁnition of absolute value, this expression is equivalent to the piecewise expression shown, which accounts for both positive and negative values

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Example 2 Simplify

Create a strategy . Notice that a4 is already a perfect 4th power, so we do not need

Split 80 into its prime factors and use to write out its factors.

Apply the idea Find prime factorization of 80

Rewrite using perfect 4th powers

Product of radicals property

4th root of a perfect 4th power

Simplify

Example 3 Assume that x is non-negative, simplify

Create a strategy To simplify this expression, we’ll ﬁrst break down of a negative number using i, the imaginary unit.

into its perfect square factors and separate the square root

Apply the idea Rewrite using perfect squares

Product of radicals property

Square root of perfect square

Deﬁnition of i

Evaluate the multiplication

Reﬂect and check Notice that if x had represented a negative number, the radicand would have been a positive number and the simpliﬁed expression would not be imaginary. However, the instructions said x represents a non-negative value, so the result cannot be real.

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Example 4 Simplify the algebraic radical:

Create a strategy First, we can rewrite the factors in the radicand as powers of 3. Then, we can remove the powers of 3 from under the radical and multiply the results to the variables already outside the radical.

Apply the idea Power rule

Product of radicals property

Evaluate cube roots

Product rule

Idea summary Radical expressions can be simpliﬁed by grouping factors and simplifying a product of radicals.

a, b are positive integers or variables n

is a positive integer

Simplifying also uses the property of taking the nth root of an nth power

is a positive integer or variable

is a positive integer

The power of powers property groups factors.

(am)n = amn a

is a positive integer or variable

m, n is a positive integer When the radicand is negative in a square root function, the simpliﬁed radical will result in an imaginary number, i.

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Practice What do you remember? 1

Complete the second statement by following the example on the ﬁrst statement: Statement 2:

Statement 1: 2

Fill in the boxes: a

Rewrite each expression in simplest form: a

Determine the value of a in the equation

Let’s practice 5

Complete the second statement by following the example on the ﬁrst statement: Statement 1: Statement 2:

Simplify each expression, assuming all variables are positive: a

Which expression is equivalent to A

4x2y

64x2y

16x2y

Assuming that all variables represent positive real numbers, write each expression in simplest form: a

Assuming that x is non-negative, simplify each expression: a

242

Determine whether each expression is written in simplest form. If not, simplify the expression. a

16x4y2

for positive x and y values?

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When a > 0 and b > 0, which expression is equivalent to A

in simplest form? C

Assuming that all variables represent positive real numbers, write each expression in simplest form: a

A student simpliﬁed the expression shown. Identify the step in which the student made the error and correct their mistake. 1

2 3

Assuming that all variables represent positive real numbers, write each expression in simplest form: a

Let’s extend our thinking 15

Write the expression

Find the value of x in the equation

The volume V of a regular hypothetical solid with edge length a is given by

as a single radical. .

Find the volume of the solid with a side length of 2 cm.

Write the formula in terms of a.

Find the side length of the solid with a volume of 72 cm3.

Find a positive integer value of ⬚ that would allow for both radical expressions to be simpliﬁed to a non-radical expression.

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Using the properties of exponents, we can express , which represents one of n equal factors whose product equals a, multiplied by itself m times, using rational exponents as:

is the base is the exponent

where, m and n are integers, and n ≠ 0. In general:

is the base or radicand

m is the numerator or power n

is the denominator or index

We can use these rules for rewriting radicals along with the properties of exponents to simplify expressions involving radicals and rational exponents. Recall that the properties of exponents can be applied to integer exponents and rational exponents. Product of powers

am ⋅ an = am + n

Quotient of powers Power of a power Power of a product

(am)n = amn (ab)m = am ⋅ bm

Power of a quotient Identity exponent Zero exponent

a1 = a a0 = 1

Negative exponent

Example 1 Use the properties of exponents to deﬁne a rational exponent that would make the statement true:

Create a strategy To get from the second step to the last step, the power of a power property is being used. This property tells us to multiply the exponent inside the parenthesis with the exponent outside the parenthesis. We need a fraction that when multiplied by 3 will result in 7.

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Apply the idea

Reﬂect and check Checking the multiplication in the exponents:

This shows the radical in the ﬁrst expression can be rewritten in rational exponent form as shown in the second expression:

Example 2 Write each expression in an equivalent form using rational exponents. Assume all variables are positive. a

Create a strategy We can write this expression using rational exponents using the fact:

Apply the idea

Reﬂect and check This image can help us visualize the rule in another way. The exponent became the numerator of the rational exponent, and the index became the denominator. Power Index

Create a strategy After rewriting the radical with rational exponents, we can use a few exponent properties to simplify the expression. First, we can use the power of a product property to apply the rational exponent to both bases. We can then use the power of a power property to simplify the powers.

Apply the idea

Rewrite using rational exponents

Power of a product

Power of a power

Simplify the exponents

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Reﬂect and check In the instructions, it said “assume all variables are positive.” This is important for this problem because the 4th root of a negative number is undeﬁned. Had it not mentioned that the variables were positive, we would have needed to consider complex values when simplifying.

Example 2 Write the following expressions in simpliﬁed radical form. a

Create a strategy We can write this expression in radical form using the fact:

Apply the idea

Reﬂect and check The instructions for this problem did not say that the variables needed to represent positive numbers. That is because the odd root of a negative number is deﬁned. For example, if b = − 4 and c = 24, this expression becomes:

We can use the fact that (−1)5 = − 1 to rewrite the radical.

Next, we can use prime factorization to simplify this further: Find prime factorization of 96

Rewrite using exponents

Product of radicals property

Evaluate the ﬁfth root

When the index of a radical is odd, the variables do not need to be limited to positive values.

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assume all variables are positive.

Create a strategy We are going to use the properties of exponents to simplify this expression. There is nothing that can be simpliﬁed within the parenthesis, so we can begin by applying the power to both the numerator and denominator.

Apply the idea

Power of a quotient

Power of a product

Rewrite in radical form

Evaluate the radicals

Reﬂect and check Using radicals to simplify this expression would result in the same answer.

Rewrite in radical form

Quotient of radicals

Product of radicals

Evaluate the radicals

Create a strategy To simplify, we will apply the power of a product property. After, we can use the power of a power property to simplify further.

Apply the idea Power of a product property

Power of a power property

Rewrite in radical form

Evaluate the radical

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assume all variables are positive.

Create a strategy To simplify, we will apply the negative exponent property, rewrite the expression in radical form, then simplify the radical if possible

Apply the idea Negative exponent property

Power of a power rule

Rewrite in radical form

Simplify each radical

Rewrite using a single radical

Idea summary We can write radicals using rational exponents, for integer values of m and n, where n ≠ 0:

a is the base or radicand m is the numerator or power n is the denominator or index The properties of integer exponents can also be applied to rational exponents: Product of powers

Power of a quotient

Quotient of powers

Identity exponent

a1 = a

Power of a power

Zero exponent

a0 = 1

Power of a product

Negative exponent

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Practice What do you remember? 1

Use the properties of exponents to deﬁne a rational exponent that would make the statement true: a

iii

b 2

Convert the following radical expressions to rational exponents: b

Consider the expression

What is the radicand of the expression in radical form?

What is the index of the expression in radical form?

Consider the radical expression

When converted to an expression with a rational exponent, what is the base?

Identify the numerator of the rational exponent.

Identify the denominator of the rational exponent.

Simplify the following expressions: a

Convert the following expressions from rational exponents to radicals:

a 4

Explain the similarities between a rational exponent and a radical expression.

a 3

Use the laws of exponents to fully simplify each expression. Leave the expression in exponential form. a

Let’s practice 8

250

Convert the following expressions from rational exponents to radicals: a

Convert the following radical expressions to rational exponents: a

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Select the expression that is equivalent to A

in radical form:

E 11

Select the expression that is equivalent to A

in radical form:

E 12

Use rational exponents to justify that each of the following equations is true: a

Write the following expressions in radical form.

Simplify the radical form if possible.

For the following expressions: Convert the expression to radical form.

For the following expressions:

Simplify the radical expression.

Consider the incorrect work and solution: 1

Rewrite the radical as an exponent

Apply the power law by adding exponents

Apply the power of a power rule

Simplify the exponent

Rewrite the exponent as a radical

Identify and explain the error.

Fully simplify the expression, showing the correct steps of work.

For the following radical expressions: i

Rewrite the radical expression with a rational exponent.

Simplify the expression and rewrite in radical form.

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For the following expressions: i

Convert from radicals to rational exponents.

Simplify the expression with rational exponents.

Let’s extend our thinking 18

Identify and correct the error in the following work:

Solve for the value of

Explain why we need to restrict the variables to non-negative numbers in some cases, and give an example of a radical involving variables that does not need a restriction at all, and one that would require a different restriction.

Using six different digits from 1 to 9, ﬁll in the boxes to make a true statement:

Using each digit from 1 to 6 once each, ﬁll in the boxes to make the largest possible value:

Determine whether the following statements accurately describe the meaning of the expression

252

means we are raising x to the power of , then taking the reciprocal of the result.

means we are taking the reciprocal of x, then raising the result to the power of .

means we are raising x to the power of , then taking the reciprocal of the result.

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4.03 Add and subtract radical expressions After this lesson, you will be able to... • add and subtract numerical and algebraic radical expressions.

Add and subtract radical expressions The same operations that apply to numeric radicals can also be applied to algebraic radical expressions. We can add or subtract like radicals (radicals with the same index and radicand) by adding the coeﬃcients and keeping the radicand the same.

If there are no like radicals, check to see if any of the radicals can be simpliﬁed ﬁrst.

Example 1 Assuming y > 0, simplify

Create a strategy Because the radicals have the same radicand, we can add the coeﬃcients and keep the radicand the same.

Apply the idea Add the coeﬃcients

Reflect and check In the provided simplification, we assume that y > 0 to ensure the expression under the square root is positive, which allows us to directly add the coefficients. However, if y were negative, say y = − 2, the expression inside the square root becomes negative, leading us to consider complex numbers. Substitute y = − 2

Evaluate the multiplication

Rewrite negative radicands with i, the imaginary unit

Add the coeﬃcients

Simplify the square root

Evaluate the multiplication

Hence, if y = − 2, simplifying the original expression would result in 48i, introducing a factor of i due to the square root of a negative number.

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Example 2 Simplify

Create a strategy Because the radicals are not alike, we cannot combine them. Instead, notice that the radicands contain perfect cube factors. First, we will simplify each term, then add the coeﬃcients of the simpliﬁed, like radicals.

Apply the idea Express 125 and 8 with exponents of 3

Simplify each cube root

Combine like terms

Reﬂect and check Remember that

. This means you cannot add radicals if the radicands are different.

For example, if we had attempted to combine the radicands first, the expression would have incorrectly simplified to , which does not lead to the same answer as what we found. Anytime the radical terms in a sum or difference are not alike, always check to see if the radicals can be simplified first. After simplifying each radical, check that the resulting radicands are the same before combining the terms.

Example 3 Simplify

Create a strategy Since the radicands are different, simplify each term, then subtract the like radicals.

Apply the idea Write the radicals as a products of their factors

Simplify the cube root using 83 = 512

Subtract the like radicals

Reﬂect and check Notice the expression is fully simpliﬁed because there are no perfect cube factors in the radicand, and no other algebraic operations need to be performed.

Example 4 Simplify

Create a strategy Group the like radicals and then add the coeﬃcients of like radicals.

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Apply the idea Group like radicals

Simplify like radicals

Reflect and check Simplifying this expression is similar to simplifying the algebraic expression 6x + 7y − 3x + 8y. We identify the like terms, then add and subtract the coefficients of similar terms. Similar to how 6x − 3x becomes 3x, we subtracted the like terms 7y + 8y simplifies to

simpliﬁes to

to get

. And similar to how

Idea summary When adding and subtracting algebraic radicals, they must have the same radicand before we can add or subtract like radicals.

Practice What do you remember? 1

Select the expression that can be simplified by adding the terms. B

A b

Fill in the blank to complete the sentence. We can only add or subtract radicals which have the same ⬚ and ⬚.

Are each pair like radicals? a

and

Select the expressions that are like radicals when simpliﬁed: •

Select the expression that can be simplified by subtracting the terms. A

•

Consider the expression a

Without simplifying, are

Simplify

Evaluate the expression

•

. and

like radicals?

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Let’s practice 5

Find the sum of each expression: a

Find the difference of each expression: a

g 7

For each radical expression: i

Simplify each term in the expression.

Find the sum or difference

Choose the option that correctly simpliﬁes the radical expression A

3x − 3x

3x − 3

Choose the option that shows the sum of A

E 10

256

In the following expressions, perform the indicated operation. a

Francesco has attempted to fully simplify an expression and showed his work: 1

Combine radicals with the same index

Combine like terms in the radicand

Take out a square factor

Identify where Francesco has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

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Samantha has attempted to fully simplify an expression and showed her work: 1

Simplify the radical terms so they have the same radicand

Multiply the coeﬃcient by the reduced radical term

Add the coeﬃcients of the like terms

Identify where Samantha has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

Simplify the following radical expressions and ﬁll in the blanks with the correct simpliﬁed forms: a

Are these equations true or false? a

Let’s extend our thinking 15

Determine if this statement is always true, sometimes true, or never true. ‘The sum of two like radicals is an expression with a radical.’ Justify your answers with examples or counter-examples.

For each expression, describe the possible values of k such that the expression will be a single radical when simpliﬁed: a

Find the difference between the perimeters of the two triangles.

An engineer is designing a piece of machinery and needs to calculate the tolerances of a support beam that can be adjusted using spacers of different thicknesses. Each spacer’s thickness can be represented by a radical expression. Help the engineer by determining the combined thickness of the spacers. a

Spacer A has a thickness represented by , and Spacer B has a thickness represented by Calculate the combined thickness of Spacers A and B.

Spacer C’s thickness is represented by when Spacer C is placed on Spacer D.

Spacer E has a thickness of , and Spacer F has a thickness represented by thickness remaining when Spacer F is removed from Spacer E.

If Spacer G’s thickness is between Spacers G and H.

, and Spacer D’s thickness is y. Determine the total thickness

and Spacer H’s thickness is

. Find the

, calculate the difference in thickness

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4.04 Multiply and divide radical expressions After this lesson, you will be able to... • multiply and divide numerical and algebraic radical expressions. • rationalize denominators.

Multiply and divide radical expressions The same operations that apply to numeric radicals can also be applied to algebraic radical expressions: • Multiplication: For radicals with the same index, multiply the coefficients, multiply the radicands, and write under a single radicand before checking to see if the radicand can be simplified further. , for x, y ≥ 0 • Division: For radicals with the same index, divide the coefficients, divide the radicands, and write under a single radicand before checking to see if the radicand can be simplified further. , for x ≥ 0, y > 0, b ≠ 0

Example 1 Assuming that each variable represents a non-negative number, fully simplify each expression, writing them as a single radical: a

Create a strategy We can use the product of radicals property to combine the radicals:

Then, multiply the numeric values by one another and multiply the variables together.

Apply the idea Use

Simplify the products

Express 64 as a power of 3

Use

Evaluate the radicals

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Reflect and check If we chose to separate the numeric and algebraic products in the radicand first, our work to find the solution would be similar to the following:

Create a strategy We can use the quotient of radicals property to combine the radicals:

Then, we can divide the numeric terms and the variables.

Apply the idea

Reﬂect and check Use

Simplify the quotient

Use

In the provided simpliﬁcation, we assume that p > 0 to ensure the expression under the square root is positive, which allows us to directly divide the terms. However, if p was negative, say p = − 2, the expression inside the square root becomes negative, leading us to consider complex numbers. Substitute p = − 2

Simplify the radical

Evaluate

Rewrite using definition of i

Quotient of radicals property

Simplify

Quotient of radicals property

Simplify

Hence, if y = − 2, simplifying the original expression would result in

, introducing a factor of i due to the square

root of a negative number. The assumption that p is non-negative also excludes p = 0, because that would lead to an undefined expression due to the denominator.

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Example 2 Fully simplify each of the following expressions, writing them as a single radical. Assume all variables are non-zero. a

Create a strategy We can use the product of radicals property to combine the radicals:

Then, multiply the numeric values by one another and multiply the variables together.

Apply the idea Use

Simplify the products

Express 144 as 24⋅32

Simplify the cube root

Further simplify

Create a strategy

Apply the idea

We can use the quotient of radicals property to combine the radicals:

Use Simplify the expression inside the radical

Then, we can divide the numeric terms and the variables.

Example 3 Fully simplify this expression, where k ≥ 0:

Create a strategy We can use the distributive property: (a + b) (c + d) = ac + ad + bc + bd and then combine like terms to simplify.

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Simplify the cube root

Apply the idea Distributive property

Simplify the products

Combine like terms

Idea summary When multiplying and dividing algebraic radicals we can approach this in a couple of different ways: • •

Combine the radicals ﬁrst, then evaluate the multiplication or division of the radicands Separate the numeric and algebraic products into their own radicals, then simplify and multiply or divide the resulting coeﬃcients and radicals

When multiplying radicals with the same index, multiply the coeﬃcients, multiply the radicands, and write under a single radical. , for x, y ≥ 0 When dividing radicals with the same index, divide the coeﬃcients, divide the radicands, and write under a single radical. , for x, y ≥ 0

Rationalize the denominator Fractions with radicals in the denominator are not considered to be in a fully simpliﬁed form. For these fractions, we can rationalize the denominator, which is a method used to rewrite the expression without radicals in the denominator.

Exploration For the expression

What type of number is

Evaluate

. Does the result have the same value as the original expression? Use your calculator to

verify your answer. 3.

What number can we multiply by that does not change the value of the original number?

Using your answer, what fraction should we multiply

by that will not change the value of the

expression, but will eliminate the radical from the denominator? 5.

Multiply

by your answer to the previous question. Does the result have the same value as the original

expression? Use your calculator to verify your answer.

When an expression has only one term in the denominator, we can rationalize the denominator by multiplying the numerator and denominator by the radical in the denominator.

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To rationalize the denominator of an expression in the form

, we want to multiply it by the fraction

Since we are multiplying the numerator and denominator by the same number, it is the same as multiplying by 1, which does not change the value of the expression.

Example 4 Express the fraction in simplest form with a rational denominator:

Create a strategy First, check if we can simplify the radical. Then, we will need to rationalize the denominator. To do this, we will need to ﬁnd a cube root that, when multiplied to the simpliﬁed radical, creates a perfect cube in the radicand.

Apply the idea Find the prime factorization of 56

Simplify the radical

Simplify the fraction

When rationalizing the denominator, the goal is for the denominator to become a non-radical expression. This is only possible when the radicand is a perfect cube. If we multiply radicand 73 which is a perfect cube.

, this would make the

Rationalize the denominator

Evaluate the multiplication

Simplify

Reﬂect and check If we had multiplied the numerator and denominator by been a radical expression in the denominator.

when rationalizing the denominator, there would still have

cannot be simpliﬁed because it does not contain a perfect cube factor in the radicand. There is still a radical in the denominator which means we need to try rationalizing again.

Since 343 = 73, the denominator can now be simplified to a non-radical expression. When rationalizing denominators that are not square root expressions, we have to think a bit more about what expression to multiply by that would create a perfect nth root in the denominator.

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Example 5 Simplify

. All variables are non-negative.

Create a strategy Notice that 3 is a factor of both −39 and 6. Because of this, we can simplify the fraction ﬁrst before rationalizing.

Apply the idea Separate into two fractions

Rewrite using the deﬁnition of i

Use the quotient of radicals rule

Evaluate the division

Rationalize the denominators

Evaluate the multiplication

Reflect and check Rationalizing the denominator first would have given us the same result: Rewrite using the definition of i

Rationalize the denominator

Evaluate the multiplication

Simplify the radicands

Evaluate the division

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Example 6 For the expression

a Convert to a radical expression.

Create a strategy To convert the given expression to a radical form, we’ll replace the fractional exponents with radicals. Remember, .

Apply the idea

Reﬂect and check

Starting with the original expression:

This conversion allows us to more clearly see the radical nature of the expression, which can be useful for simpliﬁcation and evaluation steps that follow.

We can rewrite it as:

b Evaluate the quotient. Simplify fully, including rationalizing the denominator.

Create a strategy To simplify the expression, we ﬁrst reduce the coeﬃcients and divide the radicands, if possible. To rationalize the denominator, we need to multiply the numerator and denominator by a radical that combines with to create perfect square radicand.

Apply the idea

Reﬂect and check

Reducing the coeﬃcient, we get:

Rather than using radicals to simplify, we could have used exponent laws to simplify and rationalize the expression. Divide the coeﬃcients

To rationalize the denominator, we can multiply by . This will create a radicand of h4, which is a perfect square: (h2)2.

Rationalize the denominator

Product of powers property

Simplify the exponent in the denominator

Power of a product property

Converting the numerator to a radical expression, we can see this answer is the same as the one we found previously.

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Idea summary To rationalize the denominator of an expression in the form

, we want to multiply it by the fraction

To rationalize a denominator containing an nth root expression, we multiply the numerator and denominator by a radical that will create a perfect nth power in the denominator’s radicand. If possible, we should simplify the radicals before rationalizing.

Practice What do you remember? 1

Fill in the blank to complete the rule. a

Simplify each of the following expressions by writing them as a single radical: a

For the following radical expressions, determine what you should multiply each by to eliminate the radical and create a non-radical expression: a

Select the expression(s) written in simplest form. A

Let’s practice 6

Fully simplify each of the following expressions, writing them as a single radical: a

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SOL

Fully simplify each of the following expressions, writing them as a single radical: a

Rationalize the denominator for each of the following expressions: a

Which expression is equivalent to B

A 10

Patricia has attempted to fully simplify an expression and showed her work: 1

Combine the product by multiplying indices and radicands

2 Simplify the product in the radicand 3 Cancel opposite indices

Identify where Patricia has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

Fully simplify each of the following expressions, writing them as a single radical: a

Jordan has attempted to fully simplify each expression. i

Identify where Jordan has made an error and explain what it is.

Find the correct product or quotient. b

a 13

Fully simplify each of the following expressions: a

Consider the following pairs of radical expressions known as conjugates. a

Find the product of the following pairs of conjugates: i

iii

266

ii iv

What do you notice about the products of the conjugates?

Show that this pattern holds true for all conjugate pairs by expanding the product and simplifying:

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For the following expressions: i

Convert the following rational exponents to radical expressions.

Find the product or quotient. b

Consider the expression

Write

Fill in the blank to make the equation true:

Fully simplify the expression.

in exponential form.

Let’s extend our thinking 17

Fully simplify each of the following expressions: a

Determine whether each statement is always true, sometimes true, or never true. Justify your answers with examples or counterexamples. a

The product of two like radicals is an expression with a radical.

The product of two radicals which are not like radicals is an expression with a radical.

Fully simplify each of the following. Write the results in reduced radical form. Assume all variables are positive. a

Describe the possible values of k such that

A cube has a volume of

For:

will be a single radical when simpliﬁed.

and a square has an area of 8x9 in2. Find the product of their side lengths.

Find values of a, b, c, and d, so that the expansion has four terms, none of which are like terms.

Find values of a, b, c, and d, so that the expansion simpliﬁes to two terms.

Can a fully simpliﬁed product of two binomials involving radicals ever have three terms? If so, give an example. If not, explain why not.

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Match each function to one of these graphs:

−4 −3 −2 −1 −1

−2 −3

−3 −4

Graph 1

Graph 2

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

4 3

2 x 1

1 −1

−2

−3

−4

Graph 3

5 4 3 2 1

−4 −3 −2 −1 −1

−2

x 1 2 3 4 5

x 1

Graph 4

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−4

5 4 3 2 1

−4 −3 −2 −1 −1

Consider the graph of the square root function features: • Domain: [0, ∞) • Range: [0, ∞) • x-intercept: (0, 0) • y-intercept: (0, 0) • Increasing over its domain • As x → ∞, y → ∞ • Endpoint: (0, 0) • Absolute minimum: (0, 0) • Absolute maximum: none Consider the graph of the cube root function features: • Domain: (−∞, ∞) • Range: (−∞, ∞) • x-intercept: (0, 0) • y-intercept: (0, 0) • Point of inﬂection: (0, 0) • Increasing over its domain • As x → ∞, y → ∞ • As x → −∞, y → −∞

and its key

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For cube root functions, the function increases (or decreases) at a fast rate, then the rate of change slows around a point called an inﬂection point. In other words, the function continues increasing (or decreasing), but the rate is slower around the point of inﬂection. Radical functions can be transformed similarly to any transformation of the parent function, y = af [b (x − h)] + k. Square root

Cube root

Parent function: Reflection across the x-axis: Reflection across the y-axis: Vertical stretch when ∣a∣ > 1 Vertical compression when 0 < ∣a∣ < 1: Horizontal compression when ∣b∣ > 1 Horizontal stretch when 0 < ∣b∣ < 1: Horizontal translation by h Vertical translation by k: (h, k):

Endpoint

Point of inflection

The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers. Similarly, the absolute extremum of the square root function will change location when translated. If there are no reflections, the endpoint of the domain is an absolute minimum. If a vertical reflection occurs, it becomes an absolute maximum. Square root functions do not have a relative extremum, and cube root functions have neither absolute nor relative extrema.

Example 1 For a Describe the transformation that occurred to

to give f (x).

Create a strategy

Apply the idea with a = − 1 and

The function is of the form h = − 2.

The function has been translated to the left by 2 units and reﬂected across the x-axis.

b Draw a graph of the function.

Apply the idea Using the key points of the parent function, key points of the given function.

, with the transformations identiﬁed in part (a), we can identify the

The key points of the parent function are shown in the table. x

Reﬂecting across the x-axis results in: x

270

−1

−2

−3

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Translating these points left 2 units gives us: x

−2

−1

−2

−3

−4

Now, we can graph the function using these key points. y 1 x −2

8 10 12 14

−1 −2 −3 −4

Reflect and check We can use technology to graph the given function and to confirm the transformations we identified. 4

Here is the graph of the square root parent function

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

Here is the graph of the given function, . Notice that the graph does in fact show a reﬂection across the x-axis and a translation 2 units left.

3 2 1 −4 −3 −2 −1 −1 −2 −3

x 1

This does not match the shape of our graph because we chose a different scale for the axes. This graph shows −5 < x < 5 and −5 < y < 5. To match the graph that we drew, we can adjust the viewing window on the graphing calculator to −4 < x < 16 and −5 < y < 2.

−4

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y 1 x −2

After adjusting the axes, we can see that this graph has the same shape as the graph we drew by hand. This conﬁrms that the graph was not dilated and that the graph we drew is accurate.

8 10 12 14

−1 −2 −3 −4

c Write the domain and range of f (x).

Create a strategy As the graph has been translated 2 units to the left, the domain will change. The range will also change since the function was reﬂected across the x-axis.

Apply the idea Domain: [−2, ∞) Range: (−∞, 0]

Reﬂect and check In set notation, the domain can be written as {x∣x ≥ −2}, and the range can be written as {y∣y ≤ 0}.

d Determine the intervals where f (x) is increasing, decreasing, or constant.

Create a strategy The function is increasing if the y-values increase as x increases. The function is decreasing if the y-values decrease as x increases. The function is constant if the y-values remain the same as x increases.

Apply the idea y

Observing the graph of , we note that the y-values consistently decrease as the x-values get larger.

1 x −2

8 10 12 14

−1 −2 −3 −4

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This function is strictly decreasing over its entire domain, [−2, ∞)

Example 2 Consider the graph of f (x). y 7 6 5

f (x)

4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

9 10

a Write the equation that represents f (x).

Create a strategy First, we can look at the shape of the graph and determine the function family it belongs to. This graph is a cube root function, so the parent function is . Next, we want to identify the transformations applied to the function and write the equation in the form .

Apply the idea Comparing the given function to the parent cube root function, we can see that the function has been reﬂected about the x-axis and translated up by 3 units. y 6 5

f (x)

4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

−1

Using the transformation form of the equation, • The reﬂection across the x-axis corresponds to a = − 1 • There was no horizontal shift, so h = 0. • The vertical translation up 3 units corresponds to k = 3. This gives us the equation

x 1

9 10

Reﬂect and check We can verify the equation by graphing it with technology. We can graph the equation viewing window to ensure it matches the given graph.

and adjust the

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b Write the domain and range of f (x).

Create a strategy The domain and range of the parent cube root function is all real x and y. Any type of transformation applied to the parent has no effect on the domain and range of the transformed function.

Apply the idea Domain: (−∞, ∞) Range: (−∞, ∞)

Reﬂect and check The domain and range can also be represented with set notation: • Domain: {x∣−∞ < x < ∞} • Range: {y∣−∞ < y < ∞} The symbol  represents the set of real numbers. The symbol ∈ can be read as “belongs to”. So another way of stating that the domain and range contain all real numbers in set notation is: • Domain: {x∣x ∈ } • Range: {y∣y ∈ }

c Describe the end behavior of f (x).

Create a strategy To describe the end behavior, we want to determine if the y-values of the function approach a single value or if they continue increasing or decreasing indeﬁnitely as the x-values approach positive and negative inﬁnity.

Apply the idea To see what happens as the x-values increase, we need to look at the right side of the graph. As the graph continues to the right indefinitely, the function decreases indefinitely. To describe this in words, we say “as the x-values approach positive infinity, the y-values approach negative infinity.” In symbolic form, it is written as shown: As x → ∞, f (x) → −∞ To see what happens as the x-values decrease, we need to look at the left side of the graph. As the graph continues to the left indefinitely, our function increases indefinitely. To describe this in words, we say “as the x-values approach negative infinity, the y-values approach positive infinity.” In symbolic form, it is written as shown: As x → −∞, f (x) → ∞

d Find where f (x) = 1.

Create a strategy This question is asking us to ﬁnd the value(s) of x for which the y-values equal 1.

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Apply the idea y 7 6 f (x)

5 4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

x 1

9 10

The graph has a y-value of 1 at x = 8, therefore f (x) = 1 at x = 8.

Example 3 Consider the piecewise function shown in the graph:

5 4 3 2 1 −10−8 −6 −4 −2 −1 −2 −3 −4 −5

x 2 4 6 8 10

a Identify the function families in the piecewise function.

Create a strategy We can start by using the general shape of the functions to identify the parent functions. We have worked with the following parent functions: constant, linear, quadratic, cubic, polynomial, square root, cube root, and exponential.

Apply the idea

Reﬂect and check

Looking at the graphed function to the left of the y-axis we can see it is a straight line with a constant rate of change.

To take this one step further, we can use transformations to identify the equations that make up the piecewise function.

Looking at the graphed function to the right of the y-axis, we can see it is some form of a cube root function.

The linear equation as a y-intercept of 2 and a slope of , and it is deﬁned when x < 0.

The piecewise function was created using a linear function and a cube root function.

The cube root function has been translated 3 units right and is deﬁned when x ≥ 0. Therefore, the equation of the piecewise function is:

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b Find all zeros and intercepts of the piecewise function.

Create a strategy To ﬁnd all zeros and intercepts of the piecewise function, we will analyze each piece of the function separately. Zeros of the function occur where f (x) = 0, while intercepts involve ﬁnding points where the function crosses the x-axis (zeros) and the y-axis ( y-intercept).

Apply the idea

Reﬂect and check

Observing the graph of the piecewise function:

From the graph, notice that the linear portion does not include the point (0, 2) since the circle is unﬁlled. This is due to the domain constraint x < 0, which is why it is not included as a y-intercept.

5 4 3 2 1

−10−8 −6 −4 −2 −1 −2 −3 −4 −5

2 4 6 8 10

Zeros: x = − 8, and x = 3 y-intercept:

Example 4 Compare the domain, range, and intercepts for each pair of functions. a y 2 1 x −5 −4 −3 −2 −1

−1 g(x)

−2 −3 −4

Create a strategy We can use our knowledge of the square root function family to determine the domain, range, and intercepts of f (x). We can use the given graph of g (x) to identify its domain, range, and intercepts.

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Apply the idea We can use the fact that the radicand must be non-negative to ﬁnd the domain of f (x): x+2≥0 x ≥ −2 • f (x) has a domain of [−2, ∞)

• g (x) has a domain of (−∞, 0]

The domains are very different from each other. The only similarity is that both domains contain x ∈ [−2, 0]. The square root of a real number is always non-negative, so the range of f (x) is f (x) ≥ 0. • f (x) has a range of [0, ∞) • g (x) has a range of (−∞, 0] The ranges are almost opposite of one another, but they both contain the value y = 0. To ﬁnd the x-intercept of f (x), we have to substitute f (x) = 0 into the equation and solve for x.

• f (x) has an x-intercept at (−2, 0)

• g (x) has an x-intercept at (0, 0)

The x-intercept of f (x) is 2 units to the left of the x-intercept of g (x). To ﬁnd the y-intercept of f (x), we have to substitute x = 0 into the equation and solve for y.

• f (x) has a y-intercept at The y-intercept of f (x) is

• g (x) has a y-intercept at (0, 0) units higher than the y-intercept of g (x).

Create a strategy We can compare the domain, range, and intercepts for each function algebraically or graphically. Let’s graph the functions to help us visualize the functions, then use the graphs to compare the key features. y

y 4

1 −2 −1 −1

x 1

−3 −5

2 g(x)

−8−7−6−5−4−3−2 −1 −1

−2 −4

x 1 2 3 4

−2 f (x)

−3 −4

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Apply the idea Looking at our graphs, we see • f (x) has a domain of [1, ∞)

• g (x) has a domain of (−∞, ∞)

The function g (x) has a domain of all real numbers while f (x) has a restricted domain. The only similarity is that both domains contain x ∈ [1, ∞). Now, let’s ﬁnd the range using our graphs: • f (x) has a range of (−∞, −2]

• g (x) has a range of (−∞, ∞)

The function g (x) has a range of all real numbers while f (x) has a restricted range. The only similarity is that both ranges contain y ∈ (−∞, −2]. Now, let’s compare the y-intercepts. • f (x) does not have a y-intercept

• g (x) has a y-intercept at (0, 2)

The function f (x) does not have a y-intercept, but g (x) does have a y-intercept. Finally, we will compare the x-intercepts. • f (x) does not have an x-intercept • g (x) has an x-intercept at (−8, 0) The functions are clearly different in terms of x-intercepts. f (x) does not have an x-intercept while g (x) does.

Reﬂect and check We could have also found the domain, range, and intercepts algebraically. Recall our functions:

Domain We can use the fact that the radicand of a square root function must be non-negative to find the domain of f (x): x−1≥0 x≥1 So, f (x) has a domain of [1, ∞). Since g (x) is a cube root function, there are no domain restrictions. In other words, g (x) has a domain of (−∞, ∞). Range To find the range of f (x), we can compare the transformations of the graph from the parent function, y = , which has a domain of [0, ∞). Our function, f (x) = − 2, has been reflected over the x-axis and translated right 1 unit and down 2 units. The range will be affected by the reflection and vertical translation, but not the horizontal translation. The reflection will cause the graph to flip, which changes the range to (−∞, 0]. The translation down 2 units will shift the range, so it becomes (−∞, −2]. Since g (x) is a cube root function, there is no restriction on its range. In other words, the range of g (x) is all real numbers. Intercepts To find the x-intercept of f (x), we have to substitute f (x) = 0 into the equation and solve for x.

There is no value that, when you take the square root, will result in a negative number. This equation has no solution, so f (x) has no x-intercept To ﬁnd the x-intercept of g (x), we have to substitute g (x) = 0 into the equation and solve for x.

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We have found out that g(x) has an x-intercept at (−8, 0) To ﬁnd the y-intercept of f (x), we have to substitute x = 0 into the equation and solve for y.

We cannot take the square root of a negative number, so f (x) has no y-intercept. To ﬁnd the y-intercept of g (x), we have to substitute x = 0 into the equation and solve for y.

The y-intercept for g (x) is (0, 2).

Idea summary 4

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Graph of

x 1

The graphs of the square root and cube root parent functions are similar for x > 0, but the domain of the square root function does not include negative values and the domain of the cube root function does. Radical functions can be transformed in the following ways: Square root

Cube root

Endpoint

Point of inflection

The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers. The absolute extremum of square root function will change location when translated. If there are no reflections, the endpoint of the domain is an absolute minimum. If a vertical reflection occurs, it becomes an absolute maximum. Square root functions do not have a relative extremum, and cube root functions have neither absolute nor relative extrema.

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Practice What do you remember? 1

For each function, determine whether it could be a square root function, cube root function, or neither. a

f (x) = x2 + 7

f (x) =

f (x) = x0.5

f 2

x −3 −2 −1 −1

−3 −5

−4

−3

−2

−3

−4

−4 −3 −2 −1 −1

−2

−1

−3 −2 −1

−1

1 −1

Consider the tables of values of a function. x y

−4 0

−3 −1

0 −2

5 −3

12 −4

Select the graph that could represent the function. A

B 3

2 1 −4 −2 −1

280

x 2

8 10 12

−4 −2 −1 −2 −3

−2

−4

−3

−5

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x 2

8 10 12

D 3

2 1 −4 −2 −1

−4 −2 −1

x 2

8 10 12

−2

8 10 12

−3

−2

−4

−3

−5

A cube root function has a point of inﬂection at (8, −4), an x-intercept at (16, 0) and a y-intercept at (0, −8). Select the graph that represents the function. A

4 2

x −2 −2

−2 −2

2 4 6 8 10 12 14 16 18

y 2

x −2 −2

2 4 6 8 10 12 14 16 18

x −2 −2

−4

−6

−8

−10

Consider the graphs of f (x) and g (x) = f (x) + k: a

Select the type of transformation needed to get from f (x) to g (x): A Reﬂection across the x-axis B Vertical dilation C Vertical translation D Horizontal translation

Determine the value of k.

2 4 6 8 10 12 14 16 18

4 3 2 1 −1 −1 −2

f (x) 1

2 3 4 5 6 7 8 g(x)

−3 −4

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For each of the following transformed functions, identify the type of transformation applied to f (x). a

g (x) =

j (x) =

q (x) =

s (x) =

5 3

1 2 3 4 5 6 7

−2

−4 −3 −2−1 −1

−5−4−3−2 −1 −1

f (x)

g(x)

j(x)

−3

1 2 3 4 5 6 7 8 9

−4

3 2

f (x)

1 −3 −2 −1 −1

p(x)

x 1

−2

x −16 −12 −8 −4

8 12 16

−2

−4

Draw a graph of each function given the key features: A square root function a • • Endpoint at (2, 1) • x-intercept at (3, 0) • Going through the point (6, −1)

−1

h(x)

−3

f (x)

b • A cube root function • Point of inﬂection at (−2, 0) • Going through the point (−3, −2)

Let’s practice 7

282

For y =

Draw the graph.

Identify the intercept(s).

Find the domain and range.

Find the end behavior.

Find the end behavior as x → ∞.

For y =

Find the domain and range.

State any increasing or decreasing intervals.

For f (x) =

Identify four points on the graph of the parent function,

Use those points to build the table of values for f (x) =

Describe the transformation that occurred to the parent function to give f (x).

Graph the function.

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. .

For f (x) =

Identify ﬁve points on the graph of the parent function, y =

Use those points to build the table of values for f (x) =

Describe the transformation that occurred to the parent function to give f (x).

Graph the function.

. .

Evaluate each function for the given value of x. Fully simplify each answer. a

f (x) =

when x = − 3

f (x) =

− 7 when x = 12

f (x) =

when x = − 54

f (x) =

+ 7 when x = 500

For each function i

Identify if the function is increasing or decreasing over its domain.

Identify the transformation(s) that have occurred from

iii

Write the equation of the function.

y 6

6 5

x 1

1 2 3 4 5 6 7 8 9 10

y 2

1 x

3 −4 −3 −2 −1 −1

2 3 4 5

−2

1 x −8 −7 −6 −5 −4 −3 −2 −1

−3 −4

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283

For each function i

Identify if the function is increasing or decreasing over its domain.

Identify the transformation(s) that have occurred from y =

iii

Write the equation of the function.

2 1 −4 −3 −2 −1 −1

x 1

−2 −4 −3 −2 −1

−3

−8 −6 −4 −2

−1

x 1

−2

−3

−4

284

−4 −3 −2 −1 −1

1 −1

−4

For each function: i

Draw the graph of the function.

iii

State the increasing and decreasing intervals.

f (x) =

Find the domain and range of the function.

f (x) =

Find the zeros of the function.

f (x) =

For each function: i

Draw the graph of the function.

iii

Find the intercepts of the function.

f (x) =

Mathspace Virginia SOL Algebra 2 mathspace.co

f (x) =

−1

f (x) =

−1

For each function: i

Identify any absolute maxima or minima.

State the end behavior.

iii

State the domain.

State the range.

Evaluate the function for x = 2.

Evaluate for f (x) = 1.

f (x)

−4 −3 −2 −1 −1

−2

−3

−3 −4

f (x)

x 1

f (x)

−4 −3 −2 −1 −1

−2

−3

−4

x 1

The length of a blue whale calf in its ﬁrst few months is modeled approximately by the equation where l represents its length in meters at t months of age. a

Draw a graph of the length function.

In this context, determine the domain constraints on t.

Find and interpret the value when t = 84. Give your answer correct to one decimal place.

Find and interpret the value when l = 6.

To model the rate at which an object sinks when it is dropped into water, a diver jumped from a height into the 137 ft deep Deepspot pool in Mszczonow, Poland. She continued to allow herself to naturally sink once she entered the water. After some time, she began to ﬂoat back up again.

−2

Time was measured from when the diver hit the water in seconds, so that (0, 0) represents the time and depth when the diver entered the water.

−6

State the equation for D, the diver’s depth in feet, t seconds after entering the water.

−10

Determine the diver’s depth 1 second after entering the water.

−12

State an appropriate domain constraint on this scenario, and explain your constraint.

−14

Complete the table of values, rounding to the nearest whole number where appropriate. Months (t) Length (l)

−2

−4

−4 −3 −2 −1 −1

f (x)

Depth (ft) Time (s) 0.5

1.5

2.5

−4 −8

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Three different square root functions, f (x), g (x), and h (x), are shown. State which functions have the following key features: a

A zero

A y-intercept

An absolute minimum

End behavior as x → ∞, y → ∞

f (x) 4 3 2 1 x 1

x g(x)

2 3 4 5 6 7 8 9

0 1

1 4

4 7

9 10

16 13

h(x) = 20

Two different cube root functions, f (x) and g (x), are shown. For each function, ﬁnd and compare these key features:

y 4

End behavior as x → −∞

End behavior as x → ∞

Point of inﬂection

y-intercept

f (x) x

−8 −6 −4 −2 −1

2 4

−2 −3 −4

g(x) = 21

The graphs of f (x) = x3 and g (x) =

are shown. y

y 4

4 3

f (x) = x3

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

x 1

Compare and contrast the following key features of the functions:

286

Domain and range

Intercepts

End behavior as x → −∞ and x → ∞

Increasing and decreasing intervals

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Let’s extend our thinking 22

The length of a rectangle is 2 times its width. a

Write an equation for area in terms of width.

Rearrange the equation to solve for w.

Draw the graph of the function from part (b).

Determine the area of the rectangle that has a width of 5.

Consider the equation

, where a, h, k are non-zero real numbers.

Determine the domain and range. 24

Complete this piecewise function deﬁned by the given graph: 4

3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4

Draw the graph of piecewise function:

Toy blocks come in a very tall and narrow box where the height of the box is 8 times the length of the square base. The maximum height of the box is 32 in. a

Write an equation for the volume, V (x), of the box in terms of the length of the base.

Write an equation for the side length, x (V ), of the base of the box in terms of the volume.

Draw the graphs for both functions over an appropriate domain.

Compare V (x) and x (V ).

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4.06 Solve radical equations After this lesson, you will be able to... • solve radical equations algebraically and graphically. • verify solutions to radical equations algebraically, graphically, and with technology. • justify reasonableness of solutions. • explain the steps for solving a radical equation. • interpret solutions in context. • identify extraneous solutions and justify why they occur.

Solve radical equations Radical equations An equation containing at least one radical expression Example: y

Recall that the parent function f (x) = a range of y ≥ 0.

Range : y ≥ 0

has a domain of x ≥ 0, and

When we are solving equations involving square roots and other even roots, we need to remember that on every step of our solution, any radicand must be non-negative and any radical must also be non-negative. x

The solution to the equation, however, can be negative.

Domain : x ≥ 0

For example, when solving the equation which the radical is deﬁned.

= 5, the solution must be in the interval x ≥ −4 as this is the domain for

In addition, the square root of any non-negative value is always non-negative. This means an equation such as = − 2 has no solutions as it is equal to a negative value. When solving questions with real life applications, we also need to ensure we have viable solutions, which make sense within the context of the question. A non-viable solution does not make sense within the context of the question, such as a negative value when we are solving for the length of a physical object.

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Exploration Asha solves the following radical equation and shows her work:

Given equation

Square both sides

Distribute the parentheses

Subtract x + 18 from both sides

Factor the quadratic expression

Asha then states the solutions are x = − 2 and x = − 9. Shown below is the graph of the system of equations represented by the radical equation:

−20

−10

Algebraically check the given solutions to the equation

Based on the graph, what is the solution the to the system of equations formed by the equation ?

Why do you think Asha obtained an additional solution?

When solving radical equations, it is possible to have an extraneous solution. This is why checking solutions to radical equations is an important part of solving radical equations. Extraneous solution A solution of the simpliﬁed form of an equation that does not satisfy the original equation. We can use substitution to verify solutions algebraically. We can graph a system of equations by hand or with technology to verify solutions graphically. When solving equations involving radicals with higher indices, we can use inverse operations in the same way we do when solving equations with square root expressions. It’s important to note that radical expressions with odd indices do not have domain restrictions. This will prevent the emergence of extraneous solutions.

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Example 1 Solve the following equations: a

Create a strategy To solve for y, we ﬁrst want to isolate equation, and check our solution(s).

. We can then square both sides of the equation, solve the resulting

Apply the idea

Reﬂect and check State the equation

Subtract 5 from both sides

Square both sides

Evaluate the squares

If we substitute y = 16 into the original equation, we obtain + 5 = 9 which is a true statement, so the solution is valid. Remember, we must always check our solution(s) for possible extraneous solutions.

Create a strategy The cube root expression is already isolated on one side of equation. To solve for x, we will raise each side of the equation to a power of 3.

Apply the idea

Reﬂect and check State the equation

Raise both sides to a power of 3

Evaluate the cubes

Add 9 to both sides

Divide both sides by 2

If we substitute x = 2 into the original equation, we obtain = − 1 which is a true statement. When dealing with odd valued indexes such as , we do not need to worry about the radicand being negative. Raising both sides of an equation to an odd power is reversible. If a = b, then a3 = b3 and if a3 = b3, then a = b.

Example 2 For each of the following equations: 1. Solve each equation for x and identify any extraneous solutions 2. Write the equation as a system of equations 3. Graph each system of equations a

Create a strategy To solve for x, we ﬁrst want to isolate . We can then square both sides of the equation to remove the radical. We can then solve the resulting quadratic equation. We then want to check for extraneous solutions.

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Apply the idea 1. State the equation

Subtract 1 from both sides

Square both sides

Evaluate the exponents

Subtract 4x from both sides

Factor the quadratic

Using the zero product property we get two solutions x = , x = 1. We now want to test both solutions: Substitute x = 1

Evaluate the multiplication and addition

We can see that x = 1 satisﬁes the original equation and is a valid solution. Now, let’s test x = . Substitute x = We can see that x = 2. The equation 3x = 1 +

Evaluate the multiplication and addition leads to a false statement and is therefore an extraneous solution. can be written as the system of equations:

3. We can use technology to graph the system of equations.

Reﬂect and check We can see the solution x = 1 as the point of intersection on the graph, while the extraneous solution found algebraically, x = , does not appear on the graph.

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Create a strategy To solve for x, we ﬁrst want to square both sides of the equation to remove the radical. We can then solve the resulting quadratic equation. We then want to check for extraneous solutions.

Apply the idea 1.

State the equation

Square both sides

Evaluate the exponents

Subtract x and 17 from both sides

Factor the quadratic

Using the zero product property we get two solutions x = − 8, x = − 1. We now want to test both solutions: Substitute x = − 8

Evaluate the addition and square root

We can see that x = − 8 leads to a false statement and is therefore an extraneous solution. Now, let’s test x = − 1. Substitute x = − 1

Evaluate the addition and square root

We can see that x = − 1 satisﬁes the original equation and is a valid solution. 2. The equation

= x + 5 can be written as the system of equations:

3. We can use technology to graph the system of equations.

Reﬂect and check It is important to note that the fact that our solutions were negative, x = − 8, x = − 1, does not necessarily make them extraneous. We can see after substitution that only x = − 8 is an extraneous solution and not a point of intersection on the graph of the system of equations, and x = − 1 is valid.

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Apply the idea 1.

State the equation

Square both sides

Evaluate the exponents

Distributive property

Subtract 16x from and add 16 to both sides

Factor the quadratic

Using the zero product property, we get two solutions: x = 10, x = 2. We now want to test both solutions: Substitute x = 10

Evaluate the addition and square root

We can see that x = 10 leads to a true statement and is therefore a valid solution. Now, let’s test x = 2. Substitute x = 2

Evaluate the addition and square root

We can see that x = 2 satisﬁes the original equation and is a valid solution. 2. The equation

can be written as the system of equations:

3. We can use technology to graph the system of equations.

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Reflect and check Notice that we did not solve the equation by isolating the radical first. When the radical has a coefficient, it is not necessary to isolate it before solving. In this case, isolating the radical first requires a couple more steps.

State the equation

Divide both sides by 4

Square both sides

Evaluate the exponents

Multiply both sides by 16

Distributive property

Subtract 16x from and add 16 to both side

Factor the quadratic equation

Solve for x

Following the same steps as before, we arrive at the same solutions for x, which are x = 10 and x = 2. This confirms that our original solution is correct, and the alternate method of isolating the radical first yields the same result.

Example 3 The radius, r, of a cone with a height that is twice its radius is given by

Nirmal is studying an underwater volcano with a height that is roughly twice its radius. Solve for the volume, V, of the volcano if it has a radius of 1.3 km. Round your answer to one decimal place.

Create a strategy Substitute the radius of the volcano into the formula and solve for V. State the equation

Substitute r = 1.3

Cube both sides

Evaluate the cubes

Multiply both sides by 2π

Divide both sides by 3

Evaluate, rounding to one decimal place

The volcano will have a volume of 4.6 km3.

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Reﬂect and check Since we are dealing with a real-world context, we must ensure the reasonableness of our solution. We know that height, radius and volume cannot be negative. If our ﬁnal solution had resulted in a negative volume, this would indicate that we made a mistake somewhere in solving or that it is not possible for the described volcano to have a radius of 1.3 km.

Idea summary To solve radical equations, we can: 1. Use inverse operations to solve, including raising both sides to a power 2. Rewrite the equation as a system, then graph the system and identify the x-values of any points of intersection 3. Verify the reasonableness of solutions and check for extraneous solutions When solving radical equations, it is important to algebraically substitute solutions into the original equation to identify whether solutions are valid or extraneous. We can also verify solutions by making connections to the graphs of the related equations.

Practice What do you remember? 1

Solve the following equations: a

4w2 = 64

a2 + 5a + 10 = 4

3b2 + 7b + 2 = 0

For each equation, determine if the given solutions are valid or extraneous solutions: , given x = 1 or x = 3

, given x = 1 or x = − 6

For each function, state the restrictions on the values of x: a

( y − 3)2 + 5 = 54

Solve the following equations:

a 4

f (x) =

Select the equation that has no solution. Explain why it has no solution. A

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295

Use the graph to solve each equation. a

f (x)

g(x)

−4 −2 −1

1 x 2

x 2

−2

−3

−1

iii

Let’s practice SOL

What is a solution of A

SOL

x = −5

x=1

Solve the following equations: a

For each equation, determine whether x = 4 and/or x = − 4 is a solution. a

Tom and Katrina are both given the equation

. Both have made errors in their solution.

Tom: Katrina:

296

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Describe Tom’s error in his solution.

Describe Katrina’s error in her solution.

Solve the equation for x.

xplain how checking their solutions would have E identiﬁed their errors.

Consider the radical equation: a

Solve the equation algebraically to ﬁnd all possible solutions.

Identify any viable solutions.

Identify any extraneous solutions.

Mathspace Virginia SOL Algebra 2 mathspace.co

SOL

Solve each equation for x. Identify any extraneous solutions. a

What is the solution set for

SOL

{35}

{70}

Solve the following equations: a

Which is a solution for A

? B

A 14

w=2

w = 10

w = 75

w = 80

Solve each equation for x. Identify any extraneous solutions. a

Ray is given the equation

. Here is his work:

Step 1:

Step 2:

Step 3:

Step 4:

when x = 6.

Evaluate

Determine if x = 6 satisﬁes the equation

Explain how Ray got an extraneous solution.

Valid and extraneous solutions are given for the following equations along with their graphs. For each of the following, explain why the solutions are extraneous: a

3 2

1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

• Valid solutions: x = 1, 2 • Extraneous solutions: x = − 1, −2

x −2

−1

−1 −2

• Valid solution: • Extraneous solution:

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Consider the radical equation: a

Draw the graph for

Draw the graph for y = x − 2 on the same coordinate plane.

Solve the equation

Are there any extraneous solutions to the equation? If so, state the extraneous solution(s).

algebraically. Explain how this solution is shown on the graph.

Luis is solving the radical equation . He suggests that x = 4 and x = − 1 could be solutions to the equation. Priya, who has not seen the graph of the equation, argues that only x = 4 is a valid solution. a

Solve the equation algebraically to ﬁnd all potential solutions. Show your work.

Verify the potential solutions graphically. Discuss what the graph indicates about the correctness of the solutions.

Without the graph, explain how Priya could know that x = − 1 is not a valid solution.

Explain the solution method used. Discuss any discrepancies between the algebraic and graphical solutions.

Solve the following equations for x. Identify any extraneous solutions. a

Consider the triangle shown, with AB = 10 and BC = x.

Write an expression for the length of the hypotenuse in terms of x.

Find the value of x for which the hypotenuse is 2 times the area of the triangle.

Identify any extraneous solutions and explain why they are non-viable in this context.

The function

can be used to ﬁnd the period T of a simple pendulum of length l meters.

Express the length l as a function of the period T.

Find the length of a pendulum, to the nearest centimeter, which has a period of 1.5 seconds.

The radius r of a sphere with surface area A is

. Solve for the surface area, A, of a moon with a radius

of 1080 km. Give your answer to the nearest square kilometer. 25

A civil engineering team is designing a new pedestrian bridge to span across a small river. The height, h, in meters, of the arch of the bridge at any point is modeled by the equation horizontal distance, in meters, from the center of the bridge.

, where x is the

The city’s safety regulations require that the minimum height of the arch over the river must be at least 4.5 meters to ensure boats can pass safely underneath.

298

Write an equation to represent the condition that the minimum height of the arch must be 4.5 meters.

Solve the equation for x, given the minimum height requirement.

Interpret the solution from part (b) in the context of the bridge design.

Mathspace Virginia SOL Algebra 2 mathspace.co

In a construction project, a diagonal support beam needs to be installed in a rectangular frame. The diagonal, d, can be determined using the Pythagorean theorem , where l is the length and w is the width of the frame. The length of the frame is known to be 12 meters. a

Use the graph to approximate the value when w = 9 meters. Explain what this value represents in the context of the construction project.

Use the graph to approximate the length of the diagonal beam when the width is 14 meters.

Discuss whether an approximate solution in this scenario is reasonable.

How long does the width of the frame need to be if the builder needs a diagonal beam length of at least 21 meters?

d 20 15 10 5 w 2 4 6 8 10 12 14 16 18

Let’s extend our thinking 27

For each description, write an equation that meets the description: a

Is a radical equation with radical solutions.

Is not a radical equation, but has radical solutions.

Is a radical equation with no extraneous solutions.

Determine whether or not a radical equation can have two extraneous solutions. Justify your conclusion.

Solve the following equations for x. Identify any extraneous solutions.

Solve the following equations for x. Identify any extraneous solutions. a

When cattle are to be transported, they are sometimes ﬁrst put into semicircular corrals. Julia is building a new corral. Each cow needs at least 50 square feet of space. a

Write an equation for the minimum radius required for a corral for c cows.

Describe the types of viable solutions to this equation.

Write an equation for ﬁnding the number of cows given any radius.

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The Beaufort Wind Force Scale was created in 1805 and is used to describe the wind and wave conditions at sea. Admiral Sir Francis Beaufort came up with the system. The Beaufort scale is the whole numbers from 0 to 12, or 17 in some countries, where each number represents a range of wind speeds, wave heights, and general conditions. The Beaufort scale for 0 to 3 is in the given table. Calm Light air Light breeze Gentle breeze

Beaufort number 0 1 2 3

Wind speed < 1 mph 1–3 mph 4–7 mph 8–12 mph

The median wind speed for each Beaufort number can be calculated using the equation:

300

For the Beaufort numbers above 5, each Beaufort number covers a range of wind speeds that has a spread of about 7 mph. Find the range of wind speeds for a Beaufort rating of 9. Justify your answer.

Find the Beaufort rating for a wind speed of 80 mph.

Mathspace Virginia SOL Algebra 2 mathspace.co

4.07 Function composition After this lesson, you will be able to... • determine the composition of two functions algebraically and graphically. • evaluate the composition of two functions algebraically and graphically. • explain the effect of composing functions on the characteristics of those functions (including domain and range).

Function composition We can create a composite function using an operation that combines two functions f and g and produces a function h such that h (x) = g ( f (x)). The output, or function values, of the function f (x) become the input, or x-values, of the function g (x).

Exploration Let f (x) = 2x + 5 and g(x) = x − 1 The composition of f with g is: f ( g(x)) = 2(x − 1) + 5 = 2x − 2 + 5 = 2x + 3 1.

Find the composition of g with f.

Is g( f (x)) the same as f ( g(x))? Explain.

Evaluate g(2). Then evaluate f ( g(2)).

g( f (x)) = ⬚

g(2) = ⬚

Explain what f ( g(2)) represents.

f ( g(2)) = ⬚

Composite function A function created when one function is substituted into another function The symbol ∘ can also be used to represent a composite function. f ( g (x)) = ( f ∘ g) (x) In a composition of functions, the inner function is evaluated first, followed by the outer function. For example, in the composition g ( f (x)), the function f is applied first, followed by the function g. This means that ( g ∘ f ) (x) is not necessarily equal to ( f ∘ g) (x). We can use graphs to evaluate the composition of functions. For example, consider the graphs of f (x) and g(x) shown. We will use these graphs to find g( f (8)).

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8 7 6 5 4 3 2 1 −1 −1 −2

In a composition of function, the inner function is evaluated ﬁrst.

f (x)

For this example, we need to ﬁnd f (8). Looking at the graph of f (x), we see that when the x-value is 8, the y-value is 4. f (8) = 4

x 1 2 3 4 5 6 7 8 9

9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1

The output we just found from f (x) now becomes the input for the outer function, g(x).

g(x)

g ( f (8)) = g(4) Looking at the graph, we see that when the x-value is 4, the y-value is 5. g(4) = 5 Therefore, g( f (8)) = 5 x 1 2 3 4 5

Example 1 For f (x) = x2 + 5 and g(x) = 3x − 2: a Find f (−1).

Create a strategy

Apply the idea

To evaluate f (−1), we need to substitute x = − 1 into the function f (x) = x2 + 5.

f (x) = x2 + 5 2

f (−1) = (−1) + 5

Original function Substitute x = − 1

=1+5

Evaluate the exponent

Evaluate the addition

This shows f (−1) = 6.

b Find g ( f (−1)).

Create a strategy

Apply the idea

Since we know f (−1) = 6, we will use 6 as the input for the outer function g(x).

g(x) = 3x − 2

Original function, g (x)

g(6) = 3(6) − 2

Substitute x = 6

g ( f (−1)) = g (6)

= 18 − 2

Evaluate the multiplication

= 16

Evaluate the subtraction

This shows that g( f (−1)) = 16.

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Example 2 Consider the graphs of f (x) and g(x) shown. 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

f (x)

9 8 7 6 5 4 3 2 1

x 1 2 3 4 5

−7 −6 −5 −4 −3 −2 −1 −1

g(x)

x 1 2 3

a Find g(−3).

Create a strategy Use the graph of g (x) to ﬁnd the output (the y-value) when the input (the x-value) is −3.

Apply the idea 9 8 7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1

When x = − 3, the output is 3.

g(x)

This means g (−3) = 3.

x 1 2 3

b Find f ( g(−3)).

Create a strategy The input of f (x) is the output of g (−3), which we found to be 3. f ( g (−3)) = f (3) We now need to evaluate f (3).

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Apply the idea Using the graph of f (x), we will look for the output or y-value when x = 3. When x = 3, the output is −5.

f (x)

4 3 2 1

Therefore, f ( g(−3)) = − 5 x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

1 2 3 4 5

Example 3 Consider the following pair of functions: f (x) = − 5x + 5 g (x) = 2x2 + 3x − 10 a Find ( f ∘ g) (x)

Create a strategy To ﬁnd ( f ∘ g) (x), we need to use g (x) as the input of f (x).

Apply the idea ( f ∘ g) (x) = f ( g (x)) 2

= f (2x + 3x − 10)

Deﬁnition of function composition Substitute g (x)

The notation f (2x + 3x − 10) means we need to replace the independent variable in f (x) with 2x2 + 3x − 10. f (x) = − 5x + 5 2

f (2x + 3x − 10) = − 5 (2x + 3x − 10) + 5 = − 10x2 − 15x + 50 + 5 2

= − 10x − 15x + 55 2

Therefore, ( f ∘ g) = − 10x − 15x + 55.

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Original function, f (x) Substitute (2x2 + 3x − 10) Distribute −5 Evaluate the addition

b Find ( g ∘ f ) (x)

Create a strategy To ﬁnd ( g ∘ f ) (x), we need to use f (x) as the input of f (x).

Apply the idea g (x) = − 2x2 + 3x − 10

Original function, g (x)

g (−5x + 5) = 2 (−5x + 5) + 3(−5x + 5) − 10 2

Substitute (−5x + 5)

= 2(25x − 50x + 25) + 3(−5x + 5) − 10

Expand (−5x + 5)2

= 50x2 − 100x + 50 − 15x + 15 − 10

Distribute

= 50x − 115x + 55

Evaluate

Therefore, ( g ∘ f ) = 50x − 115x + 55. c Does ( f ∘ g) (x) = ( g ∘ f ) (x)?

Create a strategy Compare ( f ∘ g) (x) = ( g ∘ f ) (x).

Apply the idea ( f ∘ g) = − 10x2 − 15x + 55 ( g ∘ f ) = 50x2 − 115x + 55 ( f ∘ g) ≠ ( g ∘ f )

Reﬂect and check It is possible to have two functions such that ( f ∘ g) (x) = ( g ∘ f ) (x). For example, consider the functions f (x) = 5x − 4 and g(x) =

Finding ( f ∘ g) (x):

Deﬁnition of function composition

Substitute g (x)

Original function, f (x)

Substitute

Distribute 5

Evaluate the subtraction

Finding ( g ∘ f ) (x): Deﬁnition of function composition Substitute g (x)

Original function, g (x)

Substitute g(5x − 4)

Combine like terms

Divide As we can see, ( f ∘ g) (x) = ( g ∘ f ) (x).

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d Compare the domain and range of ( f ∘ g) (x) and ( g ∘ f ) (x).

Create a strategy We can use the structure of the equations to determine what function family each belongs to, then use what we know about that family to identify the domain and range. Since both ( f ∘ g) (x) and ( g ∘ f ) (x) are quadratics, we will need to use the formula x = ﬁnd the range.

to ﬁnd the vertex in order to

Apply the idea Since ( f ∘ g) = − 10x2 − 15x + 55 and ( g ∘ f ) = 50x2 − 115x + 55 are both quadratic functions, the domain is all real numbers, (−∞, ∞). The range of a quadratic function depends on the y-value of its vertex and whether it is a maximum or minimum. For ( f ∘ g) = − 10x2 − 15x + 55, the x-coordinate of the vertex is computed as Substituting this into the equation to find the y-coordinate of the vertex, we get: −10(−0.75)2 − 15(−0.75) + 55 = 60.625 Since the leading coefficient is negative, the parabola opens downward, so the range is (−∞, 60.625]. For ( g ∘ f ) = 50x2 − 115x + 55, the x-coordinate of the vertex is computed as Substituting this into the equation to find the y-coordinate of the vertex, we get: 50(1.15)2 − 115(1.15) + 55 = − 11.125 Since the leading coefficient is positive, this parabola opens upward, so the range is [−11.125, ∞).

Example 4 Use the graphs of g (x) and f (x) to ﬁnd ( f ∘ g) (1) 3

g(x)

1 −3 −2 −1

−1

f (x)

x 1

−2 −3

−3 −2

−1

x 1

−2 −3

−4

−5

Create a strategy Finding ( f ∘ g) (1) can also be written as f ( g(1)). To evaluate this composite function using graphs, we must ﬁrst evaluate the inner function, g(1). The result of that will become the input for the outer function, f (x).

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Apply the idea 3

To ﬁnd f ( g(1)) we must ﬁrst evaluate the inner function, g(1), using the graph of g (x).

g(x)

When x = 1, the output is 2.

1 −3 −2 −1

g (1) = 2

x 1

−1

−2 −3 −4 −5

f (x)

1 −3 −2

−1

x 1

The input of f (x) is the output of g(1), which we found to be 2. This means we need to use the graph of f (x) to ﬁnd the y-value when x = 2. When x = 2, the output is 0. Therefore, ( f ∘ g) (1) = 0

−2 −3 −4 −5

Example 5 A cylindrical tank initially contains 200 in3 of grain and starts being ﬁlled at a constant rate of 40 in3 per second. The radius of the tank is 12 inches. Let g be the amount of grain in the container after t seconds. a State the function for h ( g), the height of the grain in the container, in terms of g.

Create a strategy As the tank ﬁlls with grain, the amount of grain takes the shape of a cylinder which has a volume given by V = π r2h. We know that: • g represents the volume of grain in cubic inches • h ( g) represents, in inches, the height of the grain in terms of g • r is given to be 12 inches Substituting these values into the volume of a cylinder, V = π r2h, we can form an equation relating g and h ( g).

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Apply the idea Volume of a cylinder

Substituting V = g, r = 12, and h = h ( g)

Evaluating the square

Divide both sides by 144π

Symmetric property of equality represents the height of the grain, in inches, in the container, in terms of the volume of grain g.

The function

b State the function for g (t), the amount of grain in the tank after t seconds.

Create a strategy We know that initially, t = 0, there are 200 in3 of grain in the tank. Each second that passes, 40 in3 is added.

Apply the idea t (sec) g (t) (in3)

0 200

1 240

2 280

3 320

The function g (t) = 40t + 200

Creating a table of values, we can see that we have a linear equation where the amount of grain is equal to 200 in3 plus 40 in3 for every second that passes.

c The function A (t) is deﬁned as A (t) = (h ∘ g) (t). Form an equation for A (t) in terms of t.

Create a strategy (h ∘ g) (t) is the same as h ( g (t)), so want to substitute g (t) = 40t + 200 into the function

Apply the idea Deﬁnition of (h ∘ g) (t)

Substitute g (t)

Substitute h ( g)

Simplifying the quotient

The unit for A(t) is inches.

Reﬂect and check In the working above we substituted g (t) = 40t + 200 into the function for h. We can also obtain the same answer by ﬁrst substituting h ( g) =

into h ( g (t)):

Substitute h ( g)

Substitute g (t)

Simplifying the quotient

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d Explain what A (t) represents.

Create a strategy g (t) represents the amount of grain in the container after t seconds, and h ( g) represents the height of grain in terms of the amount of grain. Composing the two gives us (h ∘ g) (t). This represents height as a function of time.

Apply the idea A (t) represents the height of the grain in the container, in inches, after t seconds.

e If the barrel can hold 10 000 in3 of grain, determine the domains of g (t), h ( g) and A (t).

Create a strategy The lower boundary of the domain of g (t) is 0 as the time starts at 0 seconds. This means the lower boundary of the domain of A (t) is also 0 seconds. To calculate the upper boundaries, we can use the fact that the barrel can hold a maximum of 10 000 in3 of grain. The time it takes to ﬁll the barrel will be the upper boundary of both g (t) and A (t). As g is the input for h ( g), the range of g (t) will be the domain of h ( g). So, the lower boundary of h ( g) will be the amount of grain in the barrel initially, and the upper amount will be the maximum amount of grain the barrel can hold.

Apply the idea The lower boundary of h ( g) is 200 in3 as this is how much is in the barrel initially, and the upper boundary is 10 000 in3 as this is the maximum amount of grain the barrel can hold. Calculating the total amount of time needed to ﬁll the barrel: g (t) = 40t + 200 10 000 = 40t + 200

Substitute g (t) = 10 000

9800 = 40t

Subtract 200 from both sides

245 = t

Divide both sides by 40

This means the barrel will be completely full after 245 seconds. The domain of both g (t) and A (t) is [0, 245]. • Domain of g (t) : [0, 245] • Domain of h ( g) : [200, 10 000] • Domain of A (t) : [0, 245]

Idea summary In a composition of functions, the inner function is evaluated ﬁrst, followed by the outer function. ( f ∘ g) (x) = f ( g (x)) ( g ∘ f ) (x) = g ( f (x))

Practice What do you remember? 1

Which of the following expressions represent the composition of two functions, f and g, for an input x? A

f ( g (x))

f ⋅ g (x)

f (x) ⋅ g (x)

( f ∘ g) (x)

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Consider the following mapping: a

Complete the following table −3

x y

−2

−1

−3

−2 −1

If the input is −1, what is the corresponding output?

If the input is −2, what is the corresponding output?

If the output is , what are the corresponding inputs?

0 1

Consider the functions f (x) = 2x + 3 and g (x) = x2 − 4. a

Substitute x with 2 in the function g (x). What is g (2)?

Using your result from part (a), ﬁnd f ( g (2)).

Select the expression that shows the composition of f and g. A f ( g (x)) = 2(x2 − 4) + 3

f ( g (x)) = 2x2 − 4 + 3

C f ( g (x)) = (2x + 3)2 − 4

f ( g (x)) = 2(x2 − 4 + 3)

Simplify f ( g (x)).

Use the expression from part (d) to ﬁnd f ( g (2)).

The tables below show some inputs and outputs of functions j and k. x j (x)

−4 3

−2 7

0 11

2 15

4 19

k (x)

7 −12

9 −4

11 4

13 12

15 20

j (k (9)) = ⬚

Evaluate. a 5

k ( j (2)) = ⬚

k ( j (−2)) = ⬚

j (k (11)) = ⬚

The table shows some of the outputs of the functions f, g and h.

Use the table to evaluate the following:

0 1 2 4 8 16

( f ∘ g) (4)

( g ∘ h) (2)

( f ∘ h) (0)

( g ∘ f ) (8)

( g ∘ f ∘ h) (16)

(h ∘ h ∘ f ) (8)

f (x) 1 −1 0 2 16 64

g (x) 8 8 5 0 −2 −12

h (x) 2 4 8 11 5 2

Let’s practice 6

The graphs of f (x) =

and g (x) =

are shown.

y 6

To evaluate g ( f (4)): a

310

First, we must ﬁnd A g (4)

f (4)

C g (1)

f (1)

f (x)

4 3

g(x)

Next, we need to ﬁnd: A g (4)

f (4)

C g (1)

f (1)

Evaluate f ( g (4)).

Mathspace Virginia SOL Algebra 2 mathspace.co

1 −4 −3 −2 −1 −1

x 1 2 3 4 5 6

The graphs of two functions, f (x) and g (x), are shown. Find: a

f ( g (1))

f ( g (4))

g ( f (2))

7 6 5 4 3 2 1

f (x)

g(x)

−5 −4 −3 −2 −1 −1 −2 −3

Given f (x) = x2 − 4x + 3, g (x) =

, and h (x) =

1 2 3 4 5

, evaluate the following:

( f ∘ g) (−6)

(h ∘ h) (11)

h ( f (4))

f (h (−4))

( g ∘ f ) (5)

( g ∘ h) (31)

h ( g (15))

h ( f ( g (2)))

Find an expression for B ∘ A.

Consider the functions A (x) = 3x2 + 4 and B (x) = 2x − 5. a

Find an expression for A (B (x)).

Does A ∘ B = B ∘ A?

Given the functions h (x) = 3x − 4 and j (x) =

, ﬁnd the composition h ( j (x)).

− 4

h ( j (x)) =

h ( j (x)) = 9x − 2

h ( j (x)) = 3 (x + 2) − 4

h ( j (x)) =

Let f (x) = x + 3 and g (x) = x2 − 1. A student was asked to ﬁnd ( g ∘ f ) (x). They provided the following work: ( g ∘ f ) (x) = (x2 − 1) ⋅ (x + 3)

Substitute the functions

Simplify the expression

= x + 3x − x − 3 Find and correct the student’s error. 12

Consider the functions f (x) = x − 5 and g (x) = 4x2. Determine an expression for each of the following: a

( g ∘ f ) (x)

( f ∘ f ) (x)

( g ∘ g) (x)

g (h (x))

For each of the following pairs of functions f and g, ﬁnd an expression for f ( g (x)). a

f (x) = 3x − 6 and g (x) =

f (x) = 3x + 12 and g (x) =

f (x) = 2x3 − 5 and g (x) =

f (x) =

Given f (x) = x3 − 1, g (x) = a

( f ∘ g) (x)

f ( g (x))

, and h (x) = b

g ( f (x))

and g (x) =

, ﬁnd each composition: c

f (h (x))

The graph shows two functions, f (x) and g (x). A student claims f ( g (0)) = 4. a

Describe the student’s error.

Find the correct value for f ( g (0)).

g(x) f (x)

3 2 1

−4 −3 −2 −1 −1

x 1

−2 −3 −4

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The graphs of two functions f (x) and g (x) are shown: a

Evaluate f ( g (2)).

Evaluate f ( g (−1)).

Determine an expression for the composition function f ( g (x)).

7 f (x) 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 g(x) −3

Use the graph of h (x) = x2 + 3x − 4 and k (x) = − x + 3 to ﬁnd the composition of functions. a

Find (k ∘ h) (x).

Find (h ∘ k) (x).

For what value(s) of x is k(h (x)) = 3.

For what value(s) of x is h(k (x)) = − 4.

1 2 3 4 5

y k(x) 5 h(x)

−5

5 −5

Consider the table shows values of f (x), the graph of g (x), and the equation of p (x). x −3 −2 −1 0 1 2 3

f (x) 9 4 1 0 1 4 9

g(x) 4 2 x −4

−2

2 −2 −4

Let p (x) = 2∣x − 3∣. Find:

312

y = ( p ∘ f ) (2)

y = ( p ∘ g) (−2)

y = g ( f (−2))

Find p( f ( g (0)))

Let q (x) be deﬁned by q (x) = p( f (x)). What is q (3)?

For what value(s) of x is f ( g (x)) = 9?

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Let’s extend our thinking 19

Find two functions f and g such that f ( g (x)) = g ( f (x)).

A water tank’s height in meters at a particular time can be modeled by the function h (t), where t is time in hours since the start of the day. The cost to purify the water in the tank per hour is given by the function C (h), where h is the height of the water in meters. Water Tank Height Over Time Height

h(t)

5 4

10 Time 2

Evaluate C (h (3)).

Interpret your answer in the context of the problem.

Height 5

A square has a side length of x units. a

Write a function for x in terms of the perimeter p.

Write a function for the area A in terms of the side length x.

Write a function for the area A in terms of perimeter p. What combination of functions was used to ﬁnd an expression for A ( p)?

Find the area of a square that has perimeter 22 units.

A conical container is being ﬁlled with water. The water level is increasing such that the radius r of the water’s surface is r = a

C(h)

Cost

Cost to Purify Water Per Hour

centimeters after t seconds.

Find an expression for the function A (r) which models the area of the water’s surface in terms of its radius.

Find an expression for the composite function ( A ∘ r) (t).

Describe the function ( A ∘ r) (t) in terms of the context.

Consider the quadratic parent function f (x) = x2. Apply the following transformations to create a new function g (x): a

First, shift f (x) upward by 3 units. Then, reﬂect the result across the x-axis. Let this new function be g (x).

Now, express g (x) as a composition of two functions, u (x) which represents the shift, and v (x) which represents the reﬂection.

Describe how the transformations of f (x) to create g (x) are reﬂected in the composition v (u (x)).

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4.08 Inverse functions After this lesson, you will be able to... • find the equation of the inverse of a function. • determine if a function’s inverse is also a function. • restrict domain to allow the inverse to be a function. • graph the inverse of a function. • verify that two functions are inverses using composition.

Inverse graphs and tables Inverse operations are operations that ‘undo’ each other - for example, addition and subtraction, or multiplication and division. We can extend this concept to ﬁnd the inverse of an entire function or relation.

Exploration Complete the table below showing the relationship between the length of the sides of a cube and the volume of the cube. Length of sides 1 2 3

Volume of the cube

Volume of the cube 1 8 27

Length of sides

Then, graph the data in the table on the same set of axes using a different color to draw each graph. 27

Answer the following questions:

How do you ﬁnd the volume of the cube given the length of sides?

How do you ﬁnd the length of sides given the volume?

Draw the line y = x, using a third color, on the same graph. How are the graphs of the two functions related with respect to the line y = x?

24 21

12 9 6 3

x 3 6 9 12 15 18 21 24 27

Inverses are useful for determining the input of a relation if the outputs are known. Consider a situation where a plane is traveling at a constant speed, and we want to know how long the plane has been ﬂying over certain distances. Rather than using the function d (t) = rt and dividing by the rate to ﬁnd the time for each of the distances, we can simply rewrite the equation as t =

314

. This is the inverse relation of d (t).

Mathspace Virginia SOL Algebra 2 mathspace.co

Inverse relation A relation that reverses the original relation. The graph of an inverse relation is the original graph reﬂected across the line y = x. Geometrically, this means that the relation and its inverse are mirror images of each other across the line y = x. y

In the given ﬁgure, we have the graph of the line f (x) and its reﬂection over the line y = x, labeled g (x).

g(x)

8 6 4

f (x)

2 −8 −6 −4 −2 −2

−4 y=x

x f (x) x g(x)

−6 −8

−6 −3

−4 −2

−2 −1

0 0

2 1

4 2

6 3

If we now create a table of values for f (x) and g (x), we can notice something about the relationship between the input and output pairs for each function.

−3 −6

−2 −4

−1 −2

0 0

1 2

2 4

3 6

The x and y coordinates for g (x) are just the swapped around coordinate pairs of f (x).

The function inverse to f (x) is denoted f − 1(x), so if (a, b) is an element of f, then (b, a) is an element of f − 1.

Example 1 Complete the tables. State which relations are inverses. x

−2

−1

−2

−16

−2

−1

f (x) = 2x3

Create a strategy To complete each of the tables, we will substitute each value of x into the given function. To determine which relations are inverses, we need to examine the inputs and outputs. The inputs and outputs of inverse relations will be swapped.

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Apply the idea x

−2

−1

f (x) = 2x3

−16

−2

−16

−2

−1

−2

−1

−4

−1

−2

−1

Looking at the tables, we can see that the inputs of f (x) are the outputs of h (x), and the outputs of f (x) are the inputs of h (x). Therefore, f (x) and h (x) are inverse relations. These are the only two functions where the outputs and inputs are swapped, so these are the only inverse relations.

Reﬂect and check Using technology to graph both functions on the same coordinate plane, we can see that these functions are reﬂections of each other across the line y = x. y

f (x)

3 2 1 −3 −2

−1

h(x) 1

−1 −2 −3

Example 2 Find the inverse of each function using the same representation. a

x 2

f (x) = (x + 1) + 5

−1 5

0 6

1 9

2 14

3 21

Create a strategy To ﬁnd the inverse of the function represented by the table of values, we need to swap the x and y-values. Then, create a new table of values representing the inverse function, f − 1 (x)

Apply the idea The table of values for the inverse function is: x f

316

−1

(x)

5 −1

6 0

9 1

14 2

Mathspace Virginia SOL Algebra 2 mathspace.co

21 3

Reﬂect and check To verify the functions represented by the table of values are inverses, we can plot the points on a coordinate plane and see if the points are mirror images over the line y = x. 22 20 18 16 14 12 10 8 6 4 2 −2 −2

y=x x 2 4 6 8 10 12 14 16 18 20 22

b y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Create a strategy To find the inverse of the function graphically, we need to swap the x and y-coordinates and reflect the original function across the line y = x. We first need to identify points that lie on the function. y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

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Apply the idea

Reﬂect and check

Reﬂecting the original function across the line y = x by swapping the x and y-coordinates, we get the following inverse function:

The inverse function is a polynomial function that we have seen before. The inverse of the original function, f (x) = , is f − 1(x) = x3.

y 8

f −1 (x)

6 4 2 −8 −6 −4 −2 −2

f (x) x 6 8

−4 y=x −6 −8

c f (x) = 9

8 7 6 5 4 3 2 1

x 1

2 3 4 5 6 7 8 9

Create a strategy To ﬁnd the inverse of the function graphically, we need to identify points on the original function and swap the x and y-coordinates to reﬂect it across the line y = x. 9

8 7 6 5 4 3 2 1

x 1

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2 3 4 5 6 7 8 9

Apply the idea

Reﬂect and check

Reﬂecting the original function across the line y = x, we get the following inverse function:

We have graphed the inverse function, but let’s now take a look at the domain and range of our function in comparison to the inverse.

For the function f (x), we have a domain of [0, ∞) and a range of [3, ∞).

The inverse function, f − 1(x), has a domain of [3, ∞) and a range of [0, ∞).

7 6 5

f (x)

3 2 1

−1

We can clearly see that the domain of our function has become the range of our inverse and the range of our function has become the domain of the inverse.

(x)

y=x x 1

2 3 4 5 6 7 8 9

We can also identify the inverse function equation, which is f − 1(x) = (x − 3)2 for x ≥ 3.

Idea summary We can verify a relation’s inverse by graphing the relations to show the two relations are reﬂected across the line y = x. We can ﬁnd the inverse by: • •

Swapping x and y in a table of values Graphically swapping the x and y coordinates

Inverse functions Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 4.08 to answer these questions. 1.

The inverse of a linear function is also always a function. What do you notice about the inverse of each of these parent functions?

How does restricting the domain make a relation become a function?

Any function can be reflected across the line y = x, but not all reflections will satisfy the definition of a function. Invertible function A function is invertible if its inverse is also a function. For an inverse of a function to be a function, the domain of the function may need to be restricted.

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A function such as f (x) = x2 does not have an inverse function. If we reﬂect f (x) = x2 across the line y = x we will get a relation that is not a function. y

We can see that the inverse relation, y2 = x or y = , does not pass the vertical line test. That is, a straight line drawn vertically through the graph has more than one intercept.

y = x2

8 6 4 2 −8 −6 −4 −2 −2 −4

x 2

x = y2

−6 −8

We can restrict the domain of the function in order for the inverse function to exist and pass the vertical line test. y

f (x) = x2

8 6

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

x 2

−4

−6

−8

Restrict to x ≥ 0, the inverse is f − 1(x) =

Restrict to x ≤ 0, the inverse is f − 1(x) =

When we reﬂect a function over the line y = x, we are effectively switching the x and y-values. To ﬁnd the inverse algebraically, we swap x and y in the equation, then solve for y to get f − 1(x). 1. Write f (x) as y

2. Swap x and y

3. Solve for y

4. Replace y with f − 1(x)

Swapping the x and y variables in a relationship will exchange the coordinates for any point on the graph. Thus, the domain and range will be swapped in an inverse relation compared to the original relation. That is, the domain of the function is the same as the range of the inverse function and the range of the function is the same as the domain of the inverse function.

Example 3 Consider the graphs of f (x), g (x) and h (x) and determine if they are invertible functions. Explain how you know.

y f (x) 8 6

g(x)

4 2 −8 −6 −4 −2 −2 −4 −6 −8

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Mathspace Virginia SOL Algebra 2 mathspace.co

x 2

h(x)

a f (x)

Create a strategy To determine if a function is invertible, we will reﬂect the graph over the line y = x, then use the vertical line test to determine if the inverse is a function.

Apply the idea • We can graph the inverse relation by reﬂecting f (x) across the line y = x

y f (x) 8 −1 f (x) 6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

y 8 6

f −1 (x)

4 2

−8 −6 −4 −2 −2

• We can then determine if the inverse relation is a function by drawing vertical lines through the function, and determining the number of times they intersect the function. • We can see that there is only one point of intersection, regardless of where we draw the vertical line.

−4 −6 −8

In addition, we can see that the inverse is a line, which indicates it is a linear function.

Reﬂect and check As the two functions are reﬂections of each other across the line y = x, we could apply the horizontal line test on f (x). If we draw a horizontal line anywhere on the graph and the horizontal line only intersects the function once, then the inverse of the function will also be a function, without any domain restrictions. We can see that, no matter where the horizontal line is drawn, there is only one point of intersection. This means the function is invertible.

y f (x)

8 6 4 2

−8 −6 −4 −2 −2

x 2

−4 −6 −8

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321

b g (x)

Create a strategy To determine if a function is invertible, we will reﬂect the graph over the line y = x, then use the vertical line test to determine if the inverse is a function.

Apply the idea • We can graph the inverse relation by reﬂecting f (x) across the line y = x

y 8 g(x)

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

y 8 g(x)

6 4 2

−8 −6 −4 −2 −2

x 2

• We can then determine if the inverse relation is a function by drawing a vertical line through the function, and determining the number of times they intersect • We can see that there are two points of intersection, meaning the inverse fails the vertical line test.

−4 −6 −8

Therefore, g (x) does not have an inverse function.

Reﬂect and check Alternatively, we could perform the horizontal line test on g (x), revealing the function does not have an inverse. y 8 g(x)

6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

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x 2

However, if g (x) had a restricted domain of either [0, ∞) or (−∞, 0], then it would have an inverse function, as shown for [0, ∞). y 8 6

g−1 (x)

4 2 −8 −6 −4 −2 −2

g(x) 2

−4 −6 −8

c h (x)

Create a strategy To determine if a function is invertible, we will reﬂect the graph over the line y = x, then use the vertical line test to determine if inverse is a function.

Apply the idea • We can graph the inverse relation by reﬂecting h (x) across the line y = x

y 8 6 4 2

−8 −6 −4 −2 −2

2 4 h(x)

−4 −6

h−1 (x)

−8 y 8 6 4 2 −8 −6 −4 −2 −2

x 2

• We can then determine if the inverse relation is a function by drawing a vertical line through the function, and determining the number of times they intersect • It appears that there might be more than one point of intersection at x = − 3, so we can use the shape of the graph to see if it belongs to a function family that we have studied.

−4 −6 −8

h−1 (x)

It appears that the inverse is a cube root function, which means there is only one y-value when x = − 3. This conﬁrms that the original function h (x) is invertible.

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Example 4 For each of the following functions: • Determine an expression for the inverse relation. • State whether or not the inverse is a function. • If the inverse is not a function, ﬁnd a restricted domain for the function under which the inverse is a function. a y = 7x − 4

Create a strategy To ﬁnd the inverse, we will swap x and y, then solve for y. We can determine whether the inverse is a function by using the vertical line test.

Apply the idea Swap x and y

Add 4 to both sides

Divide both sides by 7

Simplify

is the inverse of y = 7x − 4.

We now need to determine if y =

is a function. Graphing y =

using

technology, the relation passes the vertical line test and therefore is a function.

4 3 2 1 −9−8−7−6−5−4−3−2−1 −1

x 1 2 3 4

−2 −3 −4

b y = (x − 3)2 − 5

Create a strategy We want to swap x and y, then solve for y. We can then check whether the inverse is a function using the vertical line test.

Apply the idea Inverse: Swap x and y

Add 5 to both sides

Evaluate the square root of both sides

Add 3 to both sides

y=3±

324

is the inverse of y = (x − 3)2 − 5.

Mathspace Virginia SOL Algebra 2 mathspace.co

We now need to determine if y = 3 ± is a function. Graphing y = 3 ± that it does not pass the vertical line test and therefore is not a function.

using technology, it is clear to see

We can restrict the domain of y = (x − 3)2 − 5 to x ≥ 3, and it will have the inverse function y = 3 +

Reﬂect and check As the graph of a quadratic function does not pass the horizontal line test, no quadratic functions have inverses without domain restrictions.

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Alternatively, we could have restricted the domain to x ≤ 3, in which case the inverse would have been y = 3 − 5 4 3 2 1

y = (x − 3)2 − 5, for x ≤ 3

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

c f (x) =

x 1 2 3 4 5

+ 2 for x ≥ 3

Create a strategy Since this is written as a function, we will begin by writing f (x) as y. Then, we can swap x and y, solve for y, and determine whether the inverse is a function by using the vertical line test.

Apply the idea We begin by writing f (x) as y. So, we have y = Next, we swap x and y, resulting in x =

+ 2. + 2.

Now, we need to solve for y. x−2=

Subtract 2 from both sides

(x − 2) = y − 3

Square both sides 2

y = (x − 2) + 3

Add 3 to both sides

We must now take a look at domain and range. We know with inverse functions, our domain and range are swapped. • The domain of f (x) was [3, ∞), which means that this interval is also the range of f − 1(x). • The range of f (x) was [2, ∞), which means that this interval is also the domain of the inverse, f − 1(x). The inverse function is f − 1(x) = (x − 2)2 + 3 with a domain of [2, ∞) and a range of [3, ∞). We now need to determine if y = (x − 2)2 + 3 is a function. Graphing y = (x − 2)2 + 3 over x ≥ 3 using technology, it is clear to see that it passes the vertical line test and therefore is a function. 9 8 7 6 5 4 3 2 1 −4−3−2 −1 −1

−2 −3 −4

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x 1 2 3 4 5 6 7 8 9

Idea summary • •

In order to be inverses, the domain and range of a function must be the same as the range and domain of the inverse function. We can ﬁnd the inverse algebraically by swapping x and y and solving the equation for y.

Verify inverse functions To verify that two functions, f (x) and g(x), are inverses of each other, we can use function composition. Given f (x) and g(x), if: • f ( g(x)) = x • g( f (x)) = x Then, we can say that f (x) and g(x) are inverse functions. We must check the composition both ways and if both equal x, then the two functions are inverses. If only one of the compositions is equal to x, the functions are not necessarily inverses.

Example 5 Determine if the pair of given functions are inverses. Justify your answer. a f (x) = 4x − 2 and g(x) =

Create a strategy To determine if the given functions are inverses of each other, we can use the idea that two functions, f (x) and g(x), are inverses if f ( g(x)) = g( f (x)) = x. We will ﬁnd both f ( g(x)) and g( f (x)) and check if they are equal to x.

Apply the idea First, let’s ﬁnd f ( g(x)). Substitute g(x) into f (x)

Apply the deﬁnition of f (x)

Evaluate the multiplication

Simplify

Now, let’s ﬁnd g( f (x)). Substitute f (x) into g(x)

Apply the deﬁnition of g(x)

Simplify the numerator

Simplify further

Since f ( g(x)) = x and g( f (x)) = x, we can conclude that the given functions are inverses of each other.

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Reﬂect and check To verify our results graphically, we can use graphing software or a graphing calculator to plot the functions f (x) and g(x) on the same set of axes. If the functions are inverses, their graphs should be reﬂections of each other across the line y = x.

When graphing the functions, we can observe that they are indeed reﬂections of each other across the line y = x, conﬁrming that they are inverses.

b f (x) = 5(x + 4)3 and g(x) =

Create a strategy To determine if the given functions are inverses of each other, we will use the idea that two functions, f (x) and g(x), are inverses if f ( g(x)) = g( f (x)) = x. We will compute f ( g(x)) and g( f (x)) to check if they both simplify to x.

Apply the idea First, let’s compute f ( g(x)):

Substitute g(x) into f (x)

Combine like terms

Evaluate the exponent

Simplify

Since f ( g (x)) ≠ x, we do not need to compute g( f (x)). The functions are not inverses of each other.

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Reﬂect and check To verify our result graphically, we can use graphing software or a graphing calculator to plot the functions f (x) and g(x) on the same set of axes. Upon graphing the functions, we can see that they are not reﬂections of each other across the line y = x, indicating that they are not inverses.

c f (x) =

and g(x) = x2 + 5, x ≥ 5

Create a strategy To determine if the given functions are inverses, we need to verify if f ( g(x)) = g( f (x)) = x. By substituting the expression of g(x) into f (x) and vice versa, we can check if the resulting expressions simplify to x.

Apply the idea First, let’s ﬁnd f ( g(x)): Substitute g(x) into f (x)

Apply the deﬁnition of f (x)

Simplify the expression

Take the square root of x2

Now, let’s ﬁnd g( f (x)): Substitute f (x) into g(x)

Apply the deﬁnition of g(x)

Square the square root

Simplify the expression

Since f ( g(x)) = g( f (x)) = x, the given functions are inverses of each other.

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Reﬂect and check Let’s algebraically compute the inverse of each function and compare the results to verify if they are inverses of each other. First, let’s ﬁnd the inverse of f (x) =

. Original function

Swap x and y

Square both sides

Add 5 to both sides

The inverse of f (x) is f − 1(x) = x2 + 5. Now, let’s ﬁnd the inverse of g(x) = x2 + 5. Original function

Swap y and y

Subtract 5 from both sides

Take the square root of both sides −1

The inverse of g(x) is g (x) =

Comparing the original functions with their inverses, we can see that f (x) is the inverse of g(x) and vice versa. Therefore, the given functions are inverses of each other.

Idea summary Two functions, f (x) and g(x) are inverse functions, if both f ( g(x)) = x and g( f (x)) = x.

Practice What do you remember? 1

Determine whether each statement is true or false. a

A function is invertible if and only if its inverse is also a function.

The inverse of a function is always a function.

For a function to have an inverse, each output must correspond to exactly one input.

Describe how we can determine if a graph represents a function.

Determine whether each function has an inverse function without any domain restrictions: a

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

330

y 8 6 4

x 1 2 3 4 5

Mathspace Virginia SOL Algebra 2 mathspace.co

2 −8 −6 −4 −2 −2 −4 −6 −8

x 2

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −8

x 3 4 5 f (x) 1 2 3

6 2

7 1

x −3 −2 −1

f (x)

8 0

x f (x)

−8 −2

−1 −1

0 0

1 1

8 2

−1 0

0 1

1 0

2 −1

3 0

A(r) = − 2r2 + 5

Select all the functions that have an inverse function without any domain restriction. A E

1 2 3 4 5

Consider the function values shown in each table and determine if it is possible for an inverse function to exist without any domain restriction: a

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−6

5 4 3 2 1

f (x) = 12x − 5 x

h (x) = 3

g (x) = x4 + x3 − x − 1

m (x) = 13

p (x) = 2x3 + 7

Complete the following sentences: The graph of any relation can be reﬂected across the line ⬚ to produce a graph of its inverse relation.

This means that if a point (x, y) lies on the graph of a relation, the corresponding point on the graph of the inverse relation is ⬚. For each of the following graphs, determine if the pair of relations shown are inverse relations: a

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

x 1 2 3 4 5

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Let’s practice 8

A function f is deﬁned by f (x) = 2x − 3. Select the correct representation of the inverse function f − 1: A

A function which takes a value of x, adds 3, then divides the result by 2. The domain is all real numbers.

f − 1 (x) = 3x − 2 where x ∈ 

x f

−1

(x)

−4 1

−1 2

0 3

1 4

4 5

6 5 4 3 2 1

−5 −4 −3 −2 −1−1

1 2 3 4 5

−2 −3 −4 −5 −6

A function f is shown in the graph.

Select the correct representation of the inverse function f − 1:

8 6 4 2 −5 −4 −3 −2 −1 −2

x 1 2 3 4 5

−4 −6

x −8 −1 0 −1 f (x) −2 −1 0

1 1

8 2

y 8 6 4 2 −5 −4 −3 −2 −1 −2

x 1 2 3 4 5

−4 −6

C 10

332

f − 1 (x) =

where x ∈ 

f − 1 (x) = (x − 2)3 where x ∈ 

For each of the following functions: i

Sketch the graph of f (x).

State whether an inverse function exists. Justify your answer.

f (x) = x + 5

f (x) =

f (x) = (x − 5)3

f (x) = x3 − 3x

Mathspace Virginia SOL Algebra 2 mathspace.co

f (x) = (x − 2) (x + 3)

f (x) = x2 − 1

Consider the polynomial function f (x) = x2 − 2x + 3. a

Sketch the graph of the function.

Is the inverse a function?

Write an equation for the inverse of the function.

State whether the inverse is a function.

y = 5x

y=1−

y = (x − 7)2

y = x2 − 10x + 25

y = x + 6x + 10

y = − x + 4x + 5

Draw a graph of the inverse on the same coordinate plane: a

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

y 7 6 5 4

1 2 3 4 5

2 1 −1

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Find an expression for the inverse f − 1 (x).

For each of the following functions:

i 13

x 1 2 3 4 5

−1

x 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Two functions, h (x) and g (x), are said to be inverses when g (h (x)) = x and h ( g (x)) = x. Show that this is true for g (x) = + 5 and h (x) = 4x − 20.

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For each of the following pairs of functions f and g: i

Find an expression for f ( g (x)).

iii

State whether or not the two functions are inverses.

f (x) =

f (x) = 3x + 12 and g (x) =

f (x) = 2x3 − 5 and g (x) =

−4

Consider the function f (x) = a

and g (x) =

Find f

−1

Find the range of f

Find an expression for g ( f (x)).

f (x) = 3x − 6 and g (x) =

f (x) =

and g (x) = 2 −

f (x) =

and g (x) =

Find the domain of f − 1 (x).

−2

−3

deﬁned over [0, 4].

(x). −1

(x).

For each of the following functions: i

Sketch the function f (x) over its domain.

Sketch the function f − 1 (x) over its domain.

iii

State the domain of f − 1 (x).

State the range of f − 1 (x).

f (x) = x + 3 deﬁned over the interval [0, ∞).

f (x) = 7 − x deﬁned over the interval [2, 9].

f (x) =

f (x) = (x − 6) − 2 deﬁned over the interval [6, ∞).

f (x) = (x + 2)2 + 3 deﬁned over the interval [0, ∞).

deﬁned over the interval [0, 4).

Consider the function f (x) = (x − 4)2 − 6 on the restricted domain [4, ∞). a

Find the inverse function f − 1 (x) for f (x) on the restricted domain.

State the domain and range of f − 1 (x).

Find an appropriate restricted domain for the function f (x) = (x − 7)2 + 12 to have an inverse.

For each given function: i

Suggest an appropriate domain restriction for the function to ensure its inverse will also be a function.

Graph the inverse function using the suggested domain restriction on the same coordinate plane.

b 8

2 −8 −6 −4 −2 −2

x 2

−4

−6

−8

d 8

x 4

−8 −6 −4 −2 −2

−4

−6

−8

Mathspace Virginia SOL Algebra 2 mathspace.co

−8 −6 −4 −2 −2

−4

−8 −6 −4 −2 −2

334

For each of the following functions: i

State a domain restriction which results in an inverse that is a function.

State the inverse function f − 1 (x) on the restricted domain from part (i).

iii

State the domain and range of f − 1 (x).

f (x) = (x + 2)2 − 9

f (x) = − (x − 5)2

f (x) = x2 + 10x + 23

f (x) = x2 + 4x + 3

For each of the following functions: i

State the domain and range of f (x).

Is the inverse of f (x) a function?

iii

Find the inverse function f − 1 (x).

State the domain restriction on f − 1 (x), given that f (x) is only half a parabola.

Describe the relationship between the domain and range of f (x) and f − 1 (x).

f (x) =

is deﬁned for all x ≥ 2.

The function a

Find the inverse function f − 1 (x) and state its domain.

Verify your answer to part (a) by graphing both functions on the same coordinate plane.

Let’s extend our thinking 24

The largest domain over which the function f (x) = x2 + bx + c has an inverse is [3, ∞). The domain of the inverse function f − 1 (x) is [−2, ∞). a

Find the value of b.

Find the value of c.

Taylan is saving their money up so that they can buy a brand new keyboard for their computer and decides to put their birthday money in a bank account that accrues interest. The amount of money in Taylan’s bank account can be modeled by the function g (x) = dollar amount in the account after x days.

Find the inverse of the function, g − 1 (x).

Find how many days will it take for Taylan’s bank account to reach $499.

Interpret the meaning of the inverse function g − 1 (x).

+ 50, where g (x) is the

The function d(t) = 120 − 4.9t2 can be used to ﬁnd the distance d that an object dropped from a height of 120 m has fallen after t seconds. a

Determine whether or not the function has an inverse that is a function. Justify your answer.

Zheng rearranges the function to get

. He claims that this means the inverse function is

. State whether or not Zheng’s claim is correct. c 27

Determine how long it will take an object to fall 41.6 m when dropped from a height of 120 m.

Determine if the following statements are always, sometimes, or never true. Justify your answers. a

A function of the form f (x) = xn, where n is a positive integer, has an inverse that is a function.

A function that is strictly decreasing over its domain has an inverse that is a function.

A constant function, such as y = 2, has an inverse that is a function.

Determine a function f that is its own inverse. Prove that f (x) = f − 1 (x).

Prove that the inverse of a linear function f (x) = mx + b, where m ≠ 0, is also a linear function. Determine the slope and y-intercept of the inverse in terms of the constants m and b. 4.08 Inverse functions mathspace.co

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5 Rational Functions Big ideas • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • The properties of real numbers can be applied to many types of expressions. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 5.01 5.02 5.03 5.04 5.05 5.06

Rational parent functions Direct and inverse variation Other rational functions Multiply and divide rational expressions Add and subtract rational expressions Solve rational equations

338 365 377 396 406 417

5.01 Rational parent functions After this lesson, you will be able to... • identify the graphs of the rational function family. • write the equation of a rational function from a graph using transformations. • identify the transformations given an equation or a graph. • write equations of rational functions given a description of the transformations. • graph rational functions given an equation using transformations. • compare tables, graphs, and equations of rational functions. • identify domain, range, zeros, intercepts, increasing, decreasing, constant intervals, and end behavior. • compare the characteristics of rational functions to other functions. • find f (x) given x using a graph or equation. • identify the equations of any vertical and horizontal asymptotes of a rational function using a graph or equation. • graph the inverse of a rational function.

Characteristics of rational functions Exploration Tamar plans to bike 13 miles. 1.

Construct a table of values for t (r), the amount of time it takes Tamar to bike if she rides at a rate of 1 mph, 8 mph, 10 mph, and 15 mph.

Graph the function t (r).

What happens to the function as the values of r → 0?

What happens to the function as the values of r → ∞?

The equation for a rational parent function is given by

x y

independent variable dependent variable

As the value of one variable increases, the value of the other will decrease. A reciprocal function is a rational function that has a constant numerator. 4

The parent reciprocal function is f (x) = , and is shown on the graph.

Notice that the expression is undeﬁned if x = 0. Therefore, its domain is (−∞, 0) ∪ (0, ∞), which does not include x = 0. Also notice that there is no real value of x that could be substituted into the equation to create f (x) = 0, because 1 divided by any number will never result in 0. So, its range is (−∞, 0) ∪ (0, ∞), which does not include y = 0 as the function values never reach y = 0.

2 1 −4 −3 −2 −1

−1

−2 −3 −4

338

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x 1

For these reasons, the function f (x) =

has two asymptotes: a vertical asymptote of x = 0 (the y-axis), and a

horizontal asymptote of y = 0 (the x-axis). The parent reciprocal function has no x- or y-intercepts, due to its asymptotes. Vertical asymptote A vertical line that the graph of a function approaches as the function values head towards positive or negative inﬁnity

Examine the end behavior of the reciprocal function as the domain approaches the undeﬁned value of x = 0. As x approaches 0 from the negative side, x → 0 −, f (x) approaches −∞. As x approaches 0 from the positive side, x → 0+, f (x) approaches ∞. Horizontal asymptote A horizontal line that the graph of a function approaches as the domain values head towards positive or negative inﬁnity

Similarly, when we examine end behavior of the reciprocal function as the domain values approach positive or negative inﬁnity, we see that f (x) approaches zero. That is, as x approaches +∞, f (x) approaches 0 from above and as x approaches −∞, f (x) approaches 0 from below. The equation for another rational parent function is given by

x y

independent variable dependent variable

As the value of one variable increases, the value of the other will decrease. The graph of this function is shown: 5

4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1

x 1 2 3 4 5 6 7

−2 −3 −4 −5

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The function f (x) =

has the same two asymptotes as f (x) = : a vertical asymptote of x = 0 (the y-axis), and

a horizontal asymptote of y = 0 (the x-axis). However, the branches (or pieces) of the function are in different quadrants. This causes f (x) = of (0, ∞).

to have a range

Example 1 Consider the graphs of the functions f (x) and g (x). 8

2 −8 −6 −4 −2 −2 −4

x 2

f (x)

−6 −8

g(x)

−12 −10 −8 −6 −4 −2 −2

x 2

a Identify the function family to which each function belongs.

Create a strategy Recall the shape and characteristics of the parent function of each function family. We have previously seen linear, radical, exponential, absolute value, rational, polynomial, and logarithhmic functions. Characteristics to consider include: domain, range, x-intercept, y-intercept, absolute and relative extrema, asymptotes, and increasing/ decreasing intervals

Apply the idea For f (x), both logarithmic and square root functions have a minimum or maximum for their domain. Square root functions also have a minimum or maximum for their range. Since f (x), has no restriction on the range, it must belong to the logarithmic function family. For g (x), only rational functions have two intervals on either side of an asymptote. Since both intervals are above y = 0, it must belong to the rational function family with a parent function y =

b Compare the asymptotes of f (x) and g (x).

Create a strategy Recall that a function can have both horizontal or vertical asymptotes.

Apply the idea In part (a), we identiﬁed f (x) as a logarithmic function. Since there is no restriction on the range, there is no horizontal asymptote. The vertical asymptote, drawn on the graph, is at x = 0.

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Mathspace Virginia SOL Algebra 2 mathspace.co

6 4 2 −8 −6 −4 −2 −2 −4

x 2

f (x)

−6 −8

In part (a), we identiﬁed g (x) as rational, so we must consider horizontal and veritcal asymptotes. 14

12 10 8 6 4 g(x)

−12 −10 −8 −6 −4 −2 −2

x 2

The vertical asymptote, drawn on the graph, is at x = − 4. The horizontal asymptote is at y = 0.

c Describe the end behavior of each function.

Create a strategy We want to determine what happens to the y-values when the x-values are very small and very large. Consider how the asymptotes affect end behavior.

Apply the idea First, let’s consider f (x). Since f (x) is the logarithmic parent function, as x → ∞, y → ∞. For f (x), the domain is restricted to x > 0, so as x → 0 +, y → −∞. Next, let’s consider the end behavior of g (x). As x → ∞, y → 0. As x → −∞, y → 0.

Reﬂect and check When determining end behavior, it is important to consider the domain and the direction. For f (x), we cannot discuss the behavior of x → −∞ because the domain is restricted to x > 0. We also cannot discuss the behavior of the function as x → 0− because the function is restricted to values of x greater than 0.

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Example 2 The relationship between the current, C, (in amperes) and resistance, R, (in ohms) in an electrical circuit is given by:

where the voltage provided to the circuit is 200 V. a Complete the table. R

C (R)

25 8

Create a strategy

Apply the idea

We substitute each value of R into the equation

Now,

C (R) =

If R = 5, then C (5) =

= 40.

If R = 10, then C (10) =

= 20.

If R = 20, then C (20) =

= 10.

If R = 25, then C (25) =

= 8.

If R = 40, then C (40) =

= 5.

to ﬁnd the value of C.

Therefore, the complete table of values is: R C (R)

5 40

10 20

20 10

25 8

40 5

b Sketch the relationship between the current and resistance.

Create a strategy We assign the horizontal axis for variable R and the vertical axis for variable C. We will plot the coordinates (R, C) on the coordinate plane. The points are: (5, 40), (10, 20), (20, 10), (25, 8) and (40, 5) Then, we can graph the function

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Apply the idea 50 45 40 35 30 25 20 15 10 5

Reﬂect and check

C(R)

It’s important to consider what representation is best for the types of questions we want to answer. The table clearly shows us exact values of the function. In the graph, we can see a few notable things that were less clear in the table of values, such as possible asymptotes or end behavior. Depending on what information we are looking for, one representation may be preferred over another.

R 5 10 15 20 25 30 35 40 45 50

c Specify any asymptotes and the restrictions on the domain of the function.

Create a strategy

Apply the idea

Use the graph of the relationship between the current and resistance to determine the asymptotes, then use the asymptotes, equation of the function, and context to determine the domain.

We can see that the values of the function approach ∞ as the value of R → 0. There is a vertical asymptote at R = 0. The values of the function approach 0 as the value of R → +∞, and they will never reach 0. There is a horizontal asymptote at C (R) = 0. At the vertical asymptote, R = 0, the function is undeﬁned since we cannot evaluate C (0) = , so we will exclude R = 0 from the domain of the function.

Example 3 Consider the function, f (x), shown.

6 4 2 −14−12−10 −8 −6 −4 −2 −2

x 2 4 6 8

−4 −6 −8 −10 −12 −14

a Identify the domain and range.

Create a strategy The domain represents the x-values of the function, and the range represents the y-values of the function. Recall that the graph of a rational functions can have asymptotes and discontinuities.

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343

Apply the idea From the graph, we can see that x is deﬁned for all values on either side of the vertical asymptote at x = − 6. Therefore, the domain is (−∞, −6) ∪ (−6, ∞). Similarly, we can see that y is deﬁned for all values on either side of the horizontal asymptote at y = − 3. Therefore, the range is (−∞, −3) ∪ (−3, ∞).

Reﬂect and check We can also represent domain and range using set notation: Domain: {x∣x ≠ −6} or {x∣x < −6 or x > −6} Range: {y∣y ≠ −3} or {y∣y < −3 or y > −3} We can also represent domain and range using inequalities: Domain: All x such that x ≠ −6 Range: All y such that y ≠ −3 These different representations of domain and range can be used interchangeably.

b Identify the increasing and decreasing intervals.

Create a strategy We want to determine what happens to the y-values for different values of x. We must consider points where the direction of the function could change.

Apply the idea Notice our graph is separated into two intervals. Starting from the left side of the ﬁrst interval, we can see y is decreasing until we get to the vertical asymptote at x = − 6. Continuing from left to right on the second interval, we can see y is decreasing for the remainder of the domain. Therefore, f (x) has no increasing intervals and is decreasing on the intervals (−∞, −6) ∪ (−6, ∞).

Reﬂect and check Note that even though the function is decreasing at every point in its domain, the domain is formed from two disconnected intervals (which are separated by the asymptote). So, we cannot say that it has only one decreasing region. Any rational function of the form f (x) =

+ k will only increase or decrease, not both, depending on the sign of a.

c Identify the zero(s) of the function.

Create a strategy The zeros of a function occur when f (x) = 0. These are also the location of the x-intercepts.

Apply the idea There is only one zero at (−4, 0) since this is the only point where f (x) touches the x-axis.

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Example 4 Consider the function, f (x), shown.

12 11 10 9 8 7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2

x 1 2 3 4 5 6 7

a Identify the increasing and decreasing intervals.

Create a strategy This graph has two separate intervals. For each interval, move from left to right across the graph and determine whether the y-values are increasing or decreasing.

Apply the idea Starting from the left side of the graph for the ﬁrst interval (−∞, 0), we can see y is increasing until we get to the vertical asymptote at x = 0. Continuing from left to right on the second interval (0, ∞), we can see y is decreasing for the remainder of the domain. Therefore, f (x) is increasing on the interval (−∞, 0) and decreasing on the interval (0, ∞).

Reﬂect and check The behavior of increasing and decreasing intervals can help us determine whether a rational function has a parent function f (x) =

or f (x) =

From the previous example, notice that functions of the form f (x) = intervals, but functions of the form f (x) =

+ k have only increasing or decreasing

+ k have both increasing and decreasing intervals.

b Identify the intercepts of the function.

Create a strategy The x-intercept is when the line touches or goes through the x-axis and the y-intercept is when the line touches or goes through the y-axis.

Apply the idea This function has no intercepts. It crossed neither the x-axis nor the y-axis.

Reﬂect and check Given the vertical asymptote at x = 0, the function gets increasingly close to 0 but never actually reaches the y-axis. The horizontal asymptote at y = 2 means that the function can never reach y-values below 2.

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c Describe the end behavior of the function as x → −∞ and as x → ∞.

Create a strategy We want to determine what happens to the y-values when the x-values get increasingly small or increasingly large.

Apply the idea

Reﬂect and check

As x gets very small, y approaches the horizontal asymptote y = 2. So, as x → −∞, y → 2.

Understanding the end behavior helps in sketching the graph’s “tails” and gives a sense of the function’s overall end shape.

As x gets very large, y again approaches the horizontal asymptote y = 2. So, as x → ∞, y → 2.

Idea summary There are many types of rational functions. Two of the most common parent functions are f (x) = f (x) =

and

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.01 to answer these questions. 1.

How does each slider change the parent function f (x) = ?

How does each slider change the parent function f (x) =

For each parent function, which transformation slider determines the vertical asymptote? The horizontal asymptote?

Transformations of rational functions Transformations to the parent function f (x) =

create a family of rational functions, given by the following equation:

a Stretch ∣a∣ > 1, shrink 0 < ∣a∣ < 1, reﬂection across the x-axis when a < 0 h Horizontal translation k Vertical translation, horizontal asymptote at y = k The same types of transformation can occur for the rational parent function y =

a Stretch ∣a∣ > 1, shrink 0 < ∣a∣ < 1, reﬂection across the x-axis when a < 0 h Horizontal translation k Vertical translation, horizontal asymptote at y = k

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Transforming a function affects not only its equation but also its characteristics. Consider these 2 transformations of rational parent functions: The vertical asymptote of f (x) is at x = 0 and the vertical asymptote of g (x) is at x = 4.

9 y 8 7 6 5 4 3 2 1 −5−4−3−2 −1 −2 −3 −4 −5 −6 −7 −8 −9

The domain of g (x) is changed due to the shifted asymptote. Domain of f (x) = : {x∣x ≠ 0} x

1 2 3 4 5 6 7 8 9 10 11 1213

: {x∣x ≠ 4}

Both functions have two decreasing intervals on either side of their respective vertical asymptote.

Compared to the parent function g (x) has been dilated, reﬂected across the x-axis, and shifted up. These changes will affect the range.

9 y 8 7 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

Domain of g (x) =

1 2 3 4 5 6 7 8 9 10 11

Range of f (x) =

: {f (x) ∣ f (x) > 0}

Range of g (x) =

+ 4: {g (x) | g (x) < 4}

For end behavior, as x approaches ± ∞, f (x) approaches 0, while g (x) approaches 4. As x approaches 0, f (x) approaches +∞ while g (x) approaches −∞.

Example 5 Consider the function g (x) =

a What is the transformation of the parent function f (x) = ?

Create a strategy We can use the equation of the function to identify the transformation of the function, comparing the equation to f (x) =

+ k.

Apply the idea We know there is no stretch, compression, or reﬂection because the numerator for g (x) remains the same as f (x) = . The value of h = 1 shifts the function horizontally, so we can state a horizontal translation 1 unit to the right. There was no vertical translation as k = 0. Therefore, g (x) is has been translated 1 unit to the right.

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347

b Complete the table of values. −1

g (x)

Create a strategy Substitute each of the values of x in the equation to solve for the values of g (x).

Apply the idea Evaluating the expression x g (x)

−1

at each value of x, we get:

0 −1

2 −2

c Sketch a graph of the function.

Create a strategy We can translate the parent function f (x) = draw the curve.

one unit to the right, using the points from the table of values to help

Apply the idea Based on the equation, we can see there will be a vertical asymptote at x = 1, where the function is undeﬁned. The function can also be written as g (x) =

+ 0, so the horizontal asymptote is at g (x) = 0. 4

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

Looking at the graph, and in particular the location of the vertical asymptote, we can see that the function g (x) has been translated 1 unit to the right from the parent function f (x) = .

Reflect and check The points in the table of values indicate that the vertical asymptote is between x = 0.5 and x = 1.5, as the values change from decreasing negative values to decreasing positive values. We can conﬁrm that the asymptote is the line x = 1 by looking at the equation of the function, g (x) =

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, which is undeﬁned at x = 1.

Example 6 Consider the function shown in the graph: 4

3 2 1 −6 −5 −4 −3 −2 −1 −1

x 1

−2 −3 −4

a Describe the transformation(s) used to get from the graph of y =

to the graph of this function.

Create a strategy We should check for each type of transformation: translations (vertical and/or horizontal), vertical stretch or compression, and reﬂection. It may help to add the graph of y =

to the same coordinate plane: 4

3 2 1 −6 −5 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Apply the idea We can see that the curve lies in the upper-right and lower-left sections, relative to the asymptotes, which is the same as that of y = , so no reﬂections have occurred. The vertical asymptote of the graph is x = − 3, which is 3 units to the left of the vertical asymptote of y = . Their horizontal asymptotes are both the same (along the x-axis). So there has been a horizontal translation of 3 units to the left, and no vertical translation.

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349

Shifting the parent function to the left 3 units, we can compare our graph to determine if any dilation has occurred: 6 5 4 3 2 1 −9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

x 1

Notice each y-value has been doubled to get our function. So the function has been dilated by a scale factor of 2.

Reflect and check Reflections and translations of rational functions are relatively straightforward to see by looking at the graphs and the asymptotes. Vertical stretches and compressions can be less obvious, however, so make sure to check multiple points to confirm the vertical stretch.

b Determine an equation for the function shown in the graph.

Create a strategy We can use the transformations that we described in part (a) and apply them to the function y = for the function shown.

to get an equation

Starting with y = , we can apply the transformations one at a time: • Vertical stretch by a factor of 2 • Horizontal translation of 3 units to the left

Apply the idea We can also identify the values of a, h, and k in the transformation form f (x) =

+ k.

A dilation by a scale factor of 2 means a = 2. A horizontal translation 3 units left means h = − 3. Since we do not have any vertical translation, k = 0. Substituting those into the equation, we have:

Reflect and check It is important to note that the order of transformations matters in some cases. For example, applying a vertical translation and then a reflection across the x-axis will be different to reflecting first and then applying the same vertical translation. In general, we apply reflections and dilations first, then translations.

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Example 7 Consider the function y =

a Sketch a graph of the function.

Create a strategy For the parent function f (x) =

, this function can be expressed as y = − 5 ⋅ f (x − 3), which is a horizontal translation of

3 units right, a dilation with a scale factor of 5, and a reﬂection across the x-axis.

Apply the idea The graph of f (x) =

is shown on the same coordinate plane: 9 8 7 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7 8 9

b What are the equations of the asymptotes of the function?

Create a strategy We know that this function has been dilated with a scale factor of 5, reﬂected across the x-axis, and horizontally translated 3 units right. The dilation and reﬂection do not affect the asymptotes; only translations can affect asymptotes.

Apply the idea Since there was no vertical translation, the horizontal asymptote is the same as the parent function. The vertical asymptote has translated 3 units right, which we can see in the graph. The vertical asymptote has the equation x = 3 and the horizontal asymptote has the equation y = 0.

Reflect and check We can use the equation to find that asymptotes as well. Given that there is no x-value that makes the equation equal to 0, the horizontal asymptote occurs at y = k and since k = 0, we can confirm that the horizontal asymptote occurs at y = 0. Since x = 3 is a zero of the denominator, not the numerator, we can also confirm that the vertical asymptote occurs at x = 3.

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351

c Using interval notation, what is the domain and range of the function?

Create a strategy

Apply the idea

We can see that the domain of this rational function will be all values of x except for the value at the vertical asymptote, where the function is undeﬁned.

The domain of this function is all real values of x except for 3, and the range is all real values of y less than 0. We can express this using interval notation as • Domain: (−∞, 3) ∪ (3, ∞) • Range: (−∞, 0)

The function has been reﬂected, so the range will be all negative values of y.

Example 8 Consider the function f (x) =

− 2.

a Graph f (x) using transformations of the parent function.

Create a strategy Let’s begin by making a table of values for the parent function, y = : x

−6

−3

−1

Then, we will identify and apply the transformations that have occurred to get the given function.

Apply the idea The transformations from the parent function are a vertical stretch with a scale factor of 3 and a vertical shift 2 units down. The scale factor multiplies to the y-values and the translation subtracts from the y-values, in that order. We can apply these transformations to create a new table of values for f (x): x f (x)

−6

−3

−1

−3

−5

−1

As there has been no horizontal translation, the vertical asymptote will remain at x = 0. Due to the vertical translation, the horizontal asymptote will be at y = − 2. Now, we can graph the function: y 8 6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

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x 2

b Graph the inverse function f − 1 (x).

Create a strategy To graph the inverse of f (x), reﬂect the graph of f (x) over the line y = x. We can use the table of values from part (a) and interchange the inputs and outputs to create a table of values for the inverse.

Apply the idea We can set up a table where f − 1 (x) is reﬂected across the line y = x. When a coordinate pair is reﬂected across the line y = x, the x and y values are switched. −6

−3

−1

f (x)

−3

−5

−1

−3

−5

−3

−1

(x)

−6

Use the table of values to graph f

−1 3 −1

(x). 7 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

Reﬂect and check There are some special relationships between certain functions and their inverse. The inverse of

is also .

If we attempt to ﬁnd the inverse of x

−2

f (x)

−1

, we will see it is not a function. Consider the table of values for f (x) =

Switching the coordinates, we can create a table of values for f − 1 (x): x f − 1 (x)

−2

−1

However, now the same x-values map to different y-values, and f − 1 (x) is not a function.

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353

Idea summary Use the parent function to help determine transformations on f (x) = :

a Stretch ∣a∣ > 1, shrink 0 < ∣a∣ < 1, reﬂection across the x-axis when a < 0 h Horizontal translation k Vertical translation, horizontal asymptote at y = k A vertical asymptote will occur at x = n where x = n is a zero of the denominator. The parent function can also be used to help determine transformations on f (x) =

Practice What do you remember? 1

Fill in the blanks.

The graph of a rational function of the form y = horizontal asymptote y = ⬚.

Match each of the following functions to the correct equation for their asymptote(s). Select all that apply. i

y=0

x=0

iii

y=4

x=4

vii x = − 4

x = −2

viii y = − 4 y 12

2 −8 −6 −4 −2 −2

y = −2

x 2

−4 −6 −8

354

+ k, where a ≠ 0, has a vertical asymptote x = ⬚ and a

Mathspace Virginia SOL Algebra 2 mathspace.co

6 4 2 −8 −6 −4 −2 −2

x 2

y 8

6 2 −8 −6 −4 −2 −2

x 2

−4

−6 −8

Match the following graphs with their parent functions: i a

iii

4 2

x 2

−8 −6 −4 −2 −2

−4

−6

−8

d 10

x 2

8 10

−4

x 2

−6 −4 −2 −2

iv 8

−8 −6 −4 −2 −2

y = bx

−8 −6 −4 −2 −2

−2

x 2

8 10 12 14

−4 −6

Consider the graph of f (x) and g (x) = kf (x): a

Determine the type of transformation applied to f (x) to obtain g (x).

Determine the value of k.

y g(x)

3 2 1 −4 −3 −2 −1 −1

f (x) 1

x 3

−2 −3 −4

5.01 Rational parent functions mathspace.co

355

Determine the transformations of a

that result in each function:

Let’s practice 6

Consider the graph of y = . a

Identify the equations of the horizontal and vertical asymptotes.

Write the domain and range.

Identify the quadrants in which the graph lies.

Identify the intervals where the function is increasing, decreasing, or neither.

4 3 2 1

−4 −3 −2 −1 −1

Describe the end behavior as x → ∞ and as x → −∞.

−2 −3 −4

Consider the function y = a

Copy and complete the table of values. x

−2

−1

y b

undefined

Select the graph of the rational function. A

1 −4 −3 −2 −1 −1

−3

−3 −4

356

−2

−4

−4 −3 −2 −1 −1

−2

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Identify the equations of the horizontal and vertical asymptotes.

State the domain and range.

Mathspace Virginia SOL Algebra 2 mathspace.co

x 1

Identify the quadrants in which the graph lies.

Identify the intervals where the function in increasing, decreasing, or neither.

Determine the end behavior as x → ∞ and as x → −∞.

Consider the function y = a

Determine the transformation that occurs between the graph of the function and y = .

Select the graph of the rational function. A

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

The graph of the inverse function y − 1 is shown. What do you notice about this graph and the graph y =

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

x 1 2 3 4 5

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

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357

Graph the inverse of each function. a

−9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

−9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13

x 1 2 3 4 5 6 7 8 9

Use transformations to graph the function.

Find the domain.

iii

Find the range.

Determine the increasing and decreasing intervals.

Find the intercepts.

x 1 2 3 4 5 6 7 8 9

Consider each table of values that represents a transformation of y = . i

Graph the function from each table.

Determine the transformation of the graph of y =

iii

Write an equation for f (x).

−3

−2

f (x) b

−7

−6

f (x) c

−3

f (x)

−1

−3

f (x)

358

5 4 3 2 1

For each of the following functions:

a 11

9 8 7 6 5 4 3 2 1

−2

−1

undefined

−5

−4

−3

−1

undefined

−1

−3

undefined

−1

undefined

−1

Mathspace Virginia SOL Algebra 2 mathspace.co

necessary to achieve each function.

−2

−1

3 1

For each function: i

Describe the transformations from y =

to f (x).

Draw the graph of f (x).

iii

State the domain of f (x) using interval notation.

State the range of f (x) using interval notation.

Identify where f (x) is increasing and where it is decreasing.

State the x- and y-intercepts.

f (x) =

Consider the graph of f (x). a

Find the value of f (2).

Find the value of x when f (x) = 3.

f (x) = 13 12 11 10 9 8 7 6 5 4 3 2 1

−9−8−7−6−5−4−3−2 −1 −2 −3 −4 −5

x 1 2 3 4 5 6 7 8 9

Ken is studying the intensity of light from a streetlamp at various 14 intensity distances. The relationship between the intensity of the light and the 13 distance from the streetlamp is shown in the graph. 12 What is the value of f (x) when x = 2? Explain what that point means in context.

SOL

Fill in the blanks with the correct choice. The function f (x) = the graph.

is transformed to create g (x) as shown in

Determine which transformations to f (x) produced g (x). a

First, f (x) was ⬚.

A shifted up 4 units B shifted down 4 units C shifted left 4 unit` D shifted right 4 unit b

The result was then ⬚.

A reflected across y-axis. B reflected across x-axis. C reflected across y = x.

11 10 9 8 7 6 5 4 3 2 1

distance (ft) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 g(x)

8 7 6 5 4 3 2 1

−8−7−6−5−4−3−2−1 −1 (−3, −1) −2 −3 −4 −5 −6 −7 −8

x 1 2 3 4 5 6 7 8

D reﬂected across y = − x. 5.01 Rational parent functions mathspace.co

359

SOL

Fill in the blanks with the correct choice. The function f (x) = the graph.

8 7 6 5 4 3 2 1

is transformed to create g (x) as shown in

Determine which transformations to f (x) produced g (x). a

First, f (x) was ⬚.

A shifted up 3 units

−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8

B shifted down 3 units C shifted up 1 unit D shifted down 1 unit b

The result was then ⬚.

A reﬂected across y-axis.

x 1 2 3 4 5 6 7 8 g(x)

(1, −4)

B reflected across x-axis. C reflected across y = x. D reflected across y = − x. 17

Consider the following graph of a rational function, f (x).

Select the equation for f (x). A

f (x) =

3 2 1 −3 −2

−1

1 −1 −2 −3

Consider each graph: i

Determine the transformations used to transform y =

Write the equation represented by the graph.

y 8 7 6 5 4 3 2 1

−2 −1 −1 −2

360

x 1 2 3 4 5 6 7 8

Mathspace Virginia SOL Algebra 2 mathspace.co

into the given graph.

2 1 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

y x 1 2 3 4 5 6 7 8

y 5 4 3 2 1

−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

SOL

10 8 6 4 (−1, 2) 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

1 2 3 4

2 4 6 8 10 (2, −1)

Which function best represents this graph?

y 8

f (x) =

−4 −2 −2

f (x) =

−4

6 4 2

x 2

8 10

−6 −8

Select the graph that represents f (x) = A

y 10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

x 2 4 6 8 10

10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

2 4 6 8 10

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

+ 3.

10 8 6 4 2 −10−8 −6 −4 −2 −2 −4 −6 −8 −10

2 4 6 8 10

x 2 4 6 8 10

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361

Arik is asked to draw the graph of a rational function that is decreasing on both parts of its domain, with asymptotes of x = 1 and y = 2. He drew the given graph.

7 6 5 4 3 2 1

Identify and correct his error.

−3 −2 −1 −1 −2 −3

x 1 2 3 4 5 6 7

Describe and correct the error in identifying the asymptotes of the function: Equation: y =

Marty claims they found the asymptotes. Are they correct? Vertical asymptote: x = 6 Horizontal asymptote: y = 7 23

Consider each function: i

Write the equation of the vertical asymptote.

iii

Identify the zeros of the function. b

a 24

Consider the graphs of f (x) =

Write the equation of the horizontal asymptote.

and g (x).

10 8 6 g(x) 4 2

Select all of the following statements that are true about f (x) and g (x). A

f (x) has no, x-intercept or y-intercept.

Both f (x) and g (x) have a vertical asymptote at x = 0.

Both f (x) and g (x) share the parent function .

Both f (x) and g (x) approach 0 as x → ∞.

The function f (x) is displayed in the given table. For each of the following functions: i

Given the table of values for f (x), graph g (x). x f (x)

362

−12−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

−4

−2

−1

g (x) = f (x) − 2

Mathspace Virginia SOL Algebra 2 mathspace.co

Compare f (x) and g (x).

g (x) = − 2f (x)

x 2 4 6

Given the functions f (x) and g (x): −6

−2

f (x) a

SOL

undefined

6 4 2

Which function has the larger y-intercept?

Compare the vertical asymptotes of the two functions.

Compare the range of the two functions.

Compare the end behavior of the two functions.

Compare the parent functions of the two functions.

−8 −6 −4 −2 −2

x 2

−4

g(x)

−6 −8

Select the correct answers. Identify two functions with the same range as f (x) =

g (x) = x2 + 3x − 4

m (x) = 4 + 3∣x∣

+ 4.

h (x) = x3 − 4

j (x) = 2x + 4

k (x) =

−4

Consider the following functions. a

Which function is increasing over its entire domain?

Which function has no y-intercept?

Which function has a zero at

Which function approaches ∞ as x → ∞?

Let’s extend our thinking 29

Without graphing, determine the behavior of each rational function as: i

As x tends to 0 from the right, x → 0 +

iii

As x tends to positive inﬁnity, x → ∞

As x tends to 0 from the left, x → 0 −

Determine whether or not a rational function always has a vertical asymptote. Explain your conclusion.

Write a rational function whose graph has the vertical asymptote x = 6 and the horizontal asymptote y = − 9.

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Liliana is designing a slide for a new playground. She is playing around with some graphical models of the side view. Each unit on the graph represents 1 foot. y

1 x 1

Compare the two models.

Explain which model you would recommend. Justify your choice.

x 1

Ella is driving down the freeway to her home. To drive safely, the speed limit is no more than 80 mph and no less than 60 mph. Ella is 30 miles from home and has already been driving for 0.4 hours. a

The relationship between speed, distance, and time is: time = Write an equation relating y, the total time of Ella’s drive in hours, and x, Ella’s driving speed on the freeway.

Write the domain of the function using interval notation.

Write the range of the function using interval notation.

Determine the number of minutes Ella’s drive home will be if she drives at 68 mph on the freeway.

Guido is organizing a yearbook order this year for his soccer club. There is a ﬁxed fee of $400 plus a cost of $25 per yearbook. The ﬁxed fee is evenly divided across everyone who orders a yearbook. a

There is a minimum purchase of 5 yearbooks. Draw the graph which represents this context.

If the yearbook committee wants to get the price per yearbook down to $30, determine how many people would need to purchase a yearbook.

Explain why the cost will always be higher than $25.

A yoga studio offers classes on a pay what you can model. A third of the students paid $20 for the class, a third of the students paid $7 for the class, and a third of the students made various donations totaling $55. Determine the lower limit of the average amount paid per student for very large classes. Justify your answer.

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5.02 Direct and inverse variation After this lesson, you will be able to... • use a table of values to determine when two variables are directly proportional, inversely proportional, or neither. • write an equation to represent a direct or inverse variation. • create a graph to represent a direct or inverse variation.

Direct variation A direct variation represents a proportional relationship between two quantities. This means that one variable is always a constant multiple of the other. For example, if you earn $18 per hour, your earnings are directly proportional to the number of hours worked because earnings = 18 ⋅ hours worked. We would say hours worked and money earned are directly proportional. The constant of proportionality (k), is the ratio of the dependent variable to the independent variable. So k = . We can also refer to this as the constant of variation. In the hourly earnings example, the constant would be 18. A direct variation can be written in the form:

y = kx k

constant of proportionality (k ≠ 0)

A very common misconception is that two variables are directly proportional if one increases as the other increases. This is not the case. We can only say that two variables are directly proportional if the ratio between the variables stays constant. In other words, both variables increase or decrease at a constant rate. If we graph a direct variation, we will see a linear graph (straight line) that passes through the origin, (0, 0). The graph shows a direct variation. We can see this creates a linear graph, where A is directly proportional to B.

5 4 A

3 2 1 0

The ratios explain why A and B are directly proportional. • First, we can see that this is a straight line that passes through the origin. • Next, looking at the point (1, 1), we can see the ratio of A to B, is constant for both points. • If we continued this for the point (3, 3), (4, 4) or even (4.5, 4.5) we will see the ratio is always equivalent to 1 : 1.

5 4 A

3 2:2=1:1

2 1:1

1 0

The value of k can be any real number, including fractions and negative numbers.

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4 3

k=2

2 1 −5 −4 −3 −2 −1

x 1

−1

−2 −3

−1

−4 −5

y=2⋅x

1 −4

−3

−2

−1

⋅x

x 1

−1

2 3 k = −1

−4

−3

−2

−1

x 1

−1

−2

−3

−4

y = −x

Example 1 Consider P = 90t a Find the constant of proportionality.

Create a strategy

Apply the idea

Use the equations of the form y = kx, where k is the constant of proportionality.

The constant of proportionality in P = 90t is 90.

b Find the value of P when t = 2.

Create a strategy Substitute the given value into the equation.

Apply the idea P = 90 ⋅ 2 = 180

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Substitute t = 2 Evaluate

Option C: So, k = 5 So, k = 5 So, k = 5 So, k = 5 Therefore, option C is directly proportional since the value of k is constant. Option D: So, k = 100 So, k = 15 Therefore, option D is not directly proportional since the value of k is not constant.

Reflect and check Notice that we did not calculate the value of for every pair of values in all of the tables. As soon as we identify a k value that is different from the others we have enough information to determine that the relationship is not directly proportional.

Example 4 Ivan paints 10 plates every 6 hours. a Write an equation to represent this situation.

Create a strategy Remember that for every 6 hours that passes 10 additional plates will be painted. To define the variables let y represent the total number of plates painted and x represent the total number of hours.

Apply the idea First, we can find the constant of proportionality, k, by finding the number of plates painted every hour.

Since proportional equations can be represented by the equation y = kx, we have:

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b Use your equation to plot this situation on a coordinate plane.

Create a strategy

Apply the idea

Remember the constant of proportionality represents the ratio . So, for every 3 units over, we go 5 units up.

number of plates 20 15 6 : 10

10 5

hours 5

Idea summary A direct variation can be written in the form:

y = kx k is the constant of proportionality, or constant of variation (and k ≠ 0) The graph of all points describing a direct variation is a straight line passing through the origin.

Inverse variation Now that we know about direct variation we will look at inverse variation. Inverse variation means that as one amount increases the other amount decreases. For example, speed and travel time are inversely proportional because the faster you go, the shorter your travel time. We express these kinds of inversely proportional relationships generally in the form:

constant of proportionality (k ≠ 0)

While the graph of a direct variation is linear, we can see the graph of an inverse variation is nonlinear. y

5 −20−15 −10 −5 −5

x 5 10 15 20

−10

5 −20−15 −10 −5 −5 −10

−15

−20

The graph of direct variation is linear.

x 5 10 15 20

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Inverse variation is a subset of the rational function family with parent function y = . Different constants of proportionality represent dilations. A negative constant of proportionality applys a reﬂection.

Example 5 Consider a Find the constant of proportionality.

Create a strategy

Apply the idea

Use the equations of the form y = , where k is the constant of proportionality.

The constant of proportionality in s =

is 375.

b Find the value of s when t = 6. Give your answer as an exact value.

Create a strategy

Apply the idea

Substitute the given value into the equation.

Substitute t = 6

Simplify

c Find the value of s when t = 12. Give your answer as an exact value.

Create a strategy

Apply the idea

Substitute the given value into the equation.

Substitute t = 12

Simplify

Example 6 Consider the table of values. x y

1 120

2 60

3 40

4 30

5 24

a Determine whether the table of values could represent an inverse variation between x and y.

Create a strategy We determine whether the two variable quantities can be represented by the equation for the inverse variation. If the product of the two variables is always equal to the constant of variation, then the table of values represents an inverse variation.

Apply the idea Note that the two quantities have inverse variation if they can be represented by the equation y = constant of variation and the variable quantities are x and y.

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where k is the

Observe that y =

implies k = xy. So we multiply the values of x and y to ﬁnd k.

If the product of the two variables is always equal to the constant of variation k, then the table of values represents an inverse variation. Now, If x = 1 and y = 120, then xy = 1 ⋅ 120 = 120. If x = 2 and y = 60, then xy = 2 ⋅ 60 = 120. If x = 3 and y = 40, then xy = 3 ⋅ 40 = 120. If x = 4 and y = 30, then xy = 4 ⋅ 30 = 120. If x = 5 and y = 24, then xy = 5 ⋅ 24 = 120. This means that k = 120 since xy = 120 for all (x, y). Therefore, the table of values represents an inverse variation between x and y.

b Write a function relating y and x, given the table of values.

Create a strategy

Apply the idea

We simply substitute the obtained value of the constant variation in part (a) to the equation for the inverse of variation.

From part (a), we obtained the constant variation k = 120. Substituting k = 120 to the equation y = , we get

Therefore, the table of values is represented by the equation y =

c Describe the behavior of the function as x → 0 from the right, and describe the behavior of the function as x → ∞.

Create a strategy Use the table of values and graph the function using technology to determine the behavior of the function.

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Apply the idea As x → 0 from the right, the function goes to larger and larger values. The rational function is undeﬁned at 0 so 0 is excluded from the domain. As x → ∞, the function gets closer to 0.

Example 7 Is the variation relating the distance between two locations on a map and the actual distance between the two locations an example of a direct variation or an inverse variation?

Create a strategy The scale of a map is deﬁned as the ratio of a single unit of distance on the map to the corresponding distance in real life. Maps are created with a constant multiplier between real life distances and map distances. Think about whether the distance between two locations on the map increases or decreases when the actual distance between the two locations increases.

Apply the idea

Reﬂect and check

Since the ratio between the map distance and real life distance is constant, the relationship between map and real life distances represents direct variation.

There are also real life examples of inverse variation. The distance between two locations and the time taken to travel between those locations at a constant speed represents an inverse variation. The constant of proportionality, or ratio of distance to time, represents the speed of travel.

Idea summary We express inversely proportional relationships in the form:

is the constant of proportionality (k ≠ 0)

Practice What do you remember? 1

Describe a direct variation.

Describe an inverse variation.

Determine whether each statement is an example of a direct or an inverse variation.

372

The distance between two locations on a map and the actual distance between the two locations.

The number of workers hired to build a house and the time required to build the house.

The time it takes an ice cube to melt in water and the temperature of the water.

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Find whether each table represents a directly proportional relationship between x and y: a

x 1 3 5 7 y 50 40 30 20

x y

1 5

2 20

3 45

4 80

x 1 2 3 4 y 5 10 15 20

x y

1 100

5 75

6 50

20 25

Find whether each graph shows that y is directly proportional to x: a

4 2

x 1

5 x

−2

−4

−6

−8

−10

x 1

Find whether each equation represent a direct variation between the pair of variables: a

C = 300p

C = 80n

y = − 6x

y = 2x + 8

y = 2 + 0.2x

C = 90n + 1

Let’s practice 7

In which table does y vary indirectly with x? A

x 1 2 3

y 4 8 12

x 1 2 3

y 4 4 4

x 1 2 3

y 6 8 10

x 1 2 3

y 12 6 4

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A relation is shown in this table. Which statement about this relation is true?

It is a direct variation because y = − 2.5x + 22.5.

It is an inverse variation because y =

It is an inverse variation because 20 = xy.

It is a direct variation because 20 = xy.

x 1 4 8 40

+ 22.5

Which of the following represent an inverse relationship? Select all that apply. A

f (x) =

x −2 −1 1 2

F 4

1 −4 −3 −2 −1 −1

x 1

−2

−3

−4

Inverse; k = 4

Direct; k =

Direct; k = 4

Inverse; k = − 15

Determine whether each table represents direct variation, inverse variation, or neither. a

x 1 2 3 4 y 3 1.5 1 0.75

x y

1 36

2 18

3 12

4 9

x 1 5 6 y 3 75 108

x y

1 4

2 5

3 6

4 7

x 2 3 4 y 12 8 6

x y

4 12

8 24

12 36

10 300

Find the equation relating the variables in each table of values: a

n 1 2 3

s t

1 48

2 24

3 16

4 12

p 3 6 9 12

r c

374

−4 −3 −2 −1 −1

y −2 −1 1 2

Sketch each graph given the relationship and constant of proportionality. a

x −8 −4 4 8

−2

y −2 −4 4 2

y 20.00 5.00 2.50 0.50

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If a is inversely proportional to x, and a = 20 when x = 10: a

Find the constant of proportionality, k.

Find the value of a when x = 5.

Express a in terms of x.

Consider the inverse variation equation y = . a

Complete the table.

Plot the data on a coordinate plane.

If y varies directly with x, and y =

when x = 4:

Find the constant of proportionality, k.

Write the variation equation of y in terms of x.

Create a graph for the relationship.

Write the equation being described by each of the following statements. a

The number, n, of tennis balls that can ﬁt into a box is inversely proportional to the cube of the diameter, d, of each ball.

The time, t, that a car spends driving varies inversely to its speed.

The amount of energy, E, a bird has to ﬂy is directly proportional to the amount of food it eats, f.

If y varies inversely as the square root of x, what is the constant of proportionality if y = 27 when x = 9? A

243

Two relationships are described. Relationship S: Barbara drove 200 miles in 5 hours, and then she drove 60 miles in 1.5 hours. Relationship T: Vernon cooked 12 hamburgers in 20 minutes, and then he cooked 18 hamburgers in 30 minutes. Which statement is true about these relationships?

Neither relationship is a direct variation.

Both relationships are direct variations.

Only Relationship S is a direct variation.

Only Relationship T is a direct variation.

An experiment is conducted on a container of gas that is kept at a constant temperature. When the pressure on the gas is 40 pounds per square inch, the volume is 120 cubic inches. When the pressure on the gas is 60 pounds per square inch, the volume is 80 cubic inches. Let p represent the pressure on the gas. Let v represent the volume of the gas. Which statement is true about this relationship?

The volume of the gas varies directly with the pressure because v = 3p.

The volume of the gas varies directly with the pressure because vp = 4800.

The volume of the gas varies inversely with the pressure because v = 3p.

The volume of the gas varies inversely with the pressure because vp = 4800.

The time taken by a typist to type up a document is inversely proportional to his typing speed. That is, the quicker the typing speed, the less time it will take. If it takes a typist 20 minutes to type a particular document, typing at a speed of 61 words per minute: a

Find the constant of proportionality.

Write an equation that represents this relationship where y is the time to type up the document and x is the typing speed.

Find the number of minutes it takes a typist with a typing speed of 30.5 words per minute to type up the document.

Graph the inverse relationship represented.

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The number of revolutions a wheel makes varies directly with the time it rolls for. A bike wheel revolves r times in t seconds. a

Find the formula that describes the relationship between r and t, using k as constant.

If the wheel completes 40 revolutions in 8 seconds, ﬁnd the value of k.

Express r in terms of t.

Graph the direct variation relationship, and use the graph to determine the value of r when t = 5.

A school’s marching band can be arranged in 6 rows of 27 students or 9 rows of 18 students. Your friend says this situation involves direct variation. Is your friend correct? Explain your reasoning.

Let’s extend our thinking 23

Delano is sharing a jar of jelly beans evenly between his friends. The graph shows the relationship between the number of people receiving jelly beans and the number of jelly beans they receive. a

Is the graph symmetric across the line y = x?

Determine whether or not there is an inverse variation between the number of people and the number of jelly beans they receive. If so, state the constant of variation. Justify your answer.

50 45 40 35 30 25 20 15 10 5

Jelly beans

People 5 10 15 20 25 30 35 40 45 50

A company’s dog food cans have the dimensions shown. a

If the company decreases the diameter of the can to 8 centimeters, by how much will the height of the can need to increase so that it holds the same volume of dog food?

Write an equation that relates the height h (in centimeters) of the can to its diameter d (in centimeters) for the same volume of dog food.

Describe your equation by the type of variation it represents.

11 cm

Lachlan and Aaron love reading. Lachlan reads 2 books every 3 weeks. Aaron has kept a table of his reading habits which is shown: Number of books read Number of weeks a

3 6

6 12

9 18

12 24

Complete the table for Lachlan. Number of books read Number of weeks

8.5 cm

2 3

6 12

10 15

Who reads more quickly?

The gravitational force F of the moon on an object is inversely proportional to the square of the distance d from the center of the moon. a

If a person at a distance of 1737 km (on the surface of the moon) experiences the force of gravity to be 100 N (Newtons), ﬁnd the exact value of the constant of proportionality, k. Write each line of working as an equation.

376

If a person x km further away experiences a force of 64 N, how far is x? Round your answer to the nearest km.

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5.03 Other rational functions After this lesson, you will be able to... • compare tables, graphs, and equations of rational functions. • identify domain, range, zeros, intercepts, increasing, decreasing, constant intervals, and end behavior. • compare the characteristics of rational functions to other functions. • find f (x) given x using a graph or equation. • identify the equations of any vertical and horizontal asymptotes of a rational function using a graph or equation. • graph the inverse of a rational function. • identify points of discontinuity.

Removable points of discontinuity As we saw with rational parent functions f (x) = • domain • range • zeros • x- and y-intercepts • increasing and decreasing intervals • end behavior • asymptotes

and f (x) =

, their characteristics include:

These characteristics also apply to rational functions in other forms, where the numerator is an algebraic expression. A rational function is any function that can be expressed as a quotient of two polynomials, with a non- zero denominator.

Where P (x) and Q (x) are polynomials. Rational functions can sometimes be simplified by factoring the numerator and denominator and then simplifying any common factors between them. A characteristic of rational functions that have been simplified is a removable point of discontinuity. Removable point of discontinuity A point at which a function is not defined, but the graph of the function approaches that point from both sides. Sometimes called a “hole”

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A removable point of discontinuity will occur at x = a if x = a is a zero of both the numerator and the denominator. 4

Notice that the denominator of the graphed function can be factored and simpliﬁed further:

3 2 1 −4 −3 −2 −1 −1

x 1

From here, we can see that x = − 1 is only a zero of the denominator. This is why there is a vertical asymptote at x = − 1.

−2 −3

Notice that x = 1 is a zero of both the numerator and denominator, so there is a removable point of discontinuity at x = 1.

−4

To ﬁnd the y-value of a hole, substitute the x-value into the simpliﬁed form of the function. Points of discontinuity, like asymptotes, impact the domain and range of the function because the function does not exist at that point.

Example 1 Consider the function f (x) =

a What is the domain of the function?

Create a strategy As a rational function, the domain will be all real values of x except for those which make the denominator equal to 0. So we will set the denominator to be equal to 0 and solve for x.

Apply the idea x2 + 7x + 12 = 0

Set denominator equal to 0

(x + 3) (x + 4) = 0

Factor the quadratic

x+3

x+4=0

Set each factor equal to 0

− 3, x = − 4

Solve each equation for x

These are the values that make the denominator equal to 0, which would result in an undeﬁned expression. As such, these values must be removed from the domain. The domain of the function is (−∞, −4) ∪ (−4, −3) ∪ (−3, ∞).

b For each value of x not in the domain, determine whether there is a vertical asymptote or a removable point of discontinuity at that value.

Create a strategy We need to compare the factors of the numerator with the factors of the denominator. If a factor in the denominator is not in the numerator, or is in the numerator but with a lower multiplicity, then there will be a vertical asymptote at the corresponding value of x. If a factor is in the denominator and the numerator with the same or higher multiplicity, there will be a removable point of discontinuity instead.

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Apply the idea State the function

Previous factoring of the denominator

We can see that the factor (x + 3) appears in both the numerator and denominator, while the factor (x + 4) only appears in the denominator. As such, x = − 3 will correspond to a removable point of discontinuity, while x = − 4 will be the equation of a vertical asymptote.

Reﬂect and check We can see why there is a removable point of discontinuity by analyzing the graph of the simpliﬁed form of f (x) =

We know that

= 1 when x ≠ −3, so the function can be written as: Original function

Factored form

For x ≠ −3

Consider the graph of f (x) =

, x ≠ −3: 5 4 3 2 1

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

The vertical asymptote at x = − 4 can still be seen in the simpliﬁed form of the function, but x = − 3 would lead to an undeﬁned function as well.

We have to exclude both x = − 3 and x = − 4 from the domain of the original function, leaving a hole at x = − 3.

c Describe the end behavior of the function as x → −∞ and as x → ∞.

Create a strategy The end behavior of a graph is how the graph behaves as x approaches inﬁnity or negative inﬁnity. Consider what happens to f (x) for very small values of x and very large values of x. If the degree of the denominator is larger than the degree of the numerator, there is a horizontal asymptote of y = 0, which will determine the end behavior of the function.

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Apply the idea 5 4 3 2 1

Since the degree of the denominator is greater than the numerator, there is a horizontal asymptote at y = 0.

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

As x gets increasingly small, f (x) approaches 0 from below the x-axis. x 1

As x gets increasingly large, f (x) approaches 0 from above the x-axis.

Using proper notation, the function’s end behavior is as follows: • As x → −∞, f (x) → 0 • As x → ∞, f (x) → 0

Example 2 Abdul used a calculator to graph the function f (x) =

. He was

surprised to see the graph of a line instead of a rational function.

a Explain why the graph appears to be linear.

1 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

y x 1 2 3 4 5 6

Create a strategy A straight line implies that the given rational function can be simpliﬁed. Recall that common factors in the numerator and denominator can lead to removable points of discontinuity, which affect the graph of the function.

Apply the idea First, we will factor the rational expression.

If we restrict the domain so that it does not include x = − 3, we can divide out the common factor of (x + 3). This simpliﬁes to f (x) = x − 5 for all x ≠ −3, which is the equation of the line graphed.

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Reﬂect and check Notice there is one important distinction between f (x) = x − 5 and our rational function. Rational functions exclude any values of x that result in a 0 in the denominator. Therefore, x = − 3 should be excluded from the domain of the linear graph since it is a removable discontinuity. 1 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

y x 1 2 3 4 5 6

You should always be careful when graphing with technology. Not all graphing calculators will show points of discontinuity. It is important to consider the equation and context for possible domain restrictions. You can generally use the tracing tool of the graphing calculator to see that the function is, in fact, undeﬁned at x = − 3

b Identify the domain and range.

Create a strategy The domain is all the real values of x, except for the value where the removable point of discontinuity lies. The range is all the real values of y, except for the value where the removable point of discontinuity lies.

Apply the idea In part (a), we found the only value excluded from the domain is x = − 3. Therefore, the domain is (−∞, −3) ∪ (−3, ∞). To find the range algebraically, we would need to substitute the x-value of the removable point of discontinuity into the simplified function. From part (a), we know the simplified function is: f (x) = x − 5 To find the y-value of the hole, the value that should be excluded from the range, we substitute x = − 3 into f (x) = x − 5: f (−3) = (−3) − 5 = −8 This means the range is (−∞, −8) ∪ (−8, ∞).

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c Find the intercepts.

Create a strategy Look at the graph to identify the points where the line crosses the x- and y-axis.

Apply the idea

Reﬂect and check

The x-intercept is (5, 0).

We can ﬁnd the intercepts algebraically using the simpliﬁed form of the function. • x-intercept:

The y-intercept is (0, −5).

y=x–5 0=x−5 5=x • y-intercept: y=x−5 y=0−5 y = −5

Example 3 Consider the function f (x) =

and the graph of g (x).

8 7 6 5 4 3 2 1 −8−7−6−5−4−3−2−1 −1 −2 −3 g(x) −4 −5 −6 −7 −8

a Compare the increasing intervals of f (x) and g (x).

Create a strategy Consider how each function changes as you move across the domain.

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x 1 2 3 4 5 6 7 8

Apply the idea From the graph, we can see that g (x) increases over its entire domain. So, the increasing interval of g (x) is (−∞, ∞). We can use technology to graph f (x). 7 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

The function f (x) has an asymptote at x = 1, and it is decreasing on either side of the asymptote. So, the increasing intervals of f (x) are (−∞, 1) and (1, ∞).

b Compare the zeros of f (x) and g (x).

Create a strategy Look at the graph to identify the zeros of g (x). Determine when the numerator is 0 to identify possible zeros of f (x).

Apply the idea The function g (x) intercepts the x-axis just to the right of the origin, so the zero of g (x) is just to the right of, or slightly bigger than, x = 0. The function f (x) has a 0 in the numerator when x = 0 Since this value is not excluded from the domain, x = 0 is the zero of f (x). The zero of g (x) is just to the right of, or slightly bigger than, the zero of f (x).

c State a value in the domain of g (x) that is not in the domain of f (x).

Create a strategy Look for asymptotes and removable discontinuities to ﬁnd discrepancies between the domains.

Apply the idea f (x) has an asymptote at x = 3, but the domain of g (x) includes x = 3. So x = 3 is in the domain of g (x), but it is excluded from the domain of f (x).

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Idea summary The characteristics of rational functions include: • • • • • • • •

domain range zeros x- and y-intercepts increasing and decreasing intervals end behavior asymptotes removable points of discontinuity (holes)

Rational functions can sometimes be simplified by factoring the numerator and denominator and then simplifying any common factors between them. A removable point of discontinuity will occur at x = a if x = a is a zero of both the numerator and the denominator. To find the y-value of a hole, substitute the x-value into the simplified form of the function.

Asymptotes Recall that vertical asymptotes occur for values of x that make the denominator 0. We can ﬁnd the asymptotes by factoring the denominator and setting each factor equal to 0. 7 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

The graph shows this function:

f (x) = Simpliﬁed, we have: x 1 2 3 4 5 6 7

f (x) = Setting each factor in the denominator to 0 and solving, we get the asymptotes x = 3 and x = − 5.

Whether or not the function will have a horizontal asymptote depends on the degree (highest exponent) of the polynomials in the numerator and denominator of the function. 7 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

384

If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is the horizontal asymptote.

Consider the function f (x) = x 1 2 3 4 5 6 7

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• Degree of numerator is 0 • Degree of denominator is 1 Since 0 < 1, the horizontal asymptote is at y = 0.

7 6 5 4 3 2 1

Consider the function f (x) = x

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1

If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is the ratio of the leading coeﬃcients.

1 2 3 4 5 6 7

• Degree of numerator is 1 • Degree of denominator is 1 Since 1 = 1, the horizontal asymptote is at y = .

If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

Consider the function f (x) = x 1 2 3 4 5 6 7

• Degree of numerator is 2 • Degree of denominator is 1 Since 2 > 1, this function has no horizontal asymptote.

Example 4 Consider the function y = a Determine the coordinates of the intercepts of the function.

Create a strategy The y-intercept of a function occurs where x = 0. We can substitute this into the function to ﬁnd the y-coordinate. Similarly, the x-intercept(s) of a function occur where y = 0. In particular, for a rational function, this occurs at values of x which make the numerator equal to 0 (and not the denominator).

Apply the idea For the y-intercept, we have: Substitute x = 0 into the function

Simplify numerator and denominator

Evaluate the division

So the y-intercept occurs at (0, 4).

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For the y-intercepts, we have: Set y = 0

Factor the quadratic in the numerator

Multiply both sides by x + 1

Set each factor equal to 0

Solve each equation for x

The function is undeﬁned at x = − 1 (the value which makes the denominator 0). This is neither of the values we just solved for, and so we have x-intercepts at both (1, 0) and (4, 0).

Reflect and check It is important to consider values of x which make the denominator equal to 0, as the function cannot have an intercept if it is undefined at that value of x. Note that the step of work “multiply both sides by x + 1” fails if x = − 1. This is why we factored the numerator first, so that we could see that there was no corresponding factor of x + 1 in the numerator.

b Identify any asymptotes or points of discontinuity.

Create a strategy A rational function has a horizontal asymptote if the degree of the numerator is less than or equal to the degree of the denominator. Otherwise, it does not have a horizontal asymptote. A rational function has a vertical asymptote when the denominator of the simpliﬁed form is equal to zero. Lastly, a rational function has a removable discontinuity where there are any common algebraic factors between the numerator and denominator.

Apply the idea In this case, the function is y =

, which has a numerator of degree 2 and a denominator of degree 1.

The degree of the numerator is larger than the degree of the denominator, so the function has no horizontal asymptote. We can factor the function

Since the numerator and denominator have no common factor, there are no removable discontinuities. The vertical asymptote occurs when the denominator is 0 which is at x = − 1. So, x = − 1 is the vertical asymptote.

c State the domain of f (x).

Create a strategy Use the asymptotes and points of discontinuity to determine the domain.

Apply the idea Since we have a vertical asymptote at x = − 1, the domain does not include x = − 1. In set builder notation, we say the domain is {x∣x ≠ −1}.

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Example 5 Consider the function f (x) =

a Determine any asymptotes of f (x).

Create a strategy To ﬁnd the horizontal asymptotes, consider the degree of the numerator and denominator. To ﬁnd the vertical asymptotes, consider what values make the denominator equal to zero.

Apply the idea The degree of the numerator is 1, and the degree of the denominator is 1. Since the degrees are equal, the horizontal asymptote is the ratio of leading coefficients of the numerator and denominator. • Leading coefficient of numerator: −4 • Leading coefficient of the denominator: 1 The horizontal asymptote is y =

= − 4.

The denominator of f (x) is 0 when x = 5. Since f (x) does not simplify, there is a vertical asymptote at x = 5.

b Describe the end behavior.

Create a strategy To find the end behavior, we need to first identify the horizontal asymptote. Then, we can use our knowledge of parent rational functions to find the end behavior as x → −∞ and x → ∞.

Apply the idea Using technology to graph our function: The graph confirms that there is a horizontal asymptote at y = − 4, and the y-values approach it as x tends toward negative infinity and positive infinity.

y 15 10

This means that as x → −∞, f (x) → −4, and as x → ∞, f (x) → −4.

5 −15 −10 −5 −5

x 5

10 15 20

−10 −15

Reﬂect and check We can also substitute negative values of x into the function to see what the y-values approach: For x = − 100: Substitute x = − 100

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For x = − 500: Substitute x = − 500

So as the x-values get larger in the negative direction, the y-values get closer to y = − 4. The same could have been done to check the end behavior as x → ∞.

c Describe the increasing and decreasing intervals of f (x).

Create a strategy To ﬁnd the increasing and decreasing intervals of f (x), we can observe the graph from the previous part. We also know that our horizontal asymptote is at y = − 4, and the vertical asymptote is at x = 5.

Apply the idea Using technology to graph our function: y 15 10 5 −15 −10 −5 −5

x 5

10 15 20

−10 −15

The function is increasing on either side of its vertical asymptote, x = 5. Therefore, f (x) is increasing on the intervals (−∞, 5) and (5, ∞). The function has no interval where it is decreasing.

d Determine the range of f (x).

Create a strategy Use the fact that there is a horizontal asymptote at y = − 4 to determine the range.

Apply the idea Since y = − 4 is excluded from the range, the range of f (x) can be written in interval notation as (−∞, −4) ∪ (−4, ∞).

Reflect and check The range can also be written in set notation: {y∣y ≠ −4} or {y∣y < −4 or y > −4}

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Idea summary For a rational function f (x) = • • •

If the degree of P (x) < the degree of Q (x), then y = 0 (the x-axis) is the horizontal asymptote. If the degree of P (x) = the degree of Q (x), then the horizontal asymptote is the ratio of the leading coeﬃcients of P (x) and Q (x). If the degree of P (x) > the degree of Q (x), then there are no horizontal asymptotes.

To ﬁnd any vertical asymptotes, factor the denominator and determine the values that make it equal to zero. If any factors are in both the numerator and denominator, it is a point of discontinuity. Otherwise, it represents a vertical asymptote.

Practice What do you remember? 1

Determine the domain and range of the following functions. a

y 6

x −6 −4 −2 −2

−4

−2 −1 −1

−6

−2

Give an example of a function that does not include x = − 1 in the domain.

Consider the rational function

x 1

How does division by zero affect the domain of this rational function? 4

Simplify each expression. a

For each rational function: i

Fully simplify the expression.

Identify values of x that are not in the domain and state whether they will appear as a hole or an asymptote on the corresponding function.

a 6

Write the coordinates of the removable discontinuity. a

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Let’s practice 7

Select the graph that represents A

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

2 4 6 8 10

x=2

x=3

For what value(s) of x is A

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

2 4 6 8 10

x 2 4 6 8 10

When x = 2, what is the value of f (x) for the function A

10 8 6 4 2

x 2 4 6 8 10

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

x = −3

discontinuous? Select all that apply. B

x=3

x=0

Consider the graph of f (x). a

For what value of x does f (x) = 4?

Is the function increasing, decreasing, or constant over the interval (1, ∞)?

What can you say about f (x) when x = 1?

x = −1

D 9 8 7 6 5 4 3 2 1 −9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

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x 1 2 3 4 5 6 7 8 9

SOL

For each function: i

State the degree of the numerator.

State the degree of the denominator.

iii

State the equation of the horizontal asymptote.

g (x) = x2 + 2x – 3

y=0

iv d

. C

State the range.

Choose the option that best completes the following sentence. has ⬚.

two x-intercepts and no y-intercept

two x-intercepts and one y-intercept

one x-intercept and no y-intercept

one x-intercept and one y-intercept

Consider the following rational functions: i

State the coordinates of the intercepts of the function.

State the equation(s) of the vertical asymptote(s) of the function.

iii

Determine whether the graph of f (x) has a horizontal asymptote. If so, write the equation of the horizontal asymptote.

State the increasing intervals of f (x). b

a 17

State the domain.

The graph of g (x) =

iii

For each rational function:

a 15

y=1

Identify all of the functions with the same domain as

SOL

Match each rational function to the corresponding horizontal asymptote.

A 14

For the rational function a

x → −∞

x→∞

c , determine the behavior as:

14 12 10 8 6 4 2 −12−10−8 −6 −4 −2 −2

x 2 4 6 8

−4 −6 −8 −10 −12

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SOL

as x approaches negative inﬁnity?

Which of the following describes the end behavior of A

y approaches negative inﬁnity

y approaches −6

y approaches −1

y approaches 0

Consider the following rational functions: i

Determine the end behavior of f (x) as x → ∞.

Determine the end behavior of f (x) as x → −∞.

iii

Determine the equation(s) of the vertical asymptotes.

Determine the zeros of the function.

For the rational function a

x→∞

x → −∞

b , determine the behavior as:

10 8 6 4 2 −6 −5 −4 −3 −2 −1−2 −4 −6 −8 −10 −12 −14

For each rational function: i

Find the domain.

For values of x not in the domain, determine whether vertical asymptotes or removable points of discontinuity occur at these values. b

c 22

For each rational function, ﬁnd: i

The domain.

iii

The zero(s).

The y-intercept.

The increasing and decreasing intervals.

The function value when x = 2.

The range.

y 10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

392

y 15 12 9

2 4 6 8 10

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3 −4

−2

−3 −6

x 2

x 1

Consider the equations

and

Use technology to graph the functions on the same coordinate plane.

Graph the line y = x on the same coordinate plane and use that to show whether or not f (x) and g (x) are inverse functions.

Consider function

and the graph of g (x) to answer

9 8 7 6 5 4 3 2 1

the following questions.

Which function has the greater y-intercept?

Which function(s) are increasing over their entire domain?

Which function has an asymptote at x = 1?

Determine if the two graphs represent inverse functions.

Consider the graph of

−9−8−7−6−5−4−3−2−1 −1 −2 g(x) −3 −4 −5 −6 −7 −8 −9

−12−10−8−6−4−2 −2 −4 −6 −8 −10 −12

Consider the function

x 1 2 3 4 5 6 7 8 9

12 10 8 6 4 2

Graph the inverse of the function f (x).

and the graph of g (x).

Compare the end behavior of the functions as x → ∞.

Compare the zeros of f (x) and g (x).

State a value in the domain of g (x) that is not in the domain of f (x).

Compare the range of f (x) and g (x).

5 4 3 2 1

x 2 4 6 8 10 12

−6 −5 −4 −3 −2 −1 1 2 3 4 5 6 −1 −2 g(x) −3 −4 −5 −6 −7

From Ohm’s Law for circuits, it follows that the total resistance R of two components hooked in parallel is given by the equation

where A and B are the individual resistances.

Let A = 7 ohms. Write an equation for R in terms of B.

Use technology to graph R as a function of B.

State the equation of the vertical asymptote.

State the equation of the horizontal asymptote.

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100 80 60 40 20 −25

−20

−15

−10

−5

−20 −40 −60 −80 −100

P(t)

t 5

Find the population of mongooses in the area after 0, 1, 5 and 10 years by completing the table. Give your answer rounded to the nearest whole number.

P (t)

State the equation of the horizontal asymptote.

Determine the end behavior of P (t) as t → ∞. Explain its meaning in terms of the population of mongooses.

Let’s extend our thinking 29

Determine the equation of the function on the graph.

6 5 4 3 2 1

The removable point of discontinuity is at (−2, −0.125)

−5−4−3−2 −1 −1

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6

Consider

Use algebraic manipulation to rewrite the expression in the form division.

Identify the asymptotes of the corresponding function.

where r (x) is the remainder after

Explain whether or not

The domain of a function is (−∞, −3) ∪ (−3, 4) ∪ (4, ∞) with a removable point of discontinuity at x = − 3 and a vertical asymptote at x = 4. Write a possible equation for the function.

Some wolves were introduced onto a large enclosed reserve to preserve their population and study their behavior. The rational function

394

and

represent the same function.

models the number of wolves on the reserve after x years.

State the initial number of wolves introduced into the reserve.

Find the population of wolves on the reserve after 6 years. Round your answer to the nearest whole number.

Over time, what amount will the population of wolves on the reserve approach?

Mathspace Virginia SOL Algebra 2 mathspace.co

A carnivorous pitcher plant uses acid liquid to dissolve and consume its prey. When a bug enters the pitcher plant, the pH of the liquid over time is modeled by the equation

where t is the number of hours since the bug entered the plant and f (t) is the pH of the plant’s liquid at time t. At the time that the bug enters the liquid, its pH is at its normal level. 6

f (x)

5 4 3 2 1 −40 −35 −30 −25 −20 −15 −10 −5

−1

t 5

20 25 30 35 40

−2

Find the normal pH level of the plant’s liquid.

Find the function value when t = 15, rounded to three decimal places. Explain what this represents in context.

Use the graph to estimate how many hours it takes to reach a pH level of 2.

Explain the level of the liquid’s pH over time based on the end behavior of the graph.

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To multiply two (or more) rational expressions together, we multiply the numerators to form the new numerator and multiply the denominators to form the new denominator - the same process used when multiplying fractions:

To divide two rational expressions, we multiply the ﬁrst rational expression by the reciprocal of the second rational expression - the same process used when dividing fractions:

Another way to represent division is with a complex fraction. The fraction in the numerator is divided by the fraction in the denominator.

The same algebraic properties we use with real numbers apply to rational expressions: Commutative property of multiplication Associative property of multiplication Multiplicative identity Multiplicative inverse Distributive property of multiplication Finding common factors, in particular the greatest common factor (GCF), between any of the numerators and denominators can help us use the algebraic properties to simplify rational expressions. We will need to assume that no denominator is equal to zero to avoid undeﬁned expressions. We can do this by stating restrictions on the variables which will relate to restrictions on the domain of an associated rational function.

Example 1 Fully simplify the expression, justifying each step. Write any restrictions on the variables.

Create a strategy We can multiply the fractions to combine numerators and denominators, identify common factors between the numerator and denominator and then use the commutative and associative properties of multiplication to group like terms together. Any variables that lead the denominators of the rational expressions to equal zero should be excluded.

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Apply the idea The restrictions on the variables for the expression given are b ≠ 0 and c ≠ 0. Combine into a single fraction

Commutative and associative properties of multiplication

Rewrite using factors

Divide out common factors in the coeﬃcients

Product and quotient rules of exponents

Evaluate the multiplication

, b ≠ 0 and c ≠ 0.

Example 2 Fully simplify the expression, justifying each step. State any restrictions on the variables.

Create a strategy We can ﬁrst turn the division into a multiplication by taking the reciprocal of the second fraction. We then again want to identify common factors that will help us simplify the rational expression.

Apply the idea

Rewrite as multiplication by the reciprocal

Multiplicative inverse

Evaluate the multiplication

, x ≠ 0, y ≠ 0, and z ≠ 0.

Reﬂect and check Notice that whenever we mutliply or divide variables we use the laws of exponents. When we divided subtracted the exponents. When we multiplied z ⋅ z we added the exponents.

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Example 3 Fully simplify the rational expression, justifying each step. Write any restrictions on the variables.

Create a strategy When the numerator or denominator of a rational expression contains polynomials, we need to see if it’s possible to use factoring to rewrite the polynomials in terms of multiplication. Only then can we use our multiplication properties to simplify the expression. In particular, notice that three of the expressions are quadratic, and the other one is a difference of two cubes, so we have factoring techniques we can use on each part of the expression. Factoring the denominators will also help us determine any restricted values:

Apply the idea First, we will need to determine values of x for which the expression is undeﬁned before simplifying, so if we look at the factored forms in the denominators of the multiplication problem, we can see that the binomials (x − 2), (x + 5), and (x − 3) are in the denominator. The original expression will be undeﬁned for x = 2, x = − 5, and x = 3. Second, we will use the factored forms to simplify the rational expressions: Factor each expression Commutative and associative properties of multiplication Simplify all common factors between numerator and denominator

Deﬁnition of multiplying rational expressions , x ≠ 2, x ≠ −5, and x ≠ 3.

Finally, we can state that the simpliﬁed form is

Reﬂect and check We could have skipped the commutative/associative properties of multiplication step by taking the shortcut, and simpliﬁed common factors in numerator and denominator across the fraction. ( x + 5) ( x − 2) 2

⋅

( x − 3) ( x + 3)

( x − 2) ( x + 2 x + 4) ( x + 5) ( x − 3)

Note that this only works when all the terms are written in terms of multiplication, and one common factor needs to be in the numerator and the other in the denominator in order to create an

= 1 situation.

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Example 4 Rewrite the complex fraction as a simpliﬁed rational expression, assuming no denominator equals zero:

Create a strategy To simplify this to an equivalent expression, we will multiply the ﬁrst fraction by the reciprocal of the second fraction. This approach utilizes the property that dividing by a fraction is equivalent to multiplying by its reciprocal. Rewrite the complex fraction as division

Multiply by the reciprocal

Apply the idea Replace division with multiplication by the reciprocal

Multiply numerators and denominators respectively

Factor out −1 from the numerator

Divide out the common factor, u − 3

The equivalent expression after simplifying is

Reflect and check We can check that our simplified expression is equivalent by confirming that it simplifies to the same value as the original expression for various values of u, excluding those that would result in a zero denominator. Original expression with u = 2:

Equivalent expression with u = 2:

Both calculations yield the same result, conﬁrming the expressions are equivalent.

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Idea summary By the deﬁnition of multiplying rational expressions, we know

We can use the algebraic properties of multiplication to ﬁnd common factors and simplify the expressions. Division of rational expressions can be rewritten as multiplication, where

Rational expressions can also involve complex fractions which can be simpliﬁed using the same skills used with multiplication and division:

Practice What do you remember? 1

Fill in the blanks to complete the sentence.

When dividing by a rational expression, we can replace the division sign with a ⬚ sign if we replace the dividing term with its ⬚.

When multiplying two rational expressions, under what conditions would the product be undeﬁned? A

When the numerators of both expressions are zero.

When the denominator of any of the expressions is zero before simpliﬁcation.

When the product of the numerators equals the product of the denominators.

When the expressions involve different variables.

When multiplying or dividing rational expressions, why is it important to consider the presence of zeroes in the denominators of the original expressions before simpliﬁcation? A

Zeroes in the denominator indicate that the expressions are undeﬁned at those points, which can affect the domain of the ﬁnal expression.

Zeroes in the denominator can be ignored if they are eliminated during the simpliﬁcation process.

Multiplying or dividing by zero increases the value of the rational expression signiﬁcantly.

Zeroes in the denominator signify that the rational expressions are equivalent to zero.

For each expression: i

Identify any common factors in the numerators and denominators.

Simplify the expression by dividing out the common factors. You do not need to evaluate the multiplication.

a 5

Explain why dividing by a rational expression is equivalent to multiplying by its reciprocal.

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401

Rewrite each expression with division of rational expressions as an expression using multiplication. Do not simplify the resulting expression.

a 7

For each of the following expressions, determine whether or not a factor of (x − 1) can be divided out to simplify the expression. a

The product of two fractions is 1. If one of the fractions is other fraction?

, what is an expression that could represent the

Let’s practice 9

Fully simplify each rational expression: a

Keith has attempted to fully simplify an expression and showed his work: 1

Divide out any common factors

Rewrite the division by multiplying by the reciprocal

Divide out any common factors

Evaluate the multiplication

Identify where Keith has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

For each expression: i

Factor the expressions in the numerators and denominators.

Identify any common factors in the numerators and denominators.

iii

Fully simplify

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Consider the expression a

What is the quotient in lowest terms?

What values of x must we exclude from the domains of the expressions? Choose all answers that apply: A x = −4

SOL

x=1

Which of the following is equivalent to A

SOL

y2x − 3

x=2

x=8

x6y − 2

x6y2

2m + 3

2m2 + 3m

x4y − 1

Which polynomial is equivalent to this expression if m ≠ 0?

m2 + 3m

m+3

Determine whether

Which of the following expressions are equivalent to

and

are equivalent. Justify your answer. ?

Justify each selection. A SOL

Assuming the denominator does not equal zero, completely simplify the expression.

Fully simplify: a

¡ 19

Fully simplify: a

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Consider this expression: Find the product of this expression using the following methods: Method 1: Multiplying the numerators and denominators, then simplify. Method 2: Simplify ﬁrst, then multiply. Which method do you prefer? Explain.

Simplify each complex fraction: a

SOL

Which of the following expressions is not equivalent to

A 24

Which expresson is equivalent to the following expression if no denominator equals to zero?

A 23

D ?

Ping has attempted to fully simplify an expression and showed his work: 1

Rewrite the division by multiplying by the reciprocals

Divide out any common factors

Evaluate the multiplication

Identify where Ping has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

Describe and correct the error in simplifying this rational expression.

Prove the following equation is true:

Let’s extend our thinking 27

Find two rational functions that have a product of

Consider: Determine the domain of f (x). Justify your answer.

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and a quotient of

Consider: • Darrick says that the domain is all the real values of x except for x = 2 and x = 4. • Ines says that the domain is all the real values of x except for x = 2 and x = 3. • Percy says that the domain is all the real values of x except for x = 2, x = 3 and x = 4. Determine who is correct and explain why.

A parallelogram with an area of

square units has a base of

units.

Determine the height of the parallelogram. 31

An investment fund grows according to the function A (x) = x2 + 7x + 12 (in thousands of dollars), where x is the number of years of investment. This investment fund will be shared between the number of people who contribute to the initial investment, which can be modeled by the expression

Determine the expected return per year for an investor of this fund if the investment has a duration of x years.

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5.05 Add and subtract rational expressions After this lesson, you will be able to... • add rational algebraic expressions. • subtract rational algebraic expressions. • simplify a sum or difference of rational expressions.

Add and subtract rational expressions Exploration 1.

What are the similarities and differences between these two expressions and how we evaluate them? Expression 1:

Expression 2:

Create an expression in the form:

where each blank is ﬁlled with a unique, nonzero integer value. 3.

Rewrite your original expression into two new expressions by: • Multiplying one term by , where x is a positive integer, resulting in:

• Adding x to the denominator of each fraction, resulting in:

Work with a partner to determine how to add the fractions created in step 3.

The sum of two rational expressions will result in another rational expression. Recall that a common denominator is required in order to add or subtract fractions. The same is true for rational expressions A, B, and C:

In order to add or subtract rational expressions which have different denominators, we will need to ﬁnd a common multiple to rewrite the expressions so that they share a common denominator. Given are expressions, common multiple is B ⋅ D, so we have:

Multiplicative identity, since

Multiply the fractions

Commutative property of multiplication

Add the fractions with a common denominator

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where A, B, C, and D

We need to state restrictions on the variables so we do not get an expression with 0 in the denominator, leading to an undeﬁned expression.

Example 1 Fully simplify the expression, justifying each step. Write any restrictions on the variables.

Create a strategy These two rational expressions have the same denominator, so we can subtract them by subtracting their numerators (being careful with the signs). Any values for the variables that lead the denominators of the rational expressions to equal zero should be excluded.

Apply the idea First, note that the denominators are both 3k. When 3k = 0, the expressions are undeﬁned, so we can exclude k = 0. Rewrite as a single rational expression

Distributive property

Combine like terms in the numerator

Divide out a common factor of 3

Since k = 0 would also lead the denominator of the simpliﬁed expression to equal zero, we state that the solution is , k ≠ 0.

Reﬂect and check When subtracting rational expressions, we have to be careful to distribute the negative sign to each term in the numerator of the second expression. Notice that we used parentheses when combining the rational expressions. Without parentheses, you might incorrectly simplify the numerator to k − 4 − k − 22, which leads a very different answer.

By using parentheses, we can ensure each term is subtracted correctly.

Example 2 Determine whether the two expressions are equivalent, justifying your answer. Expression 1:

Expression 2:

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Create a strategy To determine if these expressions are equivalent, we can fully simplify each expression by adding or subtracting the terms, then compare the resulting fractions. Both terms in the given expressions have uncommon denominators, so we ﬁrst have to ﬁnd a common denominator before we can add or subtract. Because the denominators in the given expressions are the same, each will need to have factors of x − 2 and 5x, so the least common denominator will be 5x (x − 2).

Apply the idea First, we will exclude the values of x that lead the denominator to equal zero. By setting the factors equal to 0 and solving, we ﬁnd x ≠ 2 and x ≠ 0. Let’s start with our ﬁrst expression: Create a common denominator

Distributive property

Combine like terms

Rewrite as a single fraction

Combine like terms

The numerator is not factorable, so this expression is fully simpliﬁed. Now, we need to do the same process with the second expression: Create a common denominator

Distributive property

Combine like terms

Rewrite as a single fraction

Distribute the negative

Combine like terms in the numerator

The numerator is not factorable, so this expression is fully simplified. Now compare the simplified expressions. Simplified expression 1:

Simpliﬁed expression 2:

The numerators of the expressions are different, so the two expressions are not equivalent.

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Reﬂect and check You may have noticed that the original expressions look very similar, with one being addition and one subtraction. Recall that subtracting a positive value is equivalent to adding a negative value. If we apply that concept to this example, the second expression could have been written as the addition of the negative fraction, but we must be sure to distribute the negative to the entire numerator. Expression 2:

This is another way to see that Expression 2 is not equivalent to Expression 1:

Example 3 Fully simplify the rational expressions, justifying each step. State any restrictions on the variables. a

Create a strategy Determine the restrictions on the variable. These two rational expressions do not have the same denominator, so we will ﬁrst want to rewrite them to have a common denominator before we add them. In this case, the denominator will need to have factors of 2, p5, p2, and m2. The smallest expression which does this is 2p5m2.

Apply the idea The expression is undeﬁned when 2p5 = 0 and p2m2 = 0, so when p = 0 or m = 0, the expressions are undeﬁned. We will exclude both values of p and m. Create a common denominator of 2p5m2

Product rule of exponents

Rewrite as a single fraction

The simpliﬁed expression is

, p ≠ 0 and m ≠ 0.

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Reﬂect and check In the original rational expressions, one denominator had a factor of p5 while the other had a factor of p2. Notice that p2 is already a factor of p5, however. So the ﬁnal expression only needed to have a factor of p5 and not p7. We can compare this to adding fractions such as already a factor of 8:

and . In this case, the LCM will only be 8 and not 16, since 2 is

Although any common multiple will work, using the least common multiple will reduce the amount of simpliﬁcation required after performing the addition.

Create a strategy These two rational expressions do not have the same denominator, so we will ﬁrst want to rewrite them to have a common denominator before we add them. In this case, the denominator will need to have factors of 6 and y + 9, and so the least common denominator will be 6 ( y + 9).

Apply the idea Exclude the value of y that leads the denominator to equal zero, so y + 9 = 0 → y = − 9 will be excluded. Create a common denominator of 6 ( y + 9)

Distributive property

Rewrite as a single fraction

Combine like terms in the numerator

Factor the numerator

The fully simpliﬁed expression is

, y ≠ −9.

Reflect and check We should determine any other values of y that would make the denominator of the simplified rational expression equal to zero after evaluating the addition, but notice that when y = − 9, the expression in its simplified form is also undefined.

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Example 4 Fully simplify the expression, justifying each step. State any restrictions on the variables.

Create a strategy These two rational expressions do not have the same denominator, so we want to rewrite them to have a common denominator before we subtract them. In order to do that, we will ﬁrst factor each denominator. Before rewriting the expressions with a common denominator, we will determine the values for which the expressions are undeﬁned.

Apply the idea Factoring the denominators, we get

So the least common denominator will be (x − 3)2 (x + 1). The values for which either rational expression will be undeﬁned are when x = 3 and x = − 1. We can now use this to subtract the two rational expressions: Create a common denominator

Distributive property

Rewrite as a single fraction

Distributive property

Combine like terms in the numerator

Factor the numerator

The fully simpliﬁed expression is

, x ≠ −1 and x ≠ 3.

Example 5 Fully simplify the expression, justifying each step. State any restrictions on the variables.

Create a strategy We will ﬁnd the common denominator of the numerator and denominator separately, then simplify the numerator and denominator. The common denominator for both the numerator and denominator is x + 3.

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Then, we can turn the division into multiplication by taking the reciprocal of the second rational expression. We then want to identify common factors that can be simpliﬁed before performing the multiplication.

Apply the idea The value of x that will lead to an undeﬁned expression is x = − 3. Creating a common denominator for the numerator: Multiply by

Distributive property

Rewrite as a single term

Creating a common denominator for the denominator: Multiply by

Distributive property

Rewrite as a single term

Now, we can continue simplifying the expression: Rewrite with the simpliﬁed numerator and denominator

Rewrite as multiplication by the reciprocal

Factor out the GCFs

Divide out common factors

Simplify

Note that the denominator of the simplified expression will have a different value of x for which the expression is undefined. We can see that x = − 4 will lead to 0 in the denominator, so it will need to be excluded. The fully simplified expression is

, x ≠ −3 and x ≠ −4.

Idea summary Prior to adding or subtracting rational expressions, do the following: • •

Determine restrictions on the variables that will lead to undeﬁned expressions If necessary, rewrite rational expressions to get a common denominator, using the multiplicative identity property:

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Practice What do you remember? 1

Fill in the gap to complete the sentence:

Rational expressions can only be added or subtracted if they have the same ⬚.

Combine like terms to simplify each expression. a

2(x + 5) − 3(x − 1) + 6x

Determine the LCM of the denominators for each expression: a

For each expression: i

Find the LCM between the denominators.

Determine what the denominator of each term should be multiplied by to reach the LCM. b

a 5

Why is it important to ﬁnd the least common denominator when adding or subtracting the algebraic fractions?

Let’s practice 6

Consider: a

State the least common multiple (LCM) of the two denominators.

Write the fraction equivalent to

which has the LCM from part (a) as its denominator.

Write the fraction equivalent to

which has the LCM from part (a) as its denominator.

Fully simplify the expression.

Fully simplify each expression: a

Fully simplify:

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SOL

Flynn has attempted to fully simplify an expression and showed his work: 1

Apply the subtraction

Combine like terms

Identify where Flynn has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

Assuming the denominators do NOT equal zero, which expression is equivalent to:

A SOL

Valentina has attempted to fully simplify an expression and showed her work: 1

Divide out any common factors

Scale the fractions to have the same denominator

Expand the parentheses

Evaluate the addition

Identify where Valentina has made an error and explain what it is.

Fully simplify the expression, showing the correct steps of work.

Fully simplify: a

If x does not equal 0, is

Explain why x − 1 = − (1 − x).

Simplify

414

Which expression is equivalent to the one shown if the denominator do not equal to zero?

A 12

the same as

. Show your work.

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? Explain your reasoning.

Jane and Mark are both training for a marathon. Jane can run Mark can run

of the marathon distance in 1 hour, while

of the marathon distance in 1 hour. If Jane runs for 2 hours and Mark runs for 3 hours, how

much farther does Mark run compared to Jane? 17

Sam and Emily are working together to solve a math problem. Sam can solve Emily can solve

of the problem in 1 hour, while

of the problem in 1 hour. If they work together, what fraction of the problem can they solve

in 1 hour?

Fill in each blank to explain how to simplify the complex fraction

Step 1: ⬚

Step 2: ⬚

Find the slope of the line that passes through

Fully simplify the complex rational expressions: a

and

Step 3: ⬚

Step 4: ⬚

Simpliﬁed form .

Find the perimeter of the quadrilateral. Express in simplest form.

Let’s extend our thinking 22

Consider

if a, b, and c are real numbers. Determine whether each statement is sometimes, always, or

never true. Explain your answer. a

abc is a common denominator.

abc is the LCD.

ab is the LCD.

b is the LCD.

The sum is

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Complete the equation:

Using the digits 1 through 9 at most once each, place a digit in each box to create a true statement.

Determine whether or not the sum of two rational expressions can be a quadratic expression. If it is possible, create an example.

Find two rational expressions whose sum is

Find two rational expressions with a sum of of work.

and a difference of

Explain how the addition and subtraction of rational expressions is similar to the addition and subtraction of rational numbers.

Deﬁne the domain of the following functions:

. Explain your method

The function f (x) is the explicit sum of two rational expressions, while g (x) is the fully simpliﬁed sum of the two rational expressions Pan claims that the domain of f (x) and g (x) are always the same. Explain whether or not he is correct.

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Example 1 Consider the rational equation:

a Determine whether x = 3 is a viable solution, an extraneous solution, or not a solution at all.

Create a strategy

Apply the idea

Substitute x = 3 in the equation and evaluate the addition, subtraction, and division. x = 3 is not a solution to this equation.

b Graph the corresponding function,

, and explain how any viable solutions, extraneous solutions or

restricted domain values are displayed in the graph.

Create a strategy By solving the denominator equal to zero we can determine any domain restrictions of the function: x−3=0

Set the denominator equal to zero

x=3

Add 3 to both sides of the equation , we have:

And by solving the equation

Multiply both sides of the equation by x − 3

Subtract 5 from both sides of the equation

Since x = − 5 does not make the denominator of the function equal to zero, it is a viable solution. Use technology to graph the function and analyze the function at x = 3 and x = − 5.

Apply the idea

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We can see that as the function approaches the undeﬁned value at x = 3 from the left, f (x) → −∞, and as the function approaches x = 3 from the right, f (x) → ∞. Hence, we see a vertical asymptote. When x = − 5, the value of the function is equal to zero and we can see an x-intercept at (−5, 0) on the graph.

Reﬂect and check Recall for a rational function, if a value of x = a leads to a zero denominator but not a zero numerator or is a zero of lower multiplicity in the numerator, a vertical asymptote forms on the graph at x = a. We also have if a value of x = b leads to a zero numerator but not a zero denominator the graph of the function will have an x-intercept at x = b.

Example 2 Consider the rational equation:

a Determine whether x = 3 is a viable solution, an extraneous solution, or not a solution at all.

Apply the idea We can solve the equation for x as shown and get the solution x = 3:

Multiply both sides of the equation by x − 3

Evaluate the square root of both sides of the equation

Add 3 to both sides of the equation

It looks like x = 3 is a solution to the equation. However, we still need to examine the denominator. We can see that x = 3 is a zero of both the numerator and denominator of the rational expression:

Although x = 3 will be an algebraic solution to the equation, it is also a value which makes the equation undeﬁned. So x = 3 is an extraneous solution to this equation.

Reﬂect and check Unlike

= 0, x = 3 would be a solution of the equation

= 0 but it will lead to a zero denominator,

which makes the type of solution different.

b Graph the corresponding function, f (x) =

, and explain how any viable solutions, extraneous solutions or

restricted domain values are displayed in the graph.

Create a strategy Use technology to graph the function and analyze the function at x = 3.

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Apply the idea

We know that x = 3 is an extraneous solution by analyzing any restrictions on the function. Without that, the graph of f (x) = point is excluded.

looks as though it includes the point (3, 0), but when tracing the function, the

We consider the extraneous solution x = 3 a removable point of discontinuity because it is a zero of both the numerator and denominator, and the zero in the numerator is not of lower multiplicity than the denominator. So, the graph should include a removable point of discontinuity or a hole, which we can manually enter to show this restriction on the domain:

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Example 3 Consider the rational equation: a Solve the equation.

Create a strategy In order for a rational expression to be equal to 0, its numerator must be equal to 0. So we will factor the numerator to solve this equation. We also want to factor the denominator, however, to check restrictions on the variable that could make each solution viable or extraneous. When ﬁnding solutions to the equation, it’s best not to simplify common factors in the numerator and denominator, as that may make us miss any possible extraneous solutions.

Apply the idea The expression in the numerator is x2 + 5x − 24, which can be factored to (x + 8) (x − 3). The expression in the denominator is x2 − 9x + 18, which can be factored to (x − 6) (x − 3). So we can rewrite the equation as

From the denominator, we can see that x cannot be equal to 3 or 6. From the numerator, we can see that the possible solutions are x = − 8 and x = 3. Putting this together, we have that x = − 8 is a viable solution, while x = 3 is an extraneous solution.

Reﬂect and check If we were to solve the equation by ﬁrst multiplying both sides by x2 − 9x + 18, and then factoring the remaining quadratic, we would still get the two possible solutions of x = − 8 and x = 3. In order to determine whether they were valid or extraneous, we could substitute into the initial expression

For x = − 8:

For x = 3:

So only the solution of x = − 8 is valid. The solution of x = 3 produces an undeﬁned expression and is therefore extraneous.

b Graph the corresponding function,

, and explain how any viable solutions, extraneous solutions

or restricted domain values are displayed in the graph.

Create a strategy Use technology to graph the function and determine the behavior of the function at the points of interest, x = − 8, x = 3, and x = 6.

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Apply the idea

We know that x = − 8 is a viable solution, and we see on the graph that the x-intercept is at (−8, 0). Since x = 6 is undeﬁned, but not extraneous, there is a vertical asymptote at x = 6. We know that x = 3 is an extraneous solution, so there is a removable point of discontinuity at x = 3. This is not obvious with a visible inspection of the computer-generated graph. When we draw a sketch of the graph, we show the visible hole in the graph at x = 3 in order to draw attention to the fact that the function is undeﬁned at this point.

Notice that if we had simpliﬁed the equation in part (a) to

, we may have missed the fact that x = 3 is an

extraneous solution, and may not have recognized there was a point of discontinuity in the graph.

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Example 4 Consider the rational equation:

a Solve the equation for x.

Create a strategy Since there are two rational expressions in this equation, we will start by rearranging the equation to have both rational expressions on the same side, equal to zero, and then combine them by making a common denominator.

Apply the idea

State the equation

Subtract

from both sides

Rewrite each rational expression to have a common denominator

Subtract the numerators

Simplify the numerator

At this point, we can see that the equation has no solutions (viable or otherwise), since there is no value of x that can make the numerator of the rational expression be 0.

Reﬂect and check It is also possible to draw this conclusion by rewriting the initial equation

At this point the numerators are equal, while the functions in the denominators are linear expressions with the same slope (i.e. parallel lines). So the denominators will never be equal for any value of x, and therefore the equation has no solutions.

b Graph the corresponding function, f (x) =

, and explain how any viable solutions, extraneous solutions

or restricted domain values are displayed in the graph.

Create a strategy Before graphing with technology, we can algebraically determine key features of the function. We know there are no removable points of discontinuity because no values of x will make the numerator of the function equal to zero. When x + 6 = 0 or 2x − 5 = 0, the graph will have vertical asymptotes and the function will be undeﬁned. So, when x = − 6 and x = , the function is undeﬁned and there will be two vertical asymptotes. We determined that the equation

has no solutions, so f (x) will have no x-intercepts.

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Apply the idea

Examining the graph, we see there is a horizontal asymptote at f (x) = 0, which means there are no solutions to f (x) = 0. Two vertical asymptotes are located at x = − 6 and x = .

Example 5 The Apoleisk Company wishes to calculate the average cost per item of producing car batteries. The ﬁxed monthly cost of production is $10 000 and each car battery produced costs $10. a Write the average cost function, A (x), of producing x car batteries in a month.

Create a strategy The average cost function is

Apply the idea Since the production costs $10 000 and each car battery produced costs $10, the expression that represents the total cost for the month of producing batteries is 10 000 + 10x, for x car batteries produced that month. The average cost per battery would be calculated by dividing the total by the number of batteries, x, so the average cost function is

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b State the domain and range of the function.

Create a strategy Consider the context of the problem and what the variables represent. A (x) is the average cost per battery and x is the number of batteries. Then, consider any values of x that lead to non-viable solutions, extraneous solutions, or that make the function undeﬁned.

Apply the idea Based on the context, we can initially state that the cost must be greater than or equal to $0 and the number of batteries produced must also be greater than or equal to 0. Also in the real-life context of producing batteries, we would only have whole number values for x, since we can’t make a negative number of batteries or make a fraction of a battery. When x = 0, the function is undeﬁned since the denominator is equal to zero, so x = 0 is excluded from the domain of the function. When 10 000 + 10x = 0, the solution for x will be a negative number of car batteries. This occurs when x = − 1000. Since we’ve already stated that the domain are positive values of x, −1000 is already excluded from the domain and considered a non-viable solution. The domain of the function can be deﬁned as the integers on the interval [1, ∞). No maximum number of batteries was indicated in the problem. Since this is an inverse relationship, as more batteries are produced, the average cost will decrease. Thus, when x = 1 the cost of production will be its most expensive, at A (1) =

= $10 010. As the cost can never

be less than $0, the range of the function can be deﬁned as (0, 10 010], in interval notation. Although, in practice this would be a set of discrete values due to the domain being whole numbers.

c How many car batteries does the Apoleisk Company need to produce each month so that the average cost of production per car battery is $60?

Create a strategy Since we are attempting to ﬁnd A (x) = 60, solve the equation 60 =

Apply the idea Substitute A (x) = 60

Multiply both sides of the equation by x

Subtract 10x from both sides of the equation

Divide both sides of the equation by 50

The Apoleisk Company needs to produce 200 car batteries each month so that the average cost of production per battery is $60.

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Idea summary We can recognize key features of a rational function by examining zeros and restricted domain values, which can involve factoring the numerator and denominator. If a solution does not lead to a restriction on the rational function and makes sense within the context of the problem, it is a viable solution. If there is a value of x that makes the denominator equal to 0, that value must be excluded from the domain of the rational function, which can occur in one of two ways: •

•

The excluded value also causes the numerator to be equal to 0, and the numerator is of the same or higher multiplicity as the denominator. In this case, the excluded value is an extraneous solution and the graph of the function will have a removable point of discontinuity The excluded value does not cause the numerator to be zero, and is not a solution to the equation. The graph will have a vertical asymptote

Practice What do you remember? 1

Describe when a rational expression is undeﬁned.

State the values which make each of the rational equations undeﬁned:

For each rational function: i

State the values of x which make the numerator equal to zero.

State the values of x which make the denominator equal to zero.

iii

State the zeros of the function. b

a 4

Use the graph to determine the value of x that makes the following equations true. a

y 9 8 7 6 5 4 3 2 1

−9−8−7−6−5−4−3−2 −1 −1

1 2 3 4 5 6 7 8 9

−2 −3 −4 −5 −6 −7 −8 −9

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9 y 8 7 6 5 4 3 2 1 −4−3−2 −1 −1

−2 −3 −4 −5 −6 −7 −8 −9

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Which system of equations could be used to solve the rational equation A

? Select all that apply.

Match each equation with the graph of its related system of equations. Then, solve for x. i a

ii y 7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

iii b

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

x 1 2 3 4 5 6 7

Let’s practice 7

SOL

Solve: a

If x ≠ 0, what is the solution to the following equation?

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SOL

What nonzero value of x is a solution to the following equation?

A 10

SOL

Solve the following rational equations: a

Student A was asked to solve the equation

1 2

Their work is shown.

Describe and correct the error made.

4 5 6

SOL

Find the solution for the given equation. Show your work/thinking.

Consider the rational equation:

Draw the graph for

State the solution to the rational equation.

on the same coordinate plane.

Consider the rational equation: a

Use technology to graph the equation

Use the graph to solve the rational equation.

Consider the rational equation:

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Solve the equation algebraically.

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Verify the solution by graphing.

Martin uses a calculator to graph the equation

On the graph, x = 1 appears to be a zero, but the calculator says undeﬁned. Martin claims x = 1 is still a solution. Is he correct? Justify your thinking.

3 2 1 (1, undefined) x −2

−1

−1 −2

Consider the equation a

Graph the corresponding system of equations.

A student claims x = 3 and x = 1 are both solutions to the rational equation. Use your graph to check the student’s solutions.

When solved algebraically, the equation

gives the possible solutions x = 6 and x = 4. Determine

which of the possible solutions is viable and which is extraneous. Justify your answer. SOL

Consider the following graph: a

Is x = 7 a solution to the equation algebraically.

When does f (x) = answer.

SOL

Is x =

equal 1? Give two justiﬁcations for your

a solution to the equation

Liam is solving the rational equation

4 3 2 1

= 2? Justify your answer

= − 8? Why or why not?

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

x 1 2 3 4 5 6 7 8 9

. He believes that both x = 2 and x = − 4 can could be

solutions to the equation. Soﬁa says that he is only partially correct and that only x = − 4 is a solution.

Who is right? How do you know?

Soﬁa hasn’t seen the graph of the equation. How did she know that x = 2 wasn’t a solution?

What values of x cannot be solutions to the rational equation shown? Why?

For each of the following equations, determine whether or not x = 4 is a viable solution, an extraneous solution or not a solution at all: a

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SOL

AstroNoodle Pty. Ltd. wishes to calculate the average cost per item of producing kayaks. The ﬁxed monthly cost of production is $200 000 and each kayak produced costs $100. a

Write an expression for C (x), the cost in dollars for producing x kayaks in a single month.

Write the average cost function, A (x), of producing x kayaks.

AstroNoodle Pty. Ltd. wants the average cost of production per kayak to be $200. Write an equation representing this scenario.

Find the number of kayaks that need to be made per month for their average production cost to be $200.

Find all real numbers x, x + 2, and x + 4, such that the reciprocal of the smallest number is the sum of the reciprocals of the other two. Are there rational solutions to this problem?

Patricia goes for a run through Blue Spring State Park and the GPS function on her ﬁtness wristband tracks her speed as she runs. The function R (x) models her speed in kilometers per hour when she is x kilometers from where she started.

R(x)

8 7 6 5

Estimate the value of R (0).

Estimate Patricia’s speed when she is 20 km from where she started.

Estimate Patricia’s distance from her starting point when her speed is 6 km per hour.

d 26

4 3 2 1

Why do estimates make sense in this context?

x 2 4 6 8 10 12 14 16 18

A farmer has two rectangular chicken coops which share a common border along their longer side. One coop has an area of 60 ft2 and a width of x ft. The other has an area of 96 ft2 and is 3 ft longer than the other coop. Determine the dimensions of each coop, showing your work.

430

Area 96 ft2

x+3

Length

Two rectangular paintings are to be placed side by side. One painting has an area of 63 in2 and a height of x in. The other painting is 2 in shorter and has an area of 77 in2. a

Write an equation relating the combined width of the two paintings, y, to the variable painting height x.

Find the value of x if the combined width of the two paintings is 18 in.

A gardener has two rectangular ﬂower beds. One has an area of 32 ft2 and a length of x ft. The other has an area of 44 ft2 and is 3 feet longer than the other ﬂower bed. a

Find the value of x if the two ﬂower beds have the same width. Show your work.

Justify why the length you found is reasonable.

Let’s extend our thinking 29

Area 60 ft2

Write a rational equation that has the following characteristics: a

A viable solution of x = 3 and an extraneous solution of x = − 2.

Three rational terms, one viable solution and no extraneous solutions.

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Let p be a non-zero real number. Determine whether each of the following are always, sometimes or never true. Explain your reasoning. a

The equation

has the extraneous solution x = p.

The equation

has exactly one viable solution.

The equation

has no viable solutions.

A construction project is made up of two stages. If the project only had a single worker: the ﬁrst stage would take 100 days and the second stage would take 200 days. It is assumed that all the workers will work at the same rate. The manager also expects 10 workers to leave after the ﬁrst stage. a

A project manager is trying to ﬁnd a function which models the number of days it will take to complete a project, D (x), with x initial workers.

Write a function for D (x) in terms of x.

Assuming that the project manager has a budget for at most 50 workers, state the constraints on the domain of this function. Justify your answer.

Write the quadratic equation whose solution is the number of initial workers required to complete the project in k days.

Find the number of initial workers needed to complete the project in 25 days. Justify your answer.

Brenita has an infected cut, for which her doctor prescribed a disinfectant spray and some antibiotics. The amount of milligrams of the active ingredient remaining in the bloodstream after using the disinfectant spray is

and after using the antibiotics is

where t is number of hours after using each medication. a

Determine which medication has a greater initial amount of active ingredient. Justify your answer.

Bernita is told by her doctor to use both medications at the same time. Find the number of hours after which both medications will have the same amount of active ingredients in her bloodstream.

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Exponential & Logarithmic 6 Functions Big ideas • Many function types share similar characteristics. • Functions provide a representation for how related quantities vary. This makes functions a good way to represent many real world situations. • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made.

Chapter outline 6.01 6.02 6.03 6.04

Exponential functions Applications of exponential functions Logarithmic functions Compare functions across representations

434 463 481 503

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 6.01 to answer these questions. 1.

What happens to the graph as b increases?

What happens to the graph as b is between 0 and 1?

What happens to the graph as a decreases?

Describe the end behavior for: • a > 0 and b > 1? • a > 0 and 0 < b < 1? • a < 0 and b > 1? • a < 0 and 0 < b < 1?

Create two equations where the function is increasing over its domain.

Create two equations where the function is decreasing over its domain.

The general shape for different values of a and b is shown. 8

4 3

a > 0, b > 1

1 −2

−1

−2

y −1

−1

−2

y > 0, Decreases at a decreasing rate y −2

−1

−1 −2

−3 −4

−1

x 1

a > 0, 0 < b < 1

y > 0, Increases at an increasing rate

−2

x 1

a < 0, 0 < b < 1

−3 a < 0, b > 1

−4

−5

−6

−7

−8

y < 0, Decreases at an increasing rate

y < 0, Increases at a decreasing rate

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Example 2 and g (x) = x2 + 2.

Consider the two functions, f (x) =

a Complete the table of values and sketch f (x) =

x y

−2

−1

Create a strategy To ﬁnd the table of values for the function f (x) =

, substitute different values of x into the equation and calculate

the corresponding y values. Then, plot these points on a graph to sketch the function.

Apply the idea Evaluate f (−2):

Evaluate f (−1):

Evaluate f (0):

Evaluate f (1):

Evaluate f (2):

Table of values: x

−2

−1

Graph: 18

16 14 12 10 8 6 4 2 −3 −2 −1

x 1

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b Sketch the graph of g (x) on the same plane as f (x).

Create a strategy To sketch the graph of g (x) = x2 + 2, we can use the transformation of the parent function x2, which is a parabola. The graph of x2 is a parabola with its vertex at the origin and opening upward.

Apply the idea 18

Since g (x) = x2 + 2, the transformation from the parent function x2 is a vertical shift upward by 2 units. This means that the vertex of the parabola for g (x) is at (0, 2). The shape of the parabola remains the same as the parent function, opening upward.

16 f (x)

14 12

g(x)

10 8 6 4 2

−3 −2 −1

c Compare the y-intercepts of both functions.

Create a strategy The y-intercept of a function is the point where it intersects the y-axis. We can ﬁnd the y-intercepts from the graphs and compare their values.

Apply the idea 18

From the graphs, we can see that the y-intercept of f (x) is at the point (0, 2) and the y-intercept of g (x) is at the point (0, 2) as well.

16 f (x)

Both functions have the same y-intercept.

14 12

g(x)

10 8 6 4 2 −3 −2 −1

438

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Reﬂect and check Another way to ﬁnd the y-intercepts of the functions is to substitute x = 0 directly into their equations and evaluate them. Substitute x = 0 into f (x)

Evaluate the exponent

Evaluate the multiplication

Substitute x = 0 into g (x)

Evaluate

This conﬁrms that both functions have the same y-intercept.

d Compare the end behavior of both functions.

Create a strategy To compare the end behavior, we need to look at what happens to the y-values of each function as x approaches positive and negative inﬁnity.

Apply the idea 18

For f (x), as x gets larger and larger, the function gets closer and closer to the asymptote causing the y-values of the function to approach 0.

16 14

As x → ∞, f (x) → 0

12 10 8 6 4 2

−3 −2 −1

As x gets smaller and smaller, the y-values of the function get very large and approach positive inﬁnity.

As x → −∞, f (x) → ∞

14 12 10 8 6 4 2 −3 −2 −1

x 1

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For g (x), as x decreases and increases, the y-values get very large, causing both ends of the graph to approach positive inﬁnity.

As x → −∞, g (x) → ∞

As x → ∞, g (x) → ∞

12 10 8 6 4 2 −4 −3 −2 −1

x 1

We can describe the end behavior of the functions in mathematical notation as: As x → −∞, f (x) → ∞ and as x → ∞, f (x) → 0 As x → −∞, g (x) → ∞ and as x → ∞, g (x) → ∞ We can see that both functions have the same behavior on the left as x approaches negative inﬁnity, but their behavior on the right as x approaches positive inﬁnity differs.

Reﬂect and check To reasonably verify the behavior of the functions f (x) and g (x) algebraically, you can use a calculator to input very large and very small values for x. For f (x), we can substitute a small value, like x = − 100, into f (x) which gives us f (−100) = 1.0308 ⋅ 1048. This is a very large number which matches our end behavior of: As x → −∞, f (x) → ∞ We can also substitute a large value, like x = 100, into f (x) which gives us f (100) = 3.8807 ⋅ 10 − 48. This is a number very close to zero which matches our end behavior of: As x → ∞, f (x) → 0 For g (x), we can substitute a small value, like x = − 100, into g (x) which gives us g (−100) = 10 002. This is a very large number which matches our end behavior of: As x → −∞, g (x) → ∞ We can also substitute a large value, like x = 100, into g (x) which gives us g (100) = 10 002. This is a very large number which matches our end behavior of: As x → ∞, g (x) → ∞

e Compare the domain and range of both functions.

Create a strategy We can use the graphs to ﬁnd the domain and range of each function and compare them. The domain of a function is the set of all possible x-values, and the range is the set of all possible y-values.

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Apply the idea 18

The graph of the exponential function f (x), never touches the x-axis, but it exists for every value of x, so its domain is all real numbers, or (−∞, ∞).

16 14

The graph has an asymptote at y = 0, but the graph exists for every y-value above the asymptote. This gives us a range of (0, ∞).

12 10 8 6 4 2

−3 −2 −1

The quadratic function g (x), also exists for all x-values, so its domain is all real numbers, or (−∞, ∞).

It has a vertex at the point (0, 2), which is the lowest point of the graph. g (x) exists at its vertex and for all y-values above the vertex so the range is all real numbers greater than or equal to 2, or [2, ∞).

14 12 10 8 6 4 2 −4 −3 −2 −1

(0, 2) 1

Comparing the domain and range of both functions, we can see that both functions have the same domain, (−∞, ∞), but their ranges are different.

Idea summary Exponential functions can take the form:

f (x) = abx a b x f (x)

Leading coeﬃcient Base where b > 0, b ≠ 1 Independent variable Dependent variable

Exponential functions of the form y = abx have the following features in common: • • •

The domain is −∞ < x < ∞. The y-intercept is at (0, a). There is a horizontal asymptote at y = 0.

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Transformations of exponential functions To draw the graph of an exponential function, we can use a variety of strategies, including: • Completing a table of values for the function and drawing the curve through the points found • Using transformations • Identifying key features from the equation • Using technology, such as a physical or online graphing calculator

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 6.01 to answer these questions. 1.

What happens to the graph when k is positive or negative?

What happens to the graph when h is positive or negative?

What happens to the graph when a is positive or negative?

What happens to the graph when c is positive or negative?

What impact does c have on the graph?

What happens to the graph when 0 < b < 1?

For the function:

f (x) = a ⋅ bc(x − h) + k b

is the base

can reﬂect, stretch or compress vertically

can reﬂect, stretch or compress horizontally

h determines the horizontal translation k

determines the vertical translation

Reflection across the x-axis: Reflection across the y-axis: Vertical stretch when ∣a∣ > 1 Vertical compression when 0 < ∣a∣ < 1: Horizontal compression when ∣c∣ > 1 Horizontal stretch when 0 < ∣c∣ < 1: Horizontal translation by h Vertical translation by k:

y = − bx y = b−x y = a (b)x y = bc ⋅ x y = bx − h + k

For horizontal stretches and compressions, recall

The parent function y = bx has key features: • Increasing for b > 1, and decreasing for 0 < b < 1 • Domain of (−∞, ∞) and range of [0, ∞) • y = 0 is the horizontal asymptote • y-intercept at (0, 1) • Another point at (1, b) To get a more precise shape, we can map the asymptote and the points (0, 1) and (1, b) using the transformations. 442

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Example 3 For each graph, ﬁrst identify the transformation from the parent function f (x) = a

5 4 f (x) 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

, then write the equation of g (x).

y g(x)

x 1 2 3 4 5

Create a strategy Compare the graph of the parent function with the graph of the transformed function. Notice that the parent function is shifted to the right to create g (x).

Apply the idea 5 4 f (x) 3 2 1

This shift can be identiﬁed by selecting a point on the function f (x) and observing the movement of its corresponding point on g (x).

y g(x)

For example, the point f (0) = 1 corresponds to g (3) = 1 on the new function.

1 2 3 4 5

This shows that the x-value moved 3 units to the right while the y-value remained unchanged, indicating a horizontal shift of 3 units in the positive direction.

A horizontal translation is represented by transformed function is

. We have just found that h = 3. Therefore, the equation of the

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

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b f (x)

5 4 3 2 1

−4 −3 −2 −1 −1 −2 −3 −4 g(x) −5

x 1

Create a strategy Compare f (x) and g (x). Notice that it appears they are the same size and shape, but g (x) is a mirror image of f (x) over the x-axis.

Apply the idea

f (x)

5 4 3 2 1

−4 −3 −2 −1 −1 −2 −3 −4 g(x) −5

Let’s identify points on f (x) and compare them to the corresponding points on g (x).

For example, f (−1) = 3 corresponds to g (−1) = − 3. x 1

For each point on f (x), the corresponding point on g (x) shares the same x-value, but the y-value is negative. This indicates a reﬂection across the x-axis.

In the function notation g (x) = a ⋅ f (x), a reﬂection across the x-axis is represented by a = − 1. Therefore, the equation of g (x) is

Reﬂect and check To check our solution, we can ﬁnd points on f (x) and g (x) algebraically and compare them. Let’s compare the points on f (x) and g (x) when x = 1. When we substitute x = 1 into f (x), we get When we substitute x = 1 into g (x), we get

. This gives us the point . This gives us the point

. .

When we compare these two points, we see that they share the same x-value, but the y-value of g (x) is the negative of f (x). We can do this with every possible x-value and get the same result where the y-values of g (x) are the negative of the y-values of f (x). This conﬁrms that we have a reﬂection about the x-axis.

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10 9 8 7 6 5 4 3 2 1

f (x)

g(x)

−5 −4 −3 −2 −1

Create a strategy Compare f (x) and g (x). Notice that they have the same horizontal asymptote and y-intercept, so a translation has not occurred. The shape of the functions are different, indicating a dilation has occurred.

Apply the idea

f (x)

g(x)

10 9 8 7 6 5 4 3 2 1

Similar to the previous examples, let’s consider the corresponding points for f (−1) and f (−2).

f (−1) = 3 corresponds to g (−2) = 3. Also, f (−2) = 9 corresponds to g (−4) = 9. For each point on f (x), the corresponding point on g (x) has the same y-value, but the x-value has been doubled. This indicates a horizontal stretch by a factor of 2. x

−5 −4 −3 −2 −1

A horizontal dilation is represented by

. We found the scale factor to be 2, so c = .

where

Therefore, the equation of g (x) is

Reﬂect and check It can sometimes be diﬃcult to identify whether the dilation was vertical or horizontal, but checking more than one point on the graph can help us avoid mistakes. 10 f (x) 9 8 7 6 5 4 3 g(x) 2 1 −5 −4 −3 −2 −1

For example, we may have mistakenly said the corresponding point of f (−2) = 9 was g (−2) = 3. This would correspond to a vertical

shrink by a factor of . We can now check whether f (−1) = 3 was also compressed by . However, g (−1) ≈

which corresponds to a scale factor of .

Since these scale factors are not the same, we know that the graph was not dilated vertically. x 1

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Example 4 Given the parent function f (x) = 4x, graph g (x). a g (x) = 4 − x

Create a strategy We can use a table of values to graph the parent function, then identify the transformation that occurred to create g (x). Then, we can use the transformation to graph g (x).

Apply the idea First, create a table of values to graph the parent function f (x) = 4x: −2

x f (x)

−1

18 16 14 12 10 8 6 4 2 −4 −3 −2 −1 −2

f (x)

x 1

Notice that the only change between the parent function, f (x) = 4x, and the transformed function, g (x) = 4 − x, is the exponent of g (x) is negative. This indicates a reﬂection across the y-axis. By reﬂecting each point of f (x) across the y-axis, we can create the graph of g (x): 18 16 14 12 10 8 6 4 2 −4 −3 −2 −1 −2

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g(x) 1

x 2

Reﬂect and check Using technology, we can check that we have correctly identiﬁed and graphed the transformation.

Notice that the graph of g (x) is a reﬂection of f (x) across the y-axis. We can also see that the points on our graph match the points shown in the graphing calculator. This veriﬁes our solution. b g (x) = − 4x + 3

Create a strategy The parent function is f (x) = 4x, and the transformed function is g (x) = − 4x + 3. In the transformation notation, y = a (b)x − h + k, the negative sign corresponds to a = − 1 and constant of 3 corresponds to k = 3. Recall that the a-value indicates a vertical reﬂection or dilation, and the k-value indicates a vertical translation.

Apply the idea 8

f (x)

x 1

−4 −3 −2 −1 −2

2 −4 −3 −2 −1 −2

x 1

3 units

−4

−6

−8

When a = − 1, a reﬂection across the x-axis has occurred. This means that the graph of g (x) will be a mirrored version of the graph of f (x) with respect to the x-axis.

Since k = 3, the original function has been translated up 3 units.

Begin by graphing f (x), then reﬂect each point to the other side of the x-axis.

Note that translating the function upwards will also translate the horizontal asymptote. The asymptote was y = 0, so it will now be y = 3.

−4 −3 −2 −1 −2

f (x) x 1

−4 −6

g(x)

−8

g(x)

To graph this tranformation, we will move each point on the reﬂected graph up 3 units.

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Reflect and check Recall that when graphing transformations, we always graph reflections and dilations before translations. If we had graphed the translation first, we would have gotten a different graph. This shows f (x) after it has been translated up 3 units.

8 6 4 2

−4 −3 −2 −1 −2

−4 −6 −8

Now, applying the reﬂection to the translated function, we would get the graph shown.

8 6

Notice that this graph is not the same as the one we found, which shows that the order in which we apply the transformations matters.

4 2

−4 −3 −2 −1 −2

−4 −6 −8

Create a strategy The parent function is f (x) = 4x, and the transformed function is functions are the negative sign and the factor a-value in the transformation notation y = a (b)

. The differences between the two

in front of the exponential term. Both of these correspond to the x−h

+ k.

Apply the idea . Since a < 0, the function has been reﬂected across the x-axis.

For

Also, 0 < ∣a∣ < 1, so the function has been compressed vertically by a factor of . x f (x) g (x)

448

−2

−1

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To apply the vertical dilation, we will multiply each point on f (x) by .

6 f (x)

2 −4 −3 −2 −1 −2

x 1

−4 −3 −2 −1 −2

−4

−6

−8

x 1

g(x)

Plotting these points helps use see that the

Next, we will reﬂect each point across the x-axis.

dilation of the graph of f (x) by a factor of .

This means that the dilated graph will be a mirrored version of the graph with respect to the x-axis. Note that the horizontal asymptote will remain at y = 0.

6 4 2 −4 −3 −2 −1 −2 −4

x 1

g(x)

−6 −8

Reflect and check The order in which we apply reflections and dilations does not matter. If we had reflected the function first, then dilated it, we would have gotten the same graph. Again, we can verify our solution using technology.

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Example 5 Consider the following exponential functions: x g (x)

−2 −9

−1 −3

y −2

−1

−1 −2 −3

−1

f (x)

−4 −5

a Determine which function increases at a slower rate.

−6 −7 −8 −9

Create a strategy To help us visualize the functions, we can graph them on the same coordinate plane. x

y −2

−1

−1 −2 −3

f (x)

−4 −5 −6 g(x)

−7 −8 −9

Apply the idea When we look at the graph, we can see g (x) does not seem as steep as f (x) and its curve is more rounded. This is a visual indication that g (x) increases at a slower rate.

Reﬂect and check Since the functions have not been translated, another way to determine the answer is by comparing the constant factors of the functions. We can ﬁnd the constant factors by dividing the outputs of two consecutive values of x. For f (x), we can use the points (−1, −8) and (0, −2).

For g (x), we can use the points (−2, −9) and (−1, −3).

Since

, f (x) gets multiplied by a smaller factor than g (x). Because of this, f (x) will get closer to 0 faster, so g (x)

increases at a slower rate.

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b Identify the y-intercept for each function.

Create a strategy On the given graph of f (x), we will look for the point where the function intercepts the y-axis. In the table for g (x), we will look for the point where x = 0.

Apply the idea The y-intercept of f (x) is (0, −2). The y-intercept of g (x) is (0, −1).

Reﬂect and check Now that we know the y-intercept and constant factor for each function, we can build the equations.

c Identify and compare f (−1) and g(−1).

Create a strategy To ﬁnd f (−1), we can use the graph to locate the point (−1, y). For g(−1), we can use the table of values to ﬁnd the corresponding y-value when x = − 1. Then, we can compare the two functions.

Apply the idea x

y −2

−1

−1 −2 −3

Looking at the graph, we will ﬁnd the point on the function when x = − 1. This occurs at (−1, −8), therefore, f (−1) = − 8.

f (x)

−4 −5 −6 −7 (−1, −8)

−8 −9

Using the table of values for the function g (x), we can ﬁnd the value of g(−1) by locating the row with x = − 1 and reading its corresponding y-value. When x = − 1, g (x) = − 3. We ﬁnd that f (−1) = − 8 and g(−1) = − 3 which are different values and g(−1) > f (−1).

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Idea summary We can use transformations to graph an exponential function in the form y = a ⋅ bc(x − h) + k and identify key features: y = − bx y = b−x

Reflection across the x-axis: Reflection across the y-axis: Vertical stretch when ∣a∣ > 1

y = a(b)x

Vertical compression when 0 < ∣a∣ < 1: Horizontal compression when ∣c∣ > 1 Horizontal stretch when 0 < ∣c∣ < 1: Horizontal translation by h Vertical translation by k:

y = bc ⋅ x y = bx − h + k

For horizontal stretches and compressions, recall

The parent function y = bx has key features: • • • • •

Increasing for b > 1, and decreasing for 0 < b < 1 Domain of (−∞, ∞) and range of [0, ∞) y = 0 is the horizontal asymptote y-intercept at (0, 1) Another point at (1, b)

Inverses of exponential functions Recall that when inverse functions are graphed on the same Coordinate plane, they are reﬂections of each other about the line y = x. 4

f (x) = 4x y=x

3 2 1 −4 −3 −2 −1 −1 −2

x 1

f −1 (x)

−3 −4

Each point (x, y) on the graph of f (x) corresponds to the point ( y, x) on f − 1 (x). The key features are affected in a similar way. • The y-intercept of f (x) becomes the x-intercept of f − 1 (x). • The x-intercept of f (x) becomes the y-intercept of f − 1 (x). • The horizontal asymptote of f (x), y = a, becomes the vertical asymptote of f − 1 (x), x = a. • The domain of f (x) becomes the range of f − 1 (x). • The range of f (x) becomes the domain of f − 1 (x).

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Example 6 Consider the graph of f (x) = 3x. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

a Sketch the inverse function on the same coordinate plane.

Create a strategy Swap the x and y-values on the curve of y = 3x to ﬁnd the corresponding points on the curve of the inverse function.

Apply the idea 4

Based on the graph of y = 3x, the points (0, 1) and (1, 3) lie on the curve of y = 3x.

(1, 3)

2 1

(0, 1)

−4 −3 −2 −1 −1

x 2

−2 −3 −4

4 3 (0, 1) −4 −3 −2 −1

(1, 3)

−1

The inverse points corresponding to (0, 1) and (1, 3) are (1, 0) and (3, 1), respectively. So, these points can be used to sketch the graph of the inverse function of y = 3x.

(3, 1) x 1 2 (1, 0)

−2 −3 −4

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b Compare the key features of f (x) and its inverse including intercepts, zeros, asymptotes, domain, and range.

Create a strategy Use the graphs from part (a) to compare the y-intercepts, zeros, asymptotes, domain, range, and any increasing or decreasing intervals. Remember that the inverse function is created by swapping the inputs and outputs, so the features related to inputs and outputs will be affected in a similar way.

Apply the idea By observing the graphs of both functions, we can now compare their key features: 1. y-intercepts: The point where the graph crosses the y-axis. • f (x): y = 1 • f −1(x): none 2. Zeros: Zeros are the x-intercepts of the graph. • f (x): none • f −1(x): x = 1 3. Asymptotes: • f (x): has a horizontal asymptote at y = 0 but no vertical asymptote. • f −1(x): has a vertical asymptote at x = 0 but no horizontal asymptote

3 2 f (x)

1 (0, 1)

−4 −3 −2 −1 −1

1 2 (1, 0)

x 3

−2 f −1 (x) −3 −4

4. Domain: The set of all possible x-values for which the function is defined. • f (x): all real numbers, (−∞, ∞) • f −1(x): (0, ∞) 5. Range: The set of all possible y-values for which the function is defined. • f (x): (0, ∞) • f −1(x): all real numbers, (−∞, ∞) 6. Increasing and Decreasing: Both functions are always increasing on their domain. They are never decreasing. In general, the features of the original function are the opposite of the corresponding features on the inverse function.

Idea summary Inverse functions are reflections of each other about the line y = x. Each point (x, y) on the graph of f (x) corresponds to the point ( y, x) on f − 1(x). The key features are affected in a similar way. • • • • •

The y-intercept of f (x) becomes the x-intercept of f − 1(x). The x-intercept of f (x) becomes the y-intercept of f − 1(x). The horizontal asymptote of f(x), y = a, becomes the vertical asymptote of f − 1(x), x = a. The domain of f (x) becomes the range of f − 1(x). The range of f (x) becomes the domain of f − 1(x).

Practice What do you remember? 1

454

Consider the exponential functions shown in the following tables of values: i

Determine the base of the exponent.

Use the base of the exponent and the y-intercept from the table to write the exponential function in the form y = a ⋅ bx.

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x −1 0 1 2 y 1 3 9 27

3 81

x −1 0 1

250

x y

2 4

3 0.4

−1 4000

0 400

1 40

State whether the following are increasing or decreasing exponential functions: y = 9 ⋅ 3x

a 3

−1

y 2

Consider the table of values: x

Select the exponential function that could represent the table. B

A 4

Consider the table of values: x

−4

−3

−2

−1

112

Select the exponential function that could represent the table. B

A 5

Which of the following graphs represents an exponential function? A

m(x)

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

x 1

−2

−3

−4

p(x)

D 4

1 −4 −3 −2 −1 −1

n(x)

x 1

q(x)

1 −4 −3 −2 −1 −1

−2

−3

−4

x 1

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Describe the transformation applied to each function. a

g(x) = 2x − 3

f (x)

1 −4 −3 −2 −1 −1

x 1

g(x)

−4 −3 −2 −1 −1

x 1

−2

−2 −3

−3

−4

Let’s practice 7

Select the exponential function that represents the following graph.

5(4)x

−5(4)x

5 4 3 2 1

−4 −3 −2 −1

Select the exponential function that represents the following graph. A C

y = − 5x

y = − 9x

−5 −4 −3 −2 −1−1

x 1 2 3 4 5

−2 −3 −4 −5 −6 −7 −8 −9 −10 −11

Consider the graph of the equation y = 4x:

State the equation of the horizontal asymptote.

Explain what the horizontal asymptote means for the end behavior of the exponential function.

4 3 2 1 −3 −2

−1 −1

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x 1

Consider the table of values for the function x

−5

−4

−3

−2

−1

Describe the behavior of the function as x increases.

Determine the y-intercept of the function.

State the domain of the function.

State the range of the function.

Consider the function g(x) = 4 (2x). a

Find the y-value of the y-intercept of the curve.

As x approaches inﬁnity, determine the value that y approaches.

Identify the transformations on g(x) = 4 (2x) from the parent function f (x) = 2x.

Graph g(x) = 4 (2x).

List the domain and range for the function.

Consider the function

Find the y-value of the y-intercept of the curve.

Complete the table of values for x y

−3

−2

−1

Find the horizontal asymptote of the curve.

Graph

List the domain and range of the function.

Consider the equation g(x) = − 10x. a

Jenny thinks she has found a set of solutions for the equation as shown in the table: x

−2

−1

g(x)

−1

−10

−100

−1000

he notices that all the g(x) values are negative and concludes that for any value of x, g(x) must always be S negative. Determine whether she is correct. Explain your answer.

Graph g(x) = − 10x.

Find the values of x for which g(x) = 0.

Consider the original graph y = 3x. The function values of the graph are multiplied by 2 to form a new graph. a

For each point on the original graph, ﬁnd the point on the new graph. Point on original graph Point on new graph

(−1, ⬚)

(0, 1)

(1, 3)

(2, 9)

(0, ⬚)

(1, ⬚)

(2, ⬚)

State the equation of the new graph.

Graph both functions on the same coordinate plane and compare their key features.

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457

For each of the following, evaluate the functions for the values: i

x = −2

x=0

j(x) = 2 ⋅ 4x

j(x) = − 1 ⋅ 4x + 3

f (x)

iii

x=2

4.5 4 3.5 3 2.5 2 1.5 1 0.5

8 7

6 5 4 3 2 1 −2

−1

f (x) −2

−1

−2

−2 −3 −4

−2

−0.5

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

−1

−5

−1

g(x)

−1 −0.5

The population of a certain species of ﬁsh in a lake is modeled by the exponential function P (t) = 300(1.08)t, where P (t) is the population of the ﬁsh at time t years after a conservation effort has begun. Evaluate the function for the following values and explain what each value represents in the context. a

t=0

t=3

t=5

Write the equation of the transformed exponential function g(x) of the parent function a

6 g(x)

−8 −6 −4 −2 −2

x 2

g(x)

2 −8 −6 −4 −2 −2

−4

−6

−8

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x 2

g(x)

2 −8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

−4

−6

−8

Graph the transformed exponential function n(x) of the parent function m(x) = 3x a

n(x) = 3x + 2

e 19

n(x) = 3x − 2

n(x) = 3 − x

n(x) = 32x

n(x) = − 1 ⋅ 3x

Which function best represents this graph? A

f (x) = − 3(x − 3)

f (x) = 3(x − 2)

f (x) = 3x + 2

f (x) = − 3x + 2

5 4 3 2 1 −4 −3 −2 −1

Do either of the functions y = 9x or

Draw the graphs of the functions following questions: a

−1

have zeros? Explain your answer. ,

and

by hand or using technology, then answer the

All of the curves have a maximum value.

All of the curves pass through the point (1, 2).

iii All of the curves have the same y-intercept.

None of the curves cross the x-axis.

State the y-intercept of each curve.

Describe what happens to the values of y as x gets very large.

Consider the functions f (x) = 3x and following graph:

plotted on the

y 15

State the coordinates of the point of intersection of the two curves.

Describe what happens to the values of y for each function as x gets very large.

State whether the following statements are true for all of the functions: i

x 1

Describe what other features these functions have in common.

f (x) = 3x

3 −5 −4 −3 −2 −1 −3

x 1 2 3 4 5

−6

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Consider the following exponential functions: f (x), g(x), and h(x). y 8

f (x)

−2 −1 0 1 2

6 4 2 −8 −6 −4 −2 −2

x 2

g(x) 4 2 1 0.5 0.25

h(x) = 2x − 6

−4 −6 −8

Determine whether each function is increasing or decreasing.

Compare the end behavior of each function as x increases?

Compare and contrast the asymptotes of f (x), g(x), and h(x).

Consider the functions f (x) = 4x + 2 and a

Graph the two functions using technology.

Compare the domain and range of f (x) and g(x).

Consider the given graph of a

Complete the table of values for y = − 5 . −2

−1

and y = − 5x on the same coordinate plane.

Graph

Compare the domain and range of

−3−2 −1 −2 −4 −6 −8 −10

and y = − 5x.

x 1 2 3 4 5 6 7 8 9

For each of the following exponential functions, graph its inverse on the plane. a

2 −8 −6 −4 −2 −2

460

10 8 6 4 2

x y

x 2

2 −8 −6 −4 −2 −2

−4

−6

−8

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x 2

2 −8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

−4

−6

−8

x 2

Let’s extend our thinking 27

Consider the graphs of the two exponential functions R and S: a

One of the graphs is of of

18 16 R 14 12 10 8 S 6 4 2

and the other graph is

. Identify which is the graph of

Explain your answer. b

For x < 0, determine the relationship between

and

. Explain your reasoning.

−3 −2

x 1

A popular video tutorial on crafting starts spreading on social media. Initially, it is shared by one craft enthusiast to ﬁve of their followers. a

Assuming each follower who watches the video shares it with ﬁve new followers the next day, construct the table showing the number of new followers who receive the video each day. Day Number of new followers reached

−1

0 1

If this trend continues, write an expression for the total number of new followers reached on day n following this sharing pattern.

If the goal is to reach 1 000 000 followers, calculate how many days it will take for the video to achieve this reach, assuming the trend continues.

If a social media algorithm change increases the average share rate from four to ﬁve followers per day, how many followers will be reached by day 6 under this new rate, starting from the beginning?

A small town’s population is increasing by 3% annually due to new job opportunities. The current population is 10 000 residents. a

Create a function, P, to describe the town’s population in t years where r is the rate of increase. Write the function in the form P = a(1 + r)t.

Calculate the expected population of the town in 10 years.

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For each function, describe transformations applied from the parent function f (x) = 2x. a

h(x) = 2x + 2 − 1

f (x)

1 −4 −3 −2 −1 −1

−4 −3 −2 −1 −1

−2

−3

−4

x 1

State whether the following functions are equivalent to i

462

Consider the function a

x 1

g(x)

n(x) = 2 − x

iii

Graph the functions g(x) = 2x and

Which transformation would transform g(x) to f (x)?

Mathspace Virginia SOL Algebra 2 mathspace.co

: p(x) = − 2x

q(x) = − 2 − x

on the same coordinate plane and compare them.

Exploration Here are three exponential functions represented in three different ways: Function 1:

Function 2: y

x y

8 7

0 1

1 1.15

2 3 1.3225 1.5209

4 1.7490

6 5 4 3 2 1

x 1

Function 3: Reagan decided to get a part-time job to begin saving for a new car. He was hired by a company that pays $21 500 in the ﬁrst year, and he will receive a 5% raise each year after that. 1.

Which of the functions would you model using y = abx and why?

Which of the functions would you model using y = a (1 + r)x and why?

In what types of situations would you prefer knowing the growth factor, b, over the growth rate, r?

We can determine the key features of an exponential function from its equation or graph: • f (x) = abx, b > 1 • f (x) = a (1 + r)x, r > 0 • The graph is increasing • y approaches a minimum value of 0 • The domain is (−∞, ∞) • The range is (0, ∞) • The y-intercept is at (0, a) • The horizontal asymptote is y = 0

• f (x) = abx, 0 < b < 1 • f (x) = a (1 − r)x, r > 0 • The graph is decreasing • y approaches a minimum value of 0 • The domain is (−∞, ∞) • The range is (−∞, 0) • The y-intercept is at (0, a) • The horizontal asymptote is y = 0

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Example 1 A frog population, F, has been following the model F (n) = 480(0.952)n, where n is the number of years. a What is the initial population of frogs?

Create a strategy The initial value will be when time is 0.

Apply the idea

Reﬂect and check

When n = 0, F = 480, so the population was 480.

Since negative values for time do not make sense, the y-intercept will be the leftmost point on the graph. In this case, we refer to it as the initial value.

b What is the decay rate? Explain what it means in this context.

Create a strategy We can ﬁnd the decay rate using that for y = abx, b = 1 − r has decay rate of r.

Apply the idea Since b = 0.952: b=1−r 0.952 = 1 − r r = 1 − 0.952 r = 0.048 This means that the decay rate is 4.8%. The frog population is declining at 4.8% per year.

c Find F (3), interpret what this means in the context.

Create a strategy We can substitute n = 3, and then evaluate using a calculator rounding to the nearest whole number.

Apply the idea

F (n) = 480(0.952)n F (3) = 480(0.952)3 F (3) ≈ 414.144 675 8 F (3) ≈ 414

After 3 years, there would be 414 frogs remaining.

d Using technology, graph the frog population for 0 < n < 120, and describe the end behavior.

Create a strategy We can input the equation and then adjust the view using the zoom so we can see from n = 0 to n = 120.

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Apply the idea A good window setting is 0 < n < 120 and 0 < F (n) < 480. 480 440 400 360 320 280 240 200 160 120 80 40

x 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

Based on the graph, we can see that the frog population is decreasing over time. The end behavior of the function shows that as the number of years n increases, the frog population F (n) approaches 0 but never reaches it. This indicates that the frog population will continue to decrease.

Reﬂect and check By utilizing technology to monitor frog populations, we observe a declining trend over time, with the numbers approaching zero without actually reaching it. If this trend persists, it’s important to note that in reality, a population cannot include a fraction of a frog. Therefore, the actual number of frogs would ultimately drop to zero.

e Use your graph to determine approximately how many years it will take for the population to reach half of the original population.

Create a strategy We want to ﬁnd half of 480 on the vertical axis and then slide over to ﬁnd where the graph reaches that value.

Apply the idea = 240.

Half of 480 is 480 440 400 360 320 280 240 200 160 120 80 40

x 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

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Using our original scale we can see this is between n = 10 and n = 15, so we can zoom in to get a better estimation. 275 y 270 265 260 255 250 245 240 235 230 225 220 215 210 205

x 11

From the zoomed in graph we can see that it is just after 14 years where the population reaches half of the original value.

Reflect and check We can solve for the exponent using a graph, table of values, or writing both sides of the equation with the same base if possible.

Example 2 The population of rabbits in Lincoln county can be modeled by an exponential growth function. Conservationists have been measuring the population since 2015. After the first year, there were 46 rabbits. After the third year, there were 66 rabbits. After the fifth year, there were 95 rabbits, as shown in the table below. Year Population

2015

2016 46

2017

2018 66

2019

2020 95

a Estimate the growth rate to one decimal place.

Create a strategy To estimate the growth rate, we will create data points from the given information. If we let x = 0 represent 2015, then x = 1 will be the ﬁrst year. The associated data point is (1, 46). The data point for the third year is (3, 66). These are not consecutive outputs, so we need to set up an equation and solve for r.

Apply the idea To get from the ﬁrst year to the third year, we multiplied the population by the growth factor twice. The equation would be 46 ⋅ b ⋅ b = 66 which simpliﬁes to 46b2 = 66. Given equation

Division property of equality

Approximate by taking the square root of both sides

This represents the growth factor, so now we must solve for the growth rate. 1.198 = 1 + r r = 0.198 Next, we can convert to a percentage and round it to one decimal place. r = 19.8% = 20%

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b Estimate the initial population.

reate a strategy The initial population will be the number of rabbits present in 2015. Now that we know the growth rate, we can use the number of rabbits in the ﬁrst year, 2016, to help us solve for the initial population.

Apply the idea To set up the equation, we begin by substituting the known values r = 0.2 from part (a), x = 1, and y = 46 into the equation y = a(1 + r)x. Exponential growth equation

Substitute y = 46, r = 0.2, and x = 1

Evaluate the addition and exponent

Division property of equality

Approximate by evaluating the division

The initial population of rabbits in 2015 was approximately 38.

c Write the equation that models this situation.

Create a strategy Since we discussed the percentage growth, we will use the growth rate form of the equation, y = a (1 + r)x.

Apply the idea We found r = 20%, a = 38 in parts (a) and (b), so we can simply plug them into the formula. y = 38 (1 + 0.2)x

Reﬂect and check Because we rounded the percentage in part (a), the values will not be exact when we check our answers. But since we are working with rabbits and we cannot have a decimal of a rabbit, it makes sense to round to the nearest whole rabbit. Year 1: y = 38(1 + 0.2)1 = 45.6 ≈ 46 rabbits Year 2: y = 38(1 + 0.2)2 = 54.72 ≈ 55 rabbits Year 3: y = 38(1 + 0.2)3 = 65.664 ≈ 66 rabbits These answers match the information given in the problem.

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Example 3 The graph shows the exponential decay of Plutonium-236, highlighting a rapid decrease in its quantity, measured in grams, over time, measured in years, due to its radioactive decay. 66 60 54 48 42 36 30 24 18 12 6 −4

−3

−2

−1

a Estimate the remaining grams of a Plutonium-236 sample after 8 years, given its exponential decay.

Create a strategy To estimate the remaining grams of Plutonium-236 after 8 years, we can use the graph to observe the trend of exponential decay. By locating the point on the graph corresponding to x = 8 (8 years), we can estimate the value of y (remaining grams of Plutonium-236).

Apply the idea Looking on the graph where x = 8, we can see that the exponential decay curve is close to the y-axis gridline between 6 and 12. By estimating the position of the curve at x = 8, we can approximate the remaining grams of Plutonium-236. 66 60 54 48 42 36 30 24 18 12 6 −4

−3

−2

−1

x 1

By zooming in, at x = 8, the curve appears to be very close to gridlines of 6 and 7 but slightly below the midpoint of 6 and 7 on the y-axis. Considering that the middle point between 6 and 7 is 6.5, we can estimate that the remaining grams of Plutonium-236 after 8 years is approximately 6.5 grams.

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9 8 7 6 5 4 3 2 1

x 7

−1

Reflect and check An alternative way to find the remaining grams of Plutonium-236 after 8 years is to first determine the decay factor by examining the trend shown in the graph and then finding the equation of the function. Once we have the equation, we can substitute x = 8 to find the remaining grams. Looking at the graph, we can see that the curve goes through the points (0, 64) and (1, 48). We can use these points to find the decay factor (r). The exponential decay function is in the form y = a(1 − r)x, where a is the initial amount and r is the decay factor. Since the initial amount at x = 0 is 64, the equation becomes y = 64(1 − r)x. Using the point (1, 48), we can solve for r: 48 = 64(1 − r) 0.75 = 1 – r r = 0.25

Substitute x = 1 and y = 48 Divide by 64 on both sides Subtract 0.75 from both sides

Now that we have found the decay factor to be 0.25, we can write the equation of the function as y = 64(1 − 0.25)x. To ﬁnd the remaining grams after 8 years we can substitute into the equation and use a calculator to evaluate: y = 64(1 − 0.25)8

Substitute x = 8

y ≈ 6.41

Evaluate the expression

Using this alternative method, we can estimate that the remaining grams of Plutonium-236 after 8 years is approximately 6.41 grams which is close to our initial estimate of 6.5 grams based on the graph.

b What is the end behavior of the function and explain what that means in the context of the scenario.

Create a strategy Observe the graph to determine the end behavior of the function as x approaches positive inﬁnity. Then, explain the meaning of this behavior in the context of Plutonium-236’s radioactive decay.

Apply the idea As x approaches positive inﬁnity, the value of the function approaches 0, which means the quantity of Plutonium-236 always decreases as time goes on. In the context of the scenario, this indicates that over time, the amount of Plutonium-236 will continue to decrease due to its radioactive decay, eventually becoming negligible.

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Reﬂect and check Let’s verify the end behavior of the function by using the equation 64(1 − 0.25)x and plugging in large values of x using a calculator. Evaluating the function for large x values will help us understand how the function behaves as time goes on. For instance, let’s calculate the value of the function for x = 100, x = 500, and x = 1000: 64(1 − 0.25)100 ≈ 1.84 ⋅ 10 − 7

Evaluate for x = 100

− 35

Evaluate for x = 500

64(1 − 0.25)1000 ≈ 1.64 ⋅ 10 − 63

Evaluate for x = 1000

64(1 − 0.25)

500

≈ 1.28 ⋅ 10

As we can see, as x increases, the value of the function approaches 0. This is a horizontal asymptote of y = 0. This conﬁrms the end behavior we observed earlier, where the amount of Plutonium-236 will continue to decrease due to radioactive decay, eventually becoming negligible.

Idea summary Exponential functions can also be expressed in terms of their constant percent rate of change.

f (x) = a (1 ± r)x a r

The initial value The growth or decay rate

This means that for f (x) = abx: • •

if b > 1, the growth rate will be r = b − 1. if 0 < b < 1, the decay rate will be r = 1 − b.

Compound interest A common application of exponential growth is compound interest, where interest applies to the current balance, not the initial investment or loan. We can write the compound interest formula as:

A = P (1 + r)n A P r n

closing balance or future value of the investment principal (starting) amount periodic interest rate, as a decimal number of time periods

The compounding period is the length of time between interest payments. Here are some possible periods: Annually Semi-annually Quarterly Monthly Bi-weekly Weekly Daily

Number of period per year 1 2 4 12 26 52 365

Length of time 1 year 6 months 3 months 1 month 2 weeks 1 week 1 day

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Reflect and check Let’s analyze the amount of interest Valerie has to pay over the fifteen years. We can calculate the interest by subtracting the initial amount borrowed from the total amount owed after fifteen years. Interest = 3750.54 – 1250 = $2500.54

Subtract the initial amount borrowed Evaluate

The interest generated on the loan over the fifteen years is $2500.54. This means that Valerie has to pay an extra $2500.54 in interest over the course of the loan. As a result, it is essential for Valerie to consider this additional cost when making decisions about the loan and her financial situation. It is always important to consider the interest that accumulates over time when taking out a loan, as it can significantly impact the total amount that needs to be repaid.

Example 5 A $9450 investment earns interest at an annual rate of 2.6% compounded monthly over 14 years. a What is the value of the investment after 14 years?

Create a strategy Use the compound interest formula: A = P (1 + r)

Apply the idea n

We are given that: P = 9450, r =

and n = 14 ⋅ 12 months = 168 months

Substitute the values Evaluate The future value is $13 593.90

b The following is a graph of the investment over time.

y 18000 16000 14000 12000 10000 8000 6000 4000 2000

x 5

Estimate how much the investment will be worth in 15 years.

Create a strategy To estimate how much the investment will be worth in 15 years, we can look at the point on the graph where x = 15, and estimate the value of y.

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Apply the idea Looking at the graph at x = 15, we can see that the y-value is approximately y = 14 000.

y 18000 16000

The estimated investment in 15 years is $14 000.

14000 12000 10000 8000 6000 4000 2000

x 5

Reﬂect and check To improve the accuracy of our estimations, we can use technology to evaluate the exponential function at x = 15 and at y = 16 000. Let’s use the compound interest formula:

where A is the ﬁnal amount, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years. Substitute P = 9450, r = 0.026, n = 12, and t = 15

Calculate the value using technology

The investment will be worth approximately $13, 951.58 in 15 years.

c Estimate how long it will take the investment to be worth $16 000.

Create a strategy Locate the point on the graph where y = 16 000 and determine the corresponding x value.

Apply the idea At y = 16 000, the corresponding x-value is very close to x = 20.

y 18000

In just over 20 years, the investment will be worth $16 000.

16000 14000 12000 10000 8000 6000 4000 2000

x 5

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Reﬂect and check Let’s solve for a more precise time that it will take for the investment to reach $16, 000. Substitute P = 9450, A = 16 000, r = 0.026, and n = 12

Solve for t using technology

It will take approximately 20.27 years for the investment to be worth $16, 000. By using technology to evaluate the exponential function, we have found a more accurate approximation for the time it takes for the investment to reach $16, 000.

Idea summary We can calculate compound interest with the formula:

A = P (1 + r)n A P r n

closing balance or future value of the investment principal amount interest rate, as a decimal number of time periods

If interest is compounded daily, weekly, bi-weekly, monthly, quarterly, or half-yearly, then we need to convert the interest rate and number of time periods to the same units.

Practice What do you remember? 1

For each exponential functions, where x represents time: i

Is it exponential growth or exponential decay?

iii

What is the growth or decay rate as a percentage?

f (x) = 0.723 (1 − 0.05)x

c 2

f (x) = 5.9 (1.025)

What is the initial amount?

f (x) = 2500 (0.877)x

f (x) = 10 000 (2)x

This is a table of values for an exponential function: x f (x) a

1 5

2 7

What is the growth factor?

3 9.8 b

4 13.72

5 19.208

What is the constant percent rate of change?

Frasier is calculating the compound interest accumulated on his loan. He uses the following formula:

Identify how much Frasier borrowed in dollars.

Identify the annual interest rate as a percentage.

Identify if the interest is being compounded weekly, monthly, quarterly or annually.

Identify how many years he is calculating the interest for.

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For compound interest, are these statements true or false? a

The account balance grows exponentially.

The amount of interest accrued for the ﬁrst and second periods will be the same.

Assuming the account remains untouched, the amount of interest accrued per period increases.

The interest rate is applied to the initial principal.

Which graph could represent a savings account that earns compound interest over 50 years? a

Balance ($)

Time (years)

Balance ($)

Time (years)

Balance ($)

Time (years)

Are these situations exponential growth, exponential decay, or neither? a

A store owner checked the sales report for the previous month and found that each week the sales were of the previous week’s sales.

In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found to triple each hour.

A sequence starts with 2, 3 and the next term is always the product of the previous two terms.

A piece of land was purchased for $80, 000. The value of the land has slowly been decreasing by 2% annually.

Michael has $1, 300 in his checking account. He gets paid $600 every 2 weeks.

Let’s practice 7

476

The local rat population is changing according to the exponential function f (t) = 840 (1 − 0.04)t where t is the number of years that have passed. a

What is the initial size of the rat population?

Is the population increasing or decreasing?

Mathspace Virginia SOL Algebra 2 mathspace.co

Chromium-51 is used to label red blood cells and has a half-life of approximately 30 days. The amount of chromium-51 in micrograms (μ g) remaining in the sample over time is modeled on the graph. 220 200 180 160 140 120 100 80 60 40 20

Chromium-51 ( µ g)

Time (days) 30

120

Find the initial value and interpret this value in the context.

Find f (x) when x = 60. Explain what this value means in context.

State the equation of the horizontal asymptote.

Describe what happens to the amount of Chromium-51 over time.

This graph shows the growth of an investment compounded semi-annually. Future value ($) 120 100 80 60 40 20 Time (years) 5

State the domain and interpret in context.

State the range and interpret in context.

Find x when y = 100 and interpret in context.

The given graph shows two investments, one with simple interest (Investment A) and another with compound interest (Investment B): Investment ($1000) 10 8

6 4

2 Time (years) 2

Which investment has a higher principal amount?

Which investment has a higher ﬁnal amount after 10 years?

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The number of fungal cells, N, in a colony after t minutes is given by N = 5000 (1.023)t. a

State the initial population of fungal cells.

Find the population of fungal cells after 3 hours, rounding to the nearest integer.

Graph the fungal cell population over the domain [0, 120].

Evaluate for t = 100, rounding to the nearest integer, and interpret your solution in the context.

The mass in kilograms of a baby orangutan at n months of age is given by the function M (x) = 1.8 (1.1n), for ages up to n = 6 months. a

Evaluate for n = 3 to the nearest tenth and interpret in the context.

Determine the domain and range in context. State any restrictions, and explain your reasoning.

Does this function have any zeros? Why or why not?

For each loan: i

Write the equation that can be used to ﬁnd the future value of the loan.

Find t = 5 and interpret in context.

Tariq borrows $8000 from a loan shark at a rate of 5.4% compounded daily. Assume there are 365 days in a year.

Umberto borrows $6000 from a bank at a rate of 7.2% compounded weekly. Assume there are 52 weeks in a year.

Veena borrows $7000 from a bank at a rate of 3.4% compounded monthly. There are 12 months in a year.

For each investment: i

Write the equation that can be used to ﬁnd the future value of the investment.

Find the interest earned when t = 4 and interpret in context.

Pilar’s investment of $4200 earns interest at 2.7% compounded quarterly. There are 4 quarters in a year.

Yasser’s investment of $4920 earns interest at 5% compounded semi-annually. There are 2 semi-annual periods in a year.

Enzo’s investment of $6220 earns interest at 2.8% compounded annually.

Amon and Blair are taking part in a study on blood sugar levels and are asked to eat the same food and in equal quantity, with the only difference being that Amon incorporates lots of sugar into his diet while Blair consumes limited amounts of sugar in her diet. After eating, their blood sugar level (amount of sugar in the bloodstream) peaked, and was then continually measured, with the following functions modeling their blood sugar over time: Amon: f (t) = 179 (0.3)t, where f (t) represents blood sugar level at time t hours. Blair: g (t) = 132 (0.6)t, where g (t) represents blood sugar level at time t hours.

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Determine the value at which Amon’s blood sugar level peaked.

Determine the value at which Blair’s blood sugar level peaked.

Identify which person’s blood sugar level decreased at a slower rate.

Determine the percentage by which Amon’s blood sugar level decreased each hour.

Explain how the amount of sugar consumed effects your blood sugar levels over time, according to these models.

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The magnitude of an earthquake is measured by values on the Richter scale. The function relating the Richter scale measure (x) and microns of ground motion ( y) is given by y = 10x. a

Complete the table of values. Richter scale measure (x) Microns of ground motion ( y)

0 1

Determine the domain and range in context. State any restrictions, and explain your reasoning.

Identify any increasing or decreasing intervals.

Describe the end behavior of the function, considering any restrictions in the context.

Carbon-14 is a form of carbon that is found in all living plants and animals. When the plant or animal dies, Carbon-14 is no longer taken in. As such, it is useful in estimating the age of fossils. It has a half-life of 5730 years, which means it takes approximately 5730 years for half of any amount of Carbon-14 to decay. Suppose the fossil of an organism originally with 400 grams of Carbon-14 has been found. The amount of Carbon-14 remaining in the fossil after n years can be modeled by

Find how much Carbon-14 would be left in the fossil after 1500 years. Round your answer to two decimal places. 18

When a heated substance such as water starts to cool, its temperature T at time t minutes after being left to cool is given by an exponential function. a

The function can be written in the form T = a (1 − r)t. Explain what the variables a and r mean within context.

Substance P starts off at a temperature of 145 °C and is left to cool in a room whose temperature is a constant 0 °C. By using the table, ﬁnd the equation for the temperature of the substance. Minutes passed (t) Temperature (T )

0 145

1 87

2 52.2

3 31.32

4 18.792

Find the temperature after 10 minutes, correct to two decimal places.

Will the temperature of the substance will ever reach 0 °C? Explain your reasoning.

The temperature of another substance Q which also starts off at a temperature of 145 °C is modeled by T = 145 (1 − 0.1)t. Which substance will cool more rapidly?

Let’s extend our thinking 19

Evangelina wants to invest $1400 at 5% interest for 5 years. She has two investment options, compounding quarterly or compounding daily. a

Calculate the future value of the investment if it is compounded quarterly.

Calculate the future value of the investment if it is compounded daily.

Calculate how much extra the investment is worth if it is compounded daily rather than quarterly.

What effect does the frequency of compounding have on the total amount of interest earned?

Dietrich wants to put a deposit on a house in 5 years. In order to ﬁnance the $80 000 deposit, he decides to put some money into a high interest savings account that pays 5.5% interest compounded monthly. If P is the amount of money that he must put into his account now to accumulate enough for the deposit, ﬁnd P.

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One person in a city is infected with a virus. During the ﬁrst day, they infect ﬁve more people with the virus. a

Copy and complete the table showing the number of newly infected people for that day. Day Number of people infected

0 1

If this trend continues, write an expression for the number of people infected on day n.

There are 345 800 people in the city in total. Given that the trend continues, after how many days will the entire city have been infected?

If everyone takes a daily vitamin C supplement, they can restrict the spread so that each new person only infects an average of two new people on their ﬁrst day. How many people would be infected on day 10 if this was followed from the beginning?

A teaspoon of sugar, weighing 4 grams, is poured into a cup of hot water and immediately begins to dissolve. Each second the amount of undissolved sugar remaining is a

of the amount present in the previous second.

Complete the table of values: Seconds passed (t) Undissolved sugar in grams ( y)

480

Write an equation linking undissolved sugar, y, and time, t.

Describe the change in the amount of undissolved sugar over time.

Use the model to detemine if all the sugar will eventually dissolve. Explain your reasoning.

The water is heated more so that when the experiment is performed again, the sugar dissolves twice as quickly. Will more or less sugar dissolve between the second and third seconds? Explain your reasoning.

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6.03 Logarithmic functions After this lesson, you will be able to... • identify the graphs of the logarithmic function family. • write equations of logarithmic functions from graphs using transformations. • identify the transformation given an equation or a graph. • write equations of logarithmic functions given a description of the transformations. • graph logarithmic functions given an equation using transformations. • use technology to verify transformations. • compare tables, graphs, and equations of logarithmic functions. • identify domain, range, zeros, intercepts, increasing, decreasing, constant intervals, and end behavior. • compare the characteristics of logarithmic functions with other functions. • write the equation of the vertical asymptote of a logarithmic function using a graph or equation. • find f (x) given x using a graph or equation.

Logarithmic functions Exploration For f (x) = 2x: 1.

Sketch the graph of f (x) and f − 1 (x) on the same coordinate plane.

Compare the characteristics of f (x) and f − 1 (x). What do you notice?

f (x) = bx

3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

x 1

An inverse function is a reﬂection across the line y = x which maps each point (x, y) to ( y, x). Since y = logb(x) is the inverse of f (x) = bx: • f (x) has a point at (0, 1), so f − 1 (x) has a point at (1, 0) • f (x) has a point at (1, b), so f − 1 (x) has a point at (b, 1) • f (x) has an asymptote at y = 0, f − 1 (x) has an asymptote at x = 0

f −1(x) = logb (x)

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The points that were approaching the y-axis on the parent exponential function are now approaching the x-axis in the logarithmic function. This means the parent logarithmic function will have a vertical asymptote. 4

3 2

f (x) = log2(x)

−1 −1

1 2 3 4 5 6 7 8 9

−2 −3 −4

When the base b > 1, the graph of the parent logarithmic function increases over its domain. • The domain is (0, ∞) • The range is (−∞, ∞) • Increasing over the domain: (0, ∞) • The x-intercept is at (1, 0) • No y-intercept • The vertical asymptote is x = 0 • End behavior as x → 0 +, f (x) → −∞ • End behavior as x → ∞, f (x) → ∞

When we examine the end behavior on the left side of the graph, we cannot look at what is happening as x approaches negative inﬁnity because the graph never crosses the vertical asymptote at x = 0. Instead, the x-values approach a single value, in this case x = 0. To indicate that x is approaching 0 from the positive side, we use the notation x → 0+. 4

3 2 1 −1 −1

x 1 2 3 4 5 6 7 8 9

−2 −3

When 0 < b < 1, the graph decreases over its domain. • The domain is (0, ∞) • The range is (−∞, ∞) • Decreasing over the domain: (0, ∞) • The x-intercept is at (1, 0) • No y-intercept • The vertical asymptote is x = 0 • End behavior as x → 0 +, f (x) → ∞ • End behavior as x → ∞, f (x) → −∞

−4

Since a logarithm is deﬁned as the inverse of an exponential equation, we can use this relationship to evaluate logarithmic functions. Remember that the output of a logarithmic function represents the exponent that the base must be raised to.

y = logb (x) ⟺ by = x b

is the base and b > 0, b ≠ 1

Consider the function f (x) = log4 x. If we want to ﬁnd the output when x = 16, we can use the inverse relationship to evaluate the logarithm. log4 (16) = 2 because 42 = 16 When the base of a logarithm is not written, the understood base is 10. log (x) = log10 (x)

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Example 1 Determine if the following graphs and table represent a logarithmic function. a 4

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

Create a strategy

Apply the idea

Identify key characteristics of a logarithmic graph, such The graph shows a curve that increases as x gets larger as the approach to a vertical asymptote and increasing or and has a vertical asymptote on the left, which are typical decreasing behavior in y-values as x increases. traits of a logarithmic function. This graph represents a logarithmic function.

b 4

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

Apply the idea This graph does not match the typical shape of a logarithmic function. It shows a graph with two asymptotes and a discontinuity that divides the function into two sections, which is characteristic of a rational function instead. So this graph does not represent a logarithmic function.

x y

1 0

2 1

4 2

8 3

Create a strategy Analyze the given table of values to see if it corresponds to a logarithmic function. We can use the relationship between exponential functions and logarithmic functions to identify patterns.

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Apply the idea The y-values in the table increase by 1 each time the x-value doubles, which suggests a logarithmic relationship where the base of the logarithm is 2. We can also see that this table is a possible inverse of the function y = 2x, which further veriﬁes that the relationship is logarithmic. This table represents a logarithmic function.

Example 2 Given

is the inverse of f (x) =

. 4 3 2 1

a Use the graph of f (x) = function.

to create a table of values for the logarithmic

−2

−1

Create a strategy

Apply the idea

Identify points on the graph of f (x) to create a table of values. Each point (x, y) on f (x) will correspond to the point ( y, x) on the inverse, g (x).

To create a table of values, we will look for points on f (x) that are easy to identify. x

−2

−1

f (x)

Now, we will interchange the inputs and outputs to create a table of values for the inverse.

b Graph

g (x)

−2

−1

Create a strategy

Apply the idea

Plot the points from the table created in part (a) and draw the curve that passes through these points.

y 4 3 2 1 −1

1 −1 −2

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x 2

c Compare the asymptotes, domain, range, and intercepts of

and

Create a strategy Since these are inverse functions, the features related to inputs and outputs will be swapped. We can use this understanding and our knowledge of parent exponential and logarithmic functions to compare their key features.

Apply the idea • Asymptotes: For f (x), there is a horizontal asymptote at y = 0, and for g (x), there is a vertical asymptote at x = 0. • Domain: The domain of f (x) is all real numbers, while the domain of g (x) is x > 0. • Range: The range of f (x) is y > 0, and the range of g (x) is all real numbers. • Intercepts: For f (x), the y-intercept is at (0, 1), and there are no x-intercepts. For g (x), the x-intercept is at (1, 0), and there is no y-intercept.

Reﬂect and check Notice that both functions are decreasing over their domain. For exponential functions of the form y = bx and logarithmic functions of the form y = logb x, the functions will increase when b > 1 and decrease when 0 < b < 1.

Example 3 Consider the function f (x) = log3 (x). a Identify the increasing or decreasing intervals.

Create a strategy The base of a parent logarithmic function will tell us whether the function is increasing or decreasing over its domain. • If b > 1, the function increases over the domain (0, ∞) • If 0 < b < 1, the function decreases over the domain (0, ∞)

Apply the idea The base of the logarithm is 3. Since b > 1, this tells us it is an increasing function. The function is increasing for (0, ∞)

Reﬂect and check We can use the graph of y = 3x to ﬁnd the key points and use them to graph f (x) = log3 (x) Graph of y = 3x 9

Graph of f (x) = log3 (x) y

y 3

8 7

5 1

4 −1

3 2

−2

1 −3 −2

−1

2 3 4 5 6 7 8 9

x 1

−3

We can conﬁrm from the graph that the function f (x) = log3 (x) is increasing everywhere.

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b Determine the end behavior.

Create a strategy To determine the end behavior, we have to look at what is happening with our graph as the x-values decrease and increase. Remember that a graph can never cross a vertical asymptote, so this will impact the values x can approach as they decrease. Instead of approaching negative inﬁnity, x now approaches a single value. We can use the graph from the reﬂect and check of part (a) to visualize the end behavior.

Apply the idea This function has a vertical asymptote at x = 0 that restricts the end behavior on the left side of the graph.

y 3 2 1

x 1

2 3 4 5 6 7 8 9

−1

As the x-values decrease, they approach x = 0 from the positive (right) side. The notation we use to indicate x is approaching from the positive (right) side is written as x → 0 +. As the x-values decrease, the y-values decrease inﬁnitely. As x → 0 +, f (x) → −∞.

−2 −3

Now let’s take a look at the right side of the graph. As the x-values increase, f (x) increases inﬁnitely. As x → ∞, f (x) → ∞.

c State the domain and range of f (x) = log3 (x).

Create a strategy A function’s domain is the set of all possible inputs, and a function’s range is the set of all possible outputs.

Apply the idea

Reﬂect and check

Domain: (0, ∞)

Remember, the graph never reaches an asymptote, but rather continues approaching it iniﬁnitely. This graph has a vertical asymptote at x = 0, which means this graph will never reach x = 0. This is why we did not include x = 0 in the domain.

Range: (−∞, ∞)

d Evaluate f (x) = log3 (x) when x = 81.

Create a strategy Begin by substituting x = 81 into the function. Then ask yourself: to what exponent must the base be raised to in order to get 81?

Apply the idea

Reﬂect and check

This question is asking us to ﬁnd the output of the expression log3(81).

We can also use technology to check our answer. However, many scientiﬁc calculators will only evaluate logarithms with a base of 10, so we must be careful to use one that allows us to specify the base of the logarithm.

The base of the logarithm is 3. To what exponent must 3 be raised to get 81? We must raise 3 to the power of 4 to get 81. Because 34 = 81, log3 (81) = 4.

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Idea summary We can use the inverse relationship between logarithmic and exponential functions to ﬁnd key points with which to sketch the graph of a logarithmic function. 4 3

f (x) = bx

(1, b)

2 1 (0, 1) −4 −3 −2 −1 −1 −2 −3 −4

1 2 (1, 0)

(b, 1) x 3 4

f −1(x) = logb (x)

The graph of the parent logarithmic function has the following characteristics: • • • • • • •

The domain is (0, ∞) The range is (−∞, ∞) The x-intercept is at (1, 0) No y-intercept The vertical asymptote is x = 0 The function is increasing when b > 1 The function is decreasing when 0 < b < 1 4

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

x 1

−2

−3

−4

y = logb (x), where b > 1

y = logb (x), where 0 < b < 1

Transformations of logarithmic functions Logarithmic functions can be dilated, reﬂected, and translated in a similar way to other functions. Translation A transformation in which every point in a ﬁgure is moved in the same direction and by the same distance Translations can be categorized as horizontal (moving left or right, along the x-axis) or vertical (moving up or down, along the y-axis), or a combination of the two. Vertical translations can be represented algebraically by g (x) = f (x) + k 6.03 Logarithmic functions mathspace.co

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where k > 0 translates upwards and k < 0 translates downwards. y

g(x)

y f (x)

−4 −3 −2 −1 −1

f (x)

−2

−4 −3 −2 −1 −1

g(x)

−2

−3

−4

Vertical translation of 4 units upwards: g(x) = f (x) + 3

Vertical translation of 4 units downwards: g(x) = f (x) − 3

Similarly, horizontal translations can be represented by g (x) = f (x − h) where h > 0 translates to the right and h < 0 translates to the left. 4

f (x)

1 −3 −2 −1 −1

−2

f (x)

x 1

−4 −3 −2 −1 −1 g(x) −2

g(x)

−3

−4

Horizontal translation of 3 units to the right: g(x) = f (x − 3)

x 4

Horizontal translation of 3 units left: g(x) = f (x + 3)

Notice that a vertical translation does not affect the vertical asymptote but a horizontal translation does affect the vertical asymptote. Reflection A transformation that produces the mirror image of a figure across a line A reflection across the x-axis can be represented algebraically by g (x) = − f (x) A reflection across the y-axis can be represented algebraically by g (x) = f (−x)

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4 3

f (x)

2 1

g(x)

−2

−4 −3 −2 −1 −1

−2

g(x)

−3

f (x)

−4 −3 −2 −1 −1

−3 −4

−4

Reﬂection across the x-axis: g (x) = − f (x)

Reﬂection across the y-axis: g (x) = f (−x)

Vertical compression

Vertical stretch

A transformation that scales all of the y-values of a function by a constant factor towards the x-axis

A transformation that scales all of the y-values of a function by a constant factor away from the x-axis

g(x)

3 f (x)

2 1 −1

2 g(x)

f (x)

x 8

−1

−2

−3

−4

Vertical compression with scale factor of 0.5: g (x) = 0.5f (x)

x 1

Vertical stretch with a scale factor of 2: g (x) = 2f (x)

Horizontal compression

Horizontal stretch

A transformation that scales all of the x-values of a function by a constant factor toward the y-axis

A transformation that scales all of the x-values of a function by a constant factor away from the y-axis

Horizontal dilations can be represented algebraically by g (x) = f (cx) where ∣c∣ > 1 corresponds to a compression and 0 < ∣c∣ < 1 corresponds to a stretch. For horizontal stretches and compressions,

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4 g(x)

x 1

f (x)

1 −1

−1

−2

g(x) 1

x 6

−2

−3

−4

Horizontal compression with a scale factor of g(x) = f (2x)

Horizontal stretch with a scale factor of 2:

The logarithmic parent function f (x) = logb (x) can be transformed to f (x) = a logb [c (x − h)] + k with the correct values of a, b, h and k to apply transformations to f (x). When given a transformed function, we must convert it back to standard notation to correctly identify the transformations applied to the parent function. We can use the relationship between an equation and its transformations to write equations and sketch graphs.

Example 4 The graph of the parent function is shown as a dashed curve. Write the equation of each transformed function. a Parent function: 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Create a strategy To write the equation of the transformed function, identify the transformations applied to the parent function. Look for reﬂections, translations, stretches, or compressions.

Apply the idea The transformed graph represents a reﬂection of the parent function over the y-axis. This can be represented by negating the input variable x. The transformed function is

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b Parent function: f (x) = log4 x 5 4 3 2 1 −1 −2 −3 −4 −5

x 1

2 3 4 5 6 7 8 9

Create a strategy

Apply the idea

The shape of the transformed graph appears to be the same as the original graph. This implies that the function has not been dilated.

The transformed graph shows the parent function was shifted upwards by 2 units. This vertical translation can be represented by adding 2 to the parent function. The transformed function is f (x) = log4 (x) + 2.

Example 5 The graph of f (x) = log (x) is shown. 4 3 2 1 −12

−10

−8

−6

−4

−2

f (x)

−1 −2 −3 −4

x 2

a Describe how g (x) = log(−(x + 2)) − 1 was transformed from f (x).

Create a strategy To describe the transformation, compare g (x) to the transformation form of a function: g (x) = a log [c (x − h)] + k

Apply the idea When comparing g (x) to transformation form, we can see that a = 1, c = − 1, h = − 2, and k = − 1. • c = − 1 indicates a reﬂection over the y-axis • h = − 2 indicates a horizontal shift 2 units to the left • k = − 1 indicates a vertical shift 1 unit downward Combining these transformations, g (x) is the result of reﬂecting f (x) across the y-axis, shifting it 2 unit to the left and 1 unit down.

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b Graph g (x) on the same coordinate plane.

Create a strategy Apply the identiﬁed transformations to the original graph point by point, reﬂecting across the y-axis, shifting left 2 units and down 1 unit to graph g (x).

Apply the idea We can use the points (1, 0) and (10, 1) on the graph of f (x) to graph the transformed function. 4 3 2 1 −12

−10

−8

−6

−4

−2

−1 −2 −3 −4

f (x)

x 2

Reflect and check We can use technology to confirm that we graphed the function correctly and to verify the transformations identified in part (a).

This graph has the same points as the function we plotted, showing that we graphed the function correctly. We can also see that the function has been reﬂected across the y-axis and shifted 2 units left and 1 unit down.

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c Identify and compare the asymptotes, zeros, and end behavior of f (x) and g (x).

Create a strategy To compare these functions, we’ll first identify each feature for both functions separately. We will look for the vertical asymptote by finding the value that makes the argument of the logarithm zero, and we can find the zeros and determine the end behavior by examining the graph from part (b).

Apply the idea For f (x) = log (x), the vertical asymptote is at x = 0, since the logarithm is undefined at x = 0. For g (x) = log (−(x + 2)) − 1, we know that the vertical asymptote will be shifted left 2 units due to the horizontal translation. This function has a vertical asymptote at x = − 2 as this logarithm is undefined at x = − 2. Now, let’s use the graph from the previous part to find and compare the zeros of each function: 4

The function f (x) has a zero at x = 1 and the function g (x) has a zero at x = − 12.

f (x)

3 2 1 −12−10−8 −6 −4 −2 −1

Both functions have a zero, but the zero of f (x) occurs on the positive side of coordinate plane whereas then zero of g (x) occurs on the negative side of coordinate plane.

2 4 6 8 10

−2 −3 −4

When examining the end behavior, we must take the vertical asymptote into consideration. Looking at the graph of f (x), we see as x approaches positive infinity, the y-values approach positive infinity. As the x-values decrease, the function approaches the vertical asymptote at x = 0 from the positive (right) side, and the y-values approach negative infinity. • End behavior as x → 0 +, f (x) → −∞ • End behavior as x → ∞, f (x) → ∞ Now let’s take a look at the end behavior of g (x). As the x-values approach negative infinity, the y-values approach positive infinity. As the x-values increase, the function approaches the vertical asymptote at x = − 2 from the negative (left) side, and the y-values approach negative infinity. • End behavior as x → −∞, f (x) → ∞ • End behavior as x → −2 −, f (x) → −∞ Upon comparison, the end behavior indicates f (x) increases everywhere whereas g (x) decreases everywhere.

Reflect and check We can determine the zeros of logarithmic functions algebraically by setting the function equal to 0 and solving for x. To find the zero of f (x), we let log (x) = 0. In a logarithmic function, x is the result of raising the base to the power of 0. Any value raised to the 0th power is 1, so x = 1. Similarly, we let g (x) = 0 to find the zero of g (x). We must first isolate the logarithmic expression to determine the exponent that the base must be raised to. log(−(x + 2)) − 1 = 0 log(−(x + 2)) = 1

Equate the function to 0 Isolate the logarithmic expression

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Now, we know that the base (which in this case is 10) raised to the power of 1 must be equal to the argument of the logarithm. 101 = − (x + 2)

Rewrite the equation

−10 = x + 2

Divide both sides by −1

−12 = x

Subtract both sides by 2

Idea summary The logarithmic parent function f (x) = logb (x) can be transformed to f (x) = a logb [c (x − h)] + k • • • • • •

If a < 0, the basic curve is reﬂected across the x-axis. The graph is vertically stretched or compressed by a factor of a. If c < 0, the basic curve is reﬂected across the y-axis. The graph is horizontally stretched or compressed by a factor of c. The graph is translated horizontally by h units. The graph is translated vertically by k units.

Practice What do you remember? 1

Fill in the blank to complete the sentence:

The inverse of an exponential function is a ⬚ function of the same base.

Identify the base and the argument of the logarithmic function y = logb a. A

The base is a and the argument is b.

The base is b and the argument is a.

Both the base and the argument are a.

Both the base and the argument are b.

log4 (16) = 2 because ⬚2 = 16

log2 (⬚) = 3 because 23 = 8

log7 (1) = 0 because ⬚⬚ = ⬚

Complete each statement by ﬁlling in the blank. a c e g

⬚

log5 (5) = 1 because 5 = ⬚

because ⬚⬚ = ⬚

Given y = log3 (x) is the inverse of y = 3x:

log⬚ (25) = 2 because 52 = 25

log125 (⬚) = ⬚ because

log⬚ (⬚) = ⬚ because 3 −2 =

Use the graph of y = 3x to create a table of values for the logarithmic function.

Write the equation of the horizontal asymptote of y = 3x.

Write the equation of the vertical asymptote of y = log3 x.

Graph y = log3 x.

24 21 18 15 12 9 6 3 −1

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x 1

Given the graph of a logarithmic function log5(x), identify the graph of the inverse function. 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

3 1

−3 −4

1 −4 −3 −2 −1 −1

−2

1 2 3 4 5 6 7 8

−2

−4 −3 −2 −1 −1

2 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Determine whether or not each could represent a logarithmic function. a

1 −4 −3 −2 −1 −1

x 1

1 −4 −3 −2 −1 −1

−2

−3

−4

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d 4

1 −4 −3 −2 −1 −1

x 1

−2 −3

−4

8 3

x y

0 1

1 2

2 4

3 8

Identify the type of transformation applied to the original function f (x) = log(x) in each graph. Choose from the following options: A

Horizontal shift

g(x)

Vertical shift

Vertical dilation

9 h(x) 8 7 6 5 4 3 2 1

3 2 1 −1 −1

−3 −4

−1 −2 −3 −4 −5 −6 −7 −8 −9

1 2 3 4 5 6 7 8 9

−2

j(x)

1 −1 −1

Reﬂection

x 2

−1 −1

1 2 3 4 5 6 7 8 9

k(x)

−2

x 1 2 3 4 5 6 7 8 9

−2

−3

−4

Describe the transformation applied to the parent function y = log x. a

496

−3

−4 −3 −2 −1 −1

−2

x 1 2 4 y 0 1 2

f (x) = log(x − 4)

Mathspace Virginia SOL Algebra 2 mathspace.co

g(x) = log(3x)

h(x) = − log(x)

j(x) = log(x) + 2

Match each logarithmic function with its graph. i

f (x) = log2 (x)

f (x) = log3 (x)

−1

iii

−1

−2

−3

d 3

−1

−2

−3

Let’s practice 10

Consider the function f (x) = log2 x. a

Complete the table of values for the inverse, y = 2x. x y

−2

−1

Use the table for y = 2x to create a table for f (x) = log2 x.

Draw the graph of f (x) = log2 x.

Identify the intercept(s) of f (x) = log2 x.

State the domain and range of f (x) = log2 x.

Consider the function

. .

Graph the inverse,

Use the inverse to create a table of values for

Draw the graph of

Write the equation of the vertical asymptote of

State the domain and range of

on the same coordinate plane. .

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497

Evaluate each function at the given value of x: a

f (x) = log4 (x) when x = 64

f (x) = log125 (x) when x = 1

f (x) when x = 3

f (x)

f (x) when x = 9

iii b

3 2

−1 −1

f (x)

−1 −1

1 2 3 4 5 6 7 8 9

−2

−3

−4

x 1 2 3 4 5 6 7 8 9

For each of the given graphs of an exponential function, graph the inverse logarithmic function. a

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

6 5 4 3 2 1

5 4 3 2 1

x 1 2 3 4 5 6

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

For each of the given exponential functions: i

Graph the exponential function.

Graph the inverse logarithmic function on the same coordinate plane.

f (x) = 3x

Mathspace Virginia SOL Algebra 2 mathspace.co

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

498

x when f (x) = 0

2 1

when x = 2

For each function, ﬁnd:

f (x) = log3 (x) when x = 243

1 2 3 4 5

x 1 2 3 4 5

SOL

What is the equation of the asymptote of the graph of the given function? f (x) = log(x − 16) + 15 A

x = 16

y = 16

x = 15

y = log7 (x)

y = 15

For each of the following functions: i

State the domain.

Determine the end behavior as x → 0 and as x → ∞.

iii

Find the coordinates of the zero(s).

Determine if the function is increasing or decreasing.

y = log4 (x)

The graph of the parent function is shown as a dashed line. Write the equation of each transformed function. a

Parent function:

y 3

2 1 −1

x 1

2 3 4 5 6 7 8 9

−2 −4 −5 −6

Parent function:

y 5 4 3 2 1

−1 −2 −3 −4 −5

x 1

2 3 4 5 6 7 8 9

y 4

x 1

2 3 4 5 6 7 8 9

7 6 5 4 3 2 1

x 1

2 3 4 5 6 7 8 9

Parent function: f (x) = log5 (x) 4

x 2

−2

Parent function: f (x) = log3 x

−1 −2 −3

Parent function: f (x) = log4 (x)

−1

5 4 3 2 1 −1 −2 −3 −4 −5

−3

Parent function: f (x) = log2 x

8 10 12 14 16 18

−8 −6 −4 −2 −1

x 2

−2 −3 −4

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For the following equations: i

Identify the transformation from the parent logarithmic function.

Complete the given table of values for the transformed function.

iii

Graph the transformed logarithmic function.

y = 2 log2 (x) − 1

y = − log2 (x) + 3

x 1 2 4 8 16 y c

y = log2 (2x + 4)

x y

−10

−6

−4

−3

y = log2 (−x − 2)

x −1 0 2 6 14 y 20

x y

−18

The graph of f (x) = log3 x is shown. a

Describe how g(x) = log3(x + 2) + 1 was transformed from f (x).

Graph g(x) on the same coordinate plane.

f (x)

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

The graph of

is shown. 4

was transformed from f (x).

Describe how

Graph g(x) on the same coordinate plane.

f (x)

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

SOL

Which of the following describes the end behavior of

as x approaches 0? A 23

500

f (x) approaches 0

f (x) approaches 5

f (x) approaches ∞

f (x) = log2 (x2 − 9)

f (x) = log6 (x − 4)

f (x) =

Evaluate each function at x = 5: a

f (x) approaches −∞ f (x) = log3 (5x + 2)

(2x − 1)

For each function: i

State the equation of the asymptote.

State the domain and range in interval notation.

iii

State the increasing and decreasing intervals.

Identify the intercept(s).

f (x) = log4 (2x) − 2

f (x) = − log3 (x − 2)

Mathspace Virginia SOL Algebra 2 mathspace.co

f (x) = 2 log2 (x) − 2

Consider the three functions f (x), g(x), and h(x) shown: • f (x) = log2 x • x g(x) a

•

Determine each of the function values for when x = 1, x = 4 and x = 8. i

f (x)

g (x)

iii

h(x)

Describe the relationship between the output values of f (x), g (x) and h (x) as x → ∞.

16 h(x) 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 −1 −1

1 2 3 4 5 6 7 8 9 10 11

Consider the logarithmic function f (x) = log2(x) and the quadratic function g(x) = x2 + 2. Compare and contrast the following characteristics of each function:

The general shape and direction of the graphs.

The domain and range of each function.

The behavior as x decreases and increases.

The presence of asymptotes, intercepts, vertices, and axes of symmetry.

The Richter scale is used to measure the intensity of earthquakes. The formula for the Richter scale rating of an earthquake is given by

where a is the intensity of a minimal earthquake that can barely be detected, and x is a multiple of the minimal earthquake’s intensity. The table shows how earthquakes are categorized according to their Richter scale rating. Richter rating Category

2–3.9 Minor

4–4.9 Light

5–5.9 Moderate

6–6.9 Strong

A seismograph measures the intensity of an earthquake to be x = 6689a. Use your calculator to determine the Richter scale rating R of this earthquake, correct to one decimal place.

Identify the category in which the earthquake falls.

In an electronic system, the signal ratio D, measured in decibels, is calculated with the formula D = 10 log10

. Here, F is the output

D(decibels) 4

power and I is the input power, both in megawatts (MW). a

Calculate the signal ratio D if the output power is 10 MW and the input power is

The graph depicts how D varies with different output powers F at a constant input power of 5 MW. What is the approximate signal ratio D for an output power of F = 11 MW? Given that decibels are not negative, identify the possible output powers (F) for the function with an input power of I = 5 MW.

2 1 F(MW) 3

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Let’s extend our thinking 29

Consider:

a 30

Find the domain.

Find the range.

Consider the logarithmic function f (x) = a logb (x − h) + k where a, b, h, and k are constants, and b > 1. a

Explain how the y-intercept of this function is related to the vertical asymptote.

What must be true about the value of h for the function to have a y-intercept, and how does this relate to the location of the vertical asymptote?

Find the domain and range of the real function y = loga (x). Justify your answer.

Without evaluating, state the two integers that the value of each logarithm lies between: a

502

log2 (13)

log10 (539)

log7 (23)

Write an equation for and sketch the graph of a logarithmic function with the features: a

A vertical asymptote at x = − 2 and passes through the origin.

An x-intercept at x = 4 and is decreasing.

A y-intercept at y = − 1 and is increasing.

A vertical asymptote at x = 3 and has an x-intercept at x = − 1.

A vertical asymptote at x = − 2, an x-intercept at x = − 1 and a y-intercept at y = 1.

Mathspace Virginia SOL Algebra 2 mathspace.co

Rational-Linear

Rational-Quadratic

Exponential

Logarithmic

f (x) = b , b > 1

f (x) = logb x, b > 1

y x

Example 1 List the following functions in order from leftmost vertical asymptote to rightmost vertical asymptote. • g(x) = log (x − 5) + 6

•

• j(x) = log (x + 2) − 5

•

Create a strategy To determine the order of the functions based on their vertical asymptotes, we ﬁrst need to identify the vertical asymptotes for each function. • For rational functions, the vertical asymptote occurs where the denominator is equal to zero. • For logarithmic functions, the vertical asymptote occurs where the argument of the logarithm is equal to zero. Once the vertical asymptotes are identiﬁed, list the functions in ascending order based on the x-value of the vertical asymptote.

Apply the idea Determine the vertical asymptotes for each function: • For

, the vertical asymptote occurs when x − 3 = 0. So, the vertical asymptote is at x = 3.

• For g(x) = log (x − 5) + 6, the vertical asymptote occurs when x − 5 = 0. So, the vertical asymptote is at x = 5. • For

, the vertical asymptote occurs when x + 4 = 0. So, the vertical asymptote is at x = − 4.

• For j(x) = log (x + 2) − 5, the vertical asymptote occurs when x + 2 = 0. So, the vertical asymptote is at x = − 2. Now, list the functions in order from leftmost to rightmost vertical asymptote: h(x), j(x), f (x), g(x)

Reﬂect and check To check our answer, we can visualize the functions on a graph and observe the order of the vertical asymptotes. The graph should show that the vertical asymptotes are ordered from left to right as follows: h(x) at x = − 4, j(x) at x = − 2, f (x) at x = 3, and g(x) at x = 5. This conﬁrms our answer.

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h(x)

9 y j(x) 8 7 6 5 4 3 2 1

−9 −8 −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

g(x)

f (x)

x 1

Example 2 Consider the following square root and logarithmic functions: 4

−4 −3 −2 −1 −1

−2

−3

−4

a Compare the zeros of the functions.

Create a strategy The zero of a function occurs at y = 0, which is where the graph crosses the x-axis. We can label the x-intercepts on each graph to more clearly see where they are.

Apply the idea 4

1 −4 −3 −2 −1 −1

x 1

1 −4 −3 −2 −1 −1

−2

−3

−4

x 1

Looking at the graphs, we can see that the zero of both functions is at the point (3, 0).

Reﬂect and check Had we known the equation for each graph, we would have been able to algebraically conﬁrm our answer by substituting 3 in for x in each equation and ensure each results in a value of 0.

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b Compare the domain and range.

Create a strategy We will analyze the graphs of the functions to determine their domain and range. The domain is the set of all possible x-values, while the range is the set of all possible y-values that the functions can take.

Apply the idea

Reﬂect and check

Observing the graph of the square root function, we can see that it starts at x = − 1 and extends to the right inﬁnitely. This means the domain of the function is x ≥ −1 or [−1, ∞). Observing the y-values, the graph starts at the point (−1, −2) and increases without bound. Thus, the range of the function is y ≥ −2 or [−2, ∞).

We can also write the domain and range of the two functions in set notation:

Observing the graph of the logarithmic function, we can see that it starts at x = 2 and extends to the right inﬁnitely. This means the domain of the function is x > 2 or (2, ∞). Observing the y-values, the graph extends downward without bound and increases without bound. Thus, the range of the logarithmic function is all real numbers, denoted as (−∞, ∞).

For the square root function: • Domain: {x∣x ≥ −1} • Range: {y∣y ≥ −2} For the logarithmic function: • Domain: x > 2 • Range: {y∣y ∈ }

c Compare the absolute maxima or minima, if any.

Create a strategy Finding the absolute maxima or minima involves identifying the highest or lowest points of the function. These will correspond with the endpoints of the function’s range. Since we have already found the domain and range, we can use this information to identify any absolute extrema.

Apply the idea • Domain of square root function: [−1, ∞)

• Range of square root function: [−2, ∞)

• Domain of logarithmic function: (−∞, ∞)

• Range of logarithmic function: (−∞, ∞)

From the domain and range information, we can see that the square root function starts at the point (−1, −2) and increases indeﬁnitely. As the function continuously increases, there is no absolute maximum. The absolute minimum occurs at the point (−1, −2). For the logarithmic function, we observe from the range that the function decreases and increases without bound. This implies that there is no absolute maximum or minimum.

Reﬂect and check By analyzing the graphs of the square root function and the logarithmic function on the same Coordinate plane, we can visually identify any absolute maxima or minima. The square root function has an absolute minimum at the point (−1, −2), while the logarithmic function has neither an absolute maximum nor an absolute minimum, which conﬁrms our answer.

4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

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x 1

d Compare the end behavior.

Create a strategy Observe the graphs of the functions to determine how the functions behave as x decreases and increases. Focus on the direction and behavior of the functions as they move to the left and to the right on the graph.

Apply the idea Observing the graph of the square root function, which we will call f (x), we can see that as x approaches −1, the function value approaches −2. As x approaches positive infinity, the function f(x) increases slowly without bound, so the function approaches positive infinity. For the logarithmic function, which we will call g(x), we can see that as x approaches 2, the function approaches negative infinity. As x approaches positive infinity, the function g(x) increases slowly without bound, so the function approaches positive infinity. The end behavior can be written as follows Square root function: as x → −1, f (x) → −2 and as x → ∞, f (x) → ∞ Logarithmic function: as x → 2, g(x) → −∞ and as x → ∞, g(x) → ∞ We can see similarities amongst the functions’ end behavior. For both functions, as x decreases, both functions are restricted, so x does not decrease infinitely. Additionally, as x approaches positive infinity, both functions approach positive infinity.

Reflect and check Remember that when we look at increasing or decreasing intervals of functions, we usually consider how the function values change as we move from left to right. When we are looking at the end behavior of functions, we consider how the function values change as the input values get smaller or larger instead.

Example 3 Consider the function f (x) = − (x + 2)3 and the function g(x) represented by the table of values. x g(x)

−1 −27

0 −8

1 −1

2 0

3 1

4 8

5 27

a Compare and contrast the intervals over which the functions are increasing or decreasing.

Create a strategy

Apply the idea

To compare and contrast the intervals of increasing or decreasing for the functions, we must first determine where, and if, the graph is increasing or decreasing. Since f(x) is a cubic function, we know it will either increase everywhere or decrease everywhere.

For the function f (x) = − (x + 2)3, it is a cubic function whose graph has been reflected over the x-axis and translated left 2 units compared to the parent function. The parent function, y = x3, is increasing everywhere, so the reflection will cause our graph to flip. This means f (x) is decreasing for (−∞, ∞).

For g(x), we will examine the pattern in the outputs.

For the limited domain given for g (x), we can see that as the x-values increase, the y-values are also increasing. In addition, the function appears to have a cubic relationship. This implies g (x) is increasing everyhwere.

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b Identify and compare the zeros.

Create a strategy We can ﬁnd the zero of f (x) algebraically by setting the equation equal to zero and solving for x. To ﬁnd the zero for g(x), we will use the table of values.

Apply the idea Let’s ﬁrst ﬁnd the zero of the function f (x): −(x + 2)3 = 0

Set the function equal to 0

(x + 2) = 0

Divide both sides by −1

x+2=0

Cube root both sides

x = −2

Subtract 2 from both sides

To ﬁnd the zero of the function g(x), we can use our table of values. We know the zero of a function occurs where the y-value of the function equals 0. We can see from the table that the function has a zero at x = 2. We can see that both f (x) and g(x) have exactly one zero. The zeros of the two functions are similar, but one is 2 units to the left of the origin while the other is 2 units to the right of the origin.

Example 4 The graph of function f (x) shown follows the vertical height of Ameth’s golf ball after being hit into the air and falling on the fairway: Ameth’s golf ball Height (feet), f (x) 30 25 20

f (x)

15 10 5

Time (seconds), x 1

The equation of function g(x) shown follows the vertical height of Massiel’s golf ball. She did not get a good hit on the ball and, after a few seconds in the air, the ball hit a tree and ricocheted off into a sand trap where it got stuck.

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a Compare the intervals where each golf ball is increasing in height.

Create a strategy We can use the graph to determine when f (x) is increasing in height. For g(x), we will examine the functions in the piecewise function to determine when it is increasing.

Apply the idea Looking at the graph of f (x), we can see the function is increasing in height until it reaches its maximum at x = 5, and then is decreasing for rest of graph. f (x) is increasing in height over the interval (0, 5). For the function g(x), we must take a look at what each piece of the piecewise function is representing. The function starts with a cube root function, which is increasing everywhere. Then at x = 4, it becomes a linear function with a negative slope, which is decreasing. The ﬁnal piece is an upward facing parabola, which is decreasing until the vertex at x = 9. g(x) is increasing in height over the interval (0, 4). Ameth’s golf ball increases in height for one second longer than Massiel’s golf ball.

b Describe what g(x) = − 2.4x + 19.125 for 4 ≤ x < 7.5 represents in context.

Create a strategy Graph g (x), then use the context to interpret the piecewise function graphically.

Apply the idea Massiel’s golf ball 11 Height (feet), f (x) 10 9 8 7 6 5 4 3 2 1 Time (seconds), x 1 2 3 4 5 6 7 8 9 10 11

In this part of the function, the golf ball’s height decreases linearly with time. This occurs within the time interval 4 to 7.5 seconds. During this time, the ball hits a tree and ricochets off, causing it to lose height as it moves towards the sand trap.

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c Determine whose ball goes higher.

Create a strategy Compare the maximum values of the graphs of Ameth’s ball and Massiel’s ball. Ameth’s golf ball

Massiel’s golf ball

Height (feet), f (x) 30 25 20

f (x)

15 10 5

Time (seconds), x 1

11 Height (feet), f (x) 10 9 8 7 6 5 4 3 2 1 Time (seconds), x 1 2 3 4 5 6 7 8 9 10 11

Apply the idea Ameth’s ball reaches a maximum height of 30 ft while Massiel’s ball reaches a maximum height of just over 9 ft. Ameth’s ball clearly goes higher.

Reﬂect and check We used the graph of g(x) to approximate the maximum height, but we could have found the precise height by substituting x = 4 into the function g(x) = − 2.4x + 19.125. g(4) = − 2.4(4) + 19.125

Substitue x = 4

= − 9.6 + 19.125

Multiply

= 9.525

Add

Massiel’s ball reaches a maximum height of 9.525 ft.

Idea summary Real-world contexts can be modeled by various types of functions. We can compare different types of models for contextual situations using their key features: • • • • • • • • •

510

Domain and range Intercepts Zeros End behavior Relative maximum or minimum value(s) Absolute maximum or minimum value(s) Increasing, decreasing, and constant intervals Asymptote(s) Axis of symmetry

Mathspace Virginia SOL Algebra 2 mathspace.co

Practice What do you remember? 1

Match each equation with the name of the functions. i ii iii

y = x2 + 4x – 5

Exponential Function

Rational Function

Cube Root Function

Logarithmic Function

y = 2x − 4

Square Root Function

vii

Piecewise-Deﬁned Function

Graph an example of each type of function. a

Linear Function

Quadratic Function

Exponential Function

Square Root Function

Cube Root Function

Rational Function

Logarithmic Function

Piecewise-Deﬁned Function

Explain how to ﬁnd each function characteristic. a

Domain

Range

Intercepts

Zeros

Increasing, decreasing, or constant intervals

End behavior

Find the following characteristics of each function. i

Domain (in interval notation)

Range (in interval notation)

iii

Increasing or decreasing intervals (in interval notation)

End Behavior

f (x) = − 2x + 3

g(x) = (x − 3)2 − 2

−8 −6 −4 −2 −2

Quadratic Function

y = log3(x − 1)

x 2

−8 −6 −4 −2 −2

−4

−6

−8

x 2

Consider the function f (x) = . Describe how the position of the horizontal and vertical asymptotes will change if we perform a vertical shift upwards by 2 units.

Consider the function g(x) = x2 − 9. Describe how the coordinates of the vertex and zeros will change if we perform a vertical stretch by a factor of 2.

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Let’s practice 7

Consider the following list of parent functions: Square Root, Cube Root, Rational, Linear Polynomial, Quadratic Polynomial, Cubic Polynomial, Exponential, and Logarithmic. Determine which parent functions satisfy each of these conditions: domain of {x∣x ∈ }.

range of {y∣y ≥ 0}.

end behavior that as x → ∞, y → 0.

zero not at the origin.

increasing and decreasing intervals across the entire domain.

Identify the type of function that would best ﬁt this data. Explain your reasoning. A large bucket has a very small leak near a b A soccer ball is kicked at 20 m/s at a 50° its base. The volume of water in the bucket, angle. The height of the ball throughout the measured in cubic feet, is recorded once a ﬁrst second is plotted in the given graph. minute and the results are plotted in the given y graph. 5

100

3 95 2 90

1 x

x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 2 3 4 5 6 7 8 9 10

A computer program compares and orders the scores of all students who sit for an exam. The time taken, T in milliseconds, for the program to completely order all students is plotted for different numbers of students, n: Which model will best describe the run time of the computer program as the number of students continue to increase?

2000 1800 1600 1400 1200 1000 800 600 400 200

n 5

T 4000 3600 3200 2800 2400 2000 1600 1200 800 400

n 5 10 15 20 25 30 35 40

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Mathspace Virginia SOL Algebra 2 mathspace.co

4000 3600 3200 2800 2400 2000 1600 1200 800 400

n 5 10 15 20 25 30 35 40

T 4000 3600 3200 2800 2400 2000 1600 1200 800 400

4000 3600 3200 2800 2400 2000 1600 1200 800 400

5 10 15 20 25 30 35 40

n 5 10 15 20 25 30 35 40

For each pair of functions, determine which has a greater vertical asymptote: Function 1: • Function 2: a • y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Function 3: • Function 4: b • y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

List the following functions in order from left-most vertical asymptote to right-most vertical asymptote.

• • • g(x) = log (x − 4) + 7 • j(x) = log (x + 4) − 4

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Determine which has the highest maximum value. • Function P • Function Q x −1 1 3 y 2 6 2

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

Compare the domains of the square root functions: i

Function f (x):

Function g(x):

iii

Function h(x):

x x x f (x) g(x) h(x) 0 0 3 0 0 0 1 1 4 1 1 −1 4 2 7 2 4 −2 9 3 12 3 9 −3 14

Compare and contrast the three cube root function equations: i

f (x) =

g(x) =

iii

h(x) =

j(x) =

−5

Discuss how the transformations affect the domain and shape of the graph of each function. 15

For each pair of functions, determine which has the greater y-intercept: Function 1: • Function 2: a • y = 4x2 + 6x + 3 Function 3: • Function 4: b • 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

514

x 1 2 3 4 5

Mathspace Virginia SOL Algebra 2 mathspace.co

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

For each pair of functions, determine which has a smaller zero: • Function 2: Function 1: a • 5 4 3 2 1

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Function 3: • Function 4: b • 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Determine which function has the highest maximum value. • Function P • Function Q: 8

2 −8 −6 −4 −2 −2

x 2

2 −8 −6 −4 −2 −2

−4

−6

−8

x 2

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For each of the following pairs of functions, determine which has the greater y-intercept: a • Function 1: The line with a slope of 4 that crosses the y-axis at (0, 6). • Function 2: The line given by the equation y = x + 4. • Function 4: Function 3: b • x y

2 2

4 −2

6 −6

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

• Function 5: • Function 6: x y

2 19

4 35

6 51

Identify any maximum or minimum values for each function.

• Function A: y = − 2 (x − 7) (x − 3)

Determine the zeros of each function.

• Function B:

Compare and contrast the domains and ranges of each function.

Consider the functions:

y 12 10 8 6 4 2 −2 −2

Consider the functions: a

Determine which function has the higher function value at the y-intercept.

Determine if each function is increasing or decreasing on the domain 0 < x < 5.

x 2

10 12

• Function A:

• Function B: 8

6 4 2 −8 −6 −4 −2 −2 −4 −6 −8

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x 2

Consider two functions f (x) and g (x) shown in the table of values and graph. x f (x)

−2 36

−1 9

0 0

1 9

30 27 24 21 18 15 12 9 6 3

2 36

Determine which function has a minimum value.

Determine which function has a horizontal asymptote.

−5 −4 −3 −2 −1

Consider the function f (x) = x3 − 5 and the function g (x) whose graph is shown.

g(x)

10 8 6

Determine the domain and range of each function.

x 1

Compare and contrast the intervals over which the functions are increasing or decreasing.

g(x)

4 2 −5 −4 −3 −2 −1 −2

x 1

−4 −6

The exponential function E is given by y = 3x + 4 and the exponential function F is given by y = − 8x + 1. a

Graph the exponential functions on the same coordinate plane.

Identify how many times E and F intersect.

Describe the end behavior of each exponential function as x → ∞.

The function J is given by y = 9 (3 − x). The table shows the function values for function K:

−1

Compare and contrast the patterns of change for functions J and K.

Graph functions J and K on the same coordinate plane.

Describe the end behavior of each function as x → ∞.

Determine what transformation we can apply to the function y = 3− x to produce function K.

Consider the functions: • Function Q: • Function P: y = 6 − x + 5 x

−4

Determine what type of functions Function P and Function Q are.

Graph the functions on the same coordinate plane.

Identify which function has the higher function value at the y-intercept.

Identify how many times P and Q intersect.

−3

−2

−1

0 −11

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Some friends decide to go camping for the weekend. They cannot all ﬁt in one car so some of them catch a bus to the campground, which is 450 km from home. Those in the car started driving at 8:00 AM and arrived at the campground at 3:30 PM, driving at a constant speed. The bus also drives at a constant speed and takes the same route as the car. Its distance in kilometers ( y) from home x hours after leaving is given by the equation y = 71x.

Determine the speed of the car, in kilometers per hour.

Determine the speed of the bus, in kilometers per hour.

Determine which vehicle was traveling faster.

Matt and Sophia are saving money using different strategies. The amount each has saved after each month is given by the table of values and the plotted points. If each person continues their pattern of saving, determine who will be the ﬁrst to exceed savings of $600. • Sophia: • Matt: Number of months Sophia’s savings

1 4

2 16

3 64

Matt’s savings

300 270 240 210 180 150 120 90 60 30

Months 4

What types of function could ﬁt each of the scenarios below? Explain your reasoning. a

A certain type of plant absorbs water from the soil at a rate that decreases as the amount of water it has already absorbed increases.

Consider the spread of a virus in a small closed community over time. If the spread of the virus slows down as more people become immune.

For modeling the decay of the radioactive substance.

Let’s extend our thinking 29

When comparing two phone companies, Akeem notices that both offer a $100 monthly contract, but they have different policies on how they charge fees if the bill is overdue.

600 550 500 450 400 350 300 250 200 150 100 50

Massive Dynamic’s policy is to add $13 to the bill for every day the bill is overdue. Stark Industry’s policy is to increase the bill by a particular percentage for every day the bill is overdue, and this is displayed in the graph.

518

Price

Day 2

If an unpaid bill reaches $3000, it is referred to a debt collector. Determine which company’s policy would result in an unpaid bill ﬁrst reaching $3000.

Akeem knows from past experience that he is often up to 8 days late with his payments. Determine which company Akeem should choose if he always intends to pay within 8 days of receiving the bill. Explain your reasoning.

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• Function A:

Consider the functions: a

Determine which function has the smallest x-intercept.

Determine which has the smallest minimum.

Determine for which function y is increasing the most rapidly over x > 0.

• Function B: 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Determine how many times the graphs of P and Q intersect.

• Function P: y = x5 + 7

Determine which function tends to inﬁnity the quickest as x → ∞.

• Function Q:

Consider the functions:

9 8 7 6 5 4 3 2 1 −9−8−7−6−5−4−3−2 −1 −1

1 2 3 4 5

x 1 2 3 4 5 6 7 8 9

−2 −3 −4 −5 −6 −7 −8 −9

A supermarket uses rectangular prism cardboard boxes of different widths, x, and volumes, y. A scatterplot in which the points represent different sized boxes is shown: a

c d 33

Jack is an employee who usually packs and unpacks smaller boxes. 8000 He uses the linear model to the left on the scatterplot to represent 7500 the box sizes. Determine the interval for which this model is most accurate. 7000 Georgia is an employee who usually packs and unpacks larger boxes. She uses the linear model to the right on the scatterplot to represent the box sizes. Determine the interval for which this model is most accurate. Determine whether the two models together accurately describe all of the boxes used by the supermarket. Justify your answer.

Georgia

6500 6000 5500 5000

Find a model that represents the boxes more accurately.

Consider the functions f (x) = 2x, g (x) = log2(x) and

Jack

x 25

for x ≥ 0.

Describe the pattern of how each of the functions increases. Explain how you identiﬁed the patterns.

Graph all three functions on the same coordinate plane over the domain [0, 3].

Compare how the three functions increase over the domain

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7 Probability & Statistics Big ideas • Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample. • Many sets of bivariate data can be modeled using familiar functions. • Different representations of data highlight different characteristics of the data. • Models may be used to simulate real-world situations and can inform predictions and decisions. • Many patterns can be found in sets of data. These patterns can be useful in making inferences but any inferences made from a set of data should not be taken as fact. • Probability helps us analyze the chance that an event will occur and provides us with tools to make decisions about the future based on known information.

Chapter outline 7.01 7.02 7.03 7.04 7.05 7.06 7.07 7.08

Formulate questions and collect bivariate data Scatterplots and lines of fit Linear, quadratic, and exponential models Univariate data, center, and spread Histograms with smooth curves Normal distributions Z-scores Permutations and combinations

522 538 556 578 601 622 641 655

Statistical bias Any aspect of the data cycle process that leads to a difference between the conclusion and the actual truth for the population. Example: Sampling bias, Observer bias, Measurement bias Sampling bias can occur due to undercoverage or exclusion when a particular subgroup is under- represented or fully excluded. There are a number of ways we can avoid bias in our sample, including: • Having a sample that is large enough to represent the characteristics of the population. The larger the sample size, the closer the results will be to that of the population. • Having a sample that is selected without strategically choosing more people from a certain group. • Randomly selecting the sample.

Example 1 First identify the variables, then write an investigative question related to each scenario. a The local souvenir shop has noticed that their sweatshirt sales seem to be related to the temperature outside. They want to investigate this relationship more closely.

Create a strategy When creating an investigative question, it should focus speciﬁcally on the relationship between the two variables involved, which in this case are temperature and sweatshirt sales.

Apply the idea Independent variable: Temperature Dependent variable: Sweatshirt sales Some possible investigative questions might be: 1. Does a decrease in temperature correlate with an increase in sweatshirt sales? 2. How does temperature relate to the number of sweatshirts sold? 3. Is there a speciﬁc temperature range that results in the highest sweatshirt sales?

Reﬂect and check Each of these questions investigates a different aspect of the relationship, making them effective investigative questions. If we wanted to be even more explicit, we could add “For the local souvenir shop” to the beginning of each question to make the population clear. However, this should be clear based on the scenario.

b For a school in Fairfax, VA, the principal noticed that the number of days missed by a student in September is a good predictor of the number of total days they will be absent throughout the year. She wants to investigate this relationship.

Create a strategy An investigative question must require the collection of data to answer it. Once we have identiﬁed the variables, we can formulate the question.

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Apply the idea Independent variable: Number of days absent in September Dependent variable: Number of total days absent throughout the year Some possible investigative questions might be: 1. How does the number of days missed in September relate to the number of total days absent throughout the year? 2. Does a high number of missed days in September correlate to a high number of total days absent throughout the year? 3. Is there a threshold number of days missed in September that would indicate a high number of total days absent throughout the year?

Reﬂect and check After one iteration of the data cycle, we might come up with a different question that explores something we noticed in the analysis. If we want to be more speciﬁc, we could add “For a school in Fairfax, VA” to the beginning of each question to make the population clear.

c A baker wants to adjust his pricing model to be more competitive. He wants to look at the price he charged for a cake compared to the time it took to create. He is curious if he would need speciﬁc models for wedding cakes versus to birthday cakes or if the same model would be appropriate for all kinds of cakes.

Create a strategy The investigative question should focus on the relationship between the two numerical variables involved. There is also a categorical variable that could be used for deeper exploration.

Apply the idea

Reﬂect and check

Independent variable: Time it took to create a cake

After one iteration of the data cycle, we could ask the follow up questions by splitting the scatterplot into two or using different symbols or colors for the points to see if the relationship is the same.

Dependent variable: Price charged for a cake. Categorical variable: Type of cake. Some possible investigative questions might be: 1. Does the price charged for a cake correspond to the time taken to create it? 2. Is a higher price charged for a cake that took longer to create? 3. Is there a speciﬁc time duration for creating a cake that would result in a higher price being charged?

1. Does the price charged for a cake correspond to the time taken to create it? Is this relationship the same or different for wedding and birthday cakes? 2. Is a higher price charged for a cake that took longer to create? Is this consistent for birthday and wedding cakes? 3. Is there a speciﬁc time duration for creating a cake that would result in a higher price being charged? Is this range the same for wedding and birthday cakes?

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She could try a convenience sample by emailing the venue’s mailing list or try physical surveys handed out at concerts. Her sampling method should check that the sample is representative by reaching a diverse group of concert attendees. The answer is option E.

Reﬂect and check If she wanted to do another iteration of the data cycle, she could explore this relationship for different venues with and without tiered seating.

Idea summary Bivariate data is data which is represented by two variables. This data is typically numerical and can be organized with a scatterplot. To explore bivariate data, we ﬁrst need to formulate an investigative question and then we can determine how to collect or acquire the necessary data. Such as: • • • •

Research using secondary sources to ﬁnd existing data. Surveys can be done by asking each member of the representative sample two questions or giving them a questionnaire. Observations can be made by watching members of the sample and noting particular characteristics. Scientiﬁc experiments can be done by controlling other variables, then varying the independent variable and measuring the dependent variable.

Scatterplots and relationships We often display bivariate data using a scatterplot where the independent variable is written on the horizontal axis and the dependent variable is on the vertical axis. We can describe the relationship based on how closely the points follow a particular model using the following terms: • Form, usually described as a linear relationship or nonlinear relationship • Strength, describing how closely the data points match the model line or curve For linear relationships, we may also describe their direction as positive or negative. As a review, here are some examples: 11 10 9 8 7 6 5 4 3 2 1

300 200 100

x 1 2 3 4 5 6 7 8 9 10 11

A linear relationship that is strong and positive

−100 −200 −300 −400 −500 −600 −700 −800 −900 −1000 −1100 −1200 −1300

y x 1 2 3 4 5 6 7 8 9 10 11

A nonlinear relationship that is weak

For larger data sets, we can use technology such as spreadsheets and graphing calculators to graph scatterplots. This is especially helpful for when we also want to analyze which model or equation would be the most appropriate.

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Given the investigative question, “Is there a relationship between 100 m race time and leg length?” a

Identify the independent and dependent variables.

Label the axes of the given scatterplot with the variables and appropriate units.

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5 10 15 20 25 30 35 40 45

Determine which data collection method would best be used to collect data for each of these scenarios. a

Using a water meter to measure the amount of water needed to water a vegetable garden compared to the temperature.

Watching to see how much people spend on coffee at a cafe compared to how long they stay in the cafe.

Use census data to compare housing expenses for different housing types over time.

Planting two identical plants, giving one nitrogen fertilizer and the other only water, and observing the differences.

Is each statement true or false? a

Bias caused by the personal beliefs and expectations of researchers can be completely eliminated from any study.

Bias caused by measurement error can impact the validity and reliability of research ﬁndings.

A survey where people self-select will unbiased results.

In an experiment testing the effects of different temperatures on plant growth, maintaining consistent temperature conditions is important to minimize bias.

Justin wants to know how the students at his school might vote for the next school captain. He leaves three ﬂyers in each class to be completed by whichever students want to complete them. Are these valid reasons why this sampling might give poor results?

Sampling within each class is self-selected.

The sample includes people from outside the population.

The students are under 18 so can’t legally vote.

Is there likely to be a relationship between each pair of variables? a

Study time and test score

Length of hair and number of pets

Typing speed and dancing ability

Driving time and amount of gasoline used

Let’s practice 9

For each scenario, identify two numerical variables and then formulate an investigative question around bivariate data that the person could ask. a

Lula is interested in being a teacher. Before applying to college, Lula wants to know how long she would have to teach before making a certain amount of money.

Keisha is taking piano lessons and is told she needs to keep her nails short. She is also thinking about cutting her hair short because it is a lot of work to maintain.

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For each scenario: i

Identify possible independent and dependent variables.

Write an investigative question related to the scenario.

Anya notices that houses that are less expensive in her neighborhood get sold more quickly.

Burton is curious about factors that inﬂuence his sales at a young entrepreneur market. He thinks there might be a relationship with the weather.

Carmelita is wondering why the speed limits are usually lower in front of schools. She doesn’t think she has ever seen a car accident in front of her school.

For each scatterplot, describe any relationships, trends, patterns, or outliers. a

5 x

x 5

5 x 5

Noah is a coffee vendor. He records the maximum temperature of the day and the number of coffees sold. The results are recorded in the table: Maximum temperature (°C) Number of coffees

28 17

32 37

31 25

33 39

31 23

26 7

25 19

Construct a scatterplot for the data.

Describe what happens to the sales of coffee as the temperature increases.

29 34

35 42

A company records the data on the height and annual salary ($’000) of ten employees. Height Salary

532

180 86

186 69

161 63

166 83

195 69

Identify the independent and dependent variables.

Create a scatterplot of the data.

Describe the relationship between height and salary.

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180 74

180 89

170 72

177 80

170 62

Data was collected from a variety of seniors who had just moved into long term care. They were asked what their maximum annual income was before retiring and their age when they moved in. The data is recorded in these tables: Income Age when moved in

38 86

60 76

66 92

76 72

92 88

92 78

117 85

Income Age when moved in

155 71

157 83

157 91

164 75

179 82

192 75

195 90

119 73

136 84

151 88

Identify independent and dependent variables.

Sketch the scatterplot.

Describe the relationship between income and age when they moved into long term care.

Jaden’s lives in a one-bedroom apartment and his online electricity bill allows him to see how much electricity he used each day. For 50 days throughout the summer, he noted the maximum temperature in °F and the amount of electricity in kWh that he used the previous day. The data is recorded in the tables.

Max temperature (°F) Electricity usage (kWh)

70 11

71 11

72 11

72 9

72 7

73 7

73 9

Max temperature (°F) Electricity usage (kWh)

76 7

77 8

77 10

77 13

80 8

80 13

82 14

81 8

82 11

Max temperature (°F) Electricity usage (kWh)

82 8

84 15

85 12

84 9

87 12

91 16

90 13

91 13

93 10

Max temperature (°F) Electricity usage (kWh)

97 18

100 19

99 11

100 19

103 12

103 16

102 12

105 16

104 12

Max temperature (°F) Electricity usage (kWh)

106 16

106 12

109 13

108 17

111 22

111 13

112 22

112 30

Select the correct scatterplot that represents the data set. A

Electricity usage 30

B 115

Electricity usage

110 105

100 95

90 15

85 80

75 70

5 Temperature 65 70 75 80 85 90 95 100105 110 115

Temperature 5

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Electricity usage

10 5

b 16

Electricity usage

Temperature

65 70 75 80 85 90 95 100105 110 115

Describe the relationship between temperature and electricity usage.

Draw an example of a scatterplot and line of ﬁt that demonstrates the following types of relationship: a

Weak

Strong

Positive

Negative

No relationship

Strong, negative relationship with one outlier

Mathias’s parents just sold their old car and bought a new one. He formulates the question “How does the age of a vehicle affect its resale value? Does this vary by type of vehicle?” and he organized the data he collected using these two scatterplots. Cars 36000

Trucks

36000

32000

28000

24000

20000

16000

12000

8000

4000

x 1

534

4000

Describe the trend, pattern, and any outliers for cars.

Describe the trend, pattern, and any outliers for trucks.

What other categories of vehicles could Mathias have explored?

x 1

A researcher is conducting a study on the relationship between sleep and academic performance. They ﬁnd that students who sleep more hours per night have better academic performance. However, the researcher also ﬁnds that students who sleep more hours per night tend to have less demanding extracurricular activities. Which of these statements is true based on the study? A

The improved academic performance is solely due to increased sleep.

The improved academic performance is solely due to less demanding extracurricular activities.

The researcher should control for extracurricular activity demands in their analysis to determine the true effect of sleep on academic performance.

The relationship between sleep and academic performance is not affected by extracurricular activity demands.

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Let’s extend our thinking 19

For a topic that interests you, consider a relationship that might exist. a

Formulate an investigative question that could be used to explore the relationship.

Describe a sample and sampling method that could be used to collect data.

Lousia does an internship at a health insurance company and notices that the expenses seem to vary a lot by person. With permission, she anonymizes some data to create a model for how much to account for with annual medical expenses at different ages. 36000 32000 28000 24000 20000 16000 12000 8000 4000

Annual medical expenses

Age 10

Female

Male

Compare the trends of medical expenses for males and females based on the given data. Include a discussion of whether the trends are the same for the whole data set or if different age brackets tell a different story.

Identify at least one outlier and give a possible explanation.

Temi formulates the question “Is there a relationship between average internet download speed and population density?” She collected data and organized it into two scatterplots, one with the outlier and one without. State Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas

Average internet Population download density (people speed (Mbps) per square mile) 102 250 1 134 66 234 59 198 250 253 57 235 749 294 536 266

Average internet Population download density (people speed (Mbps) per square mile) Montana 8 133 Nebraska 26 225 Nevada 29 263 New Hampshire 157 273 New Jersey 1267 275 New Mexico 17 168 New York 413 265 North Carolina 226 269 State

11 262

224

North Dakota

240

428 194 223 24 225 192 58 36

273 217 251 173 366 241 193 254

Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee

289 60 44 289 1062 182 12 175

254 227 235 241 289 257 221 267

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State Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri

Average internet Population download density (people speed (Mbps) per square mile) 115 244 106 221 45 251 638 255 900 260 178 244 72 227 63 207 90 245

State Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming

Average internet Population download density (people speed (Mbps) per square mile) 119 256 42 248 70 172 222 246 118 230 73 214 110 237 6 139

Including outlier Average internet download speed (Mbps) 350 300 District of Columbia

250 200 150 100 50

Population density (people/mi2) 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Excluding outlier Average internet download speed (Mbps) 350 300 250 200 150 100 50 Population density (people/mi2) 100

536

200

300

400

500

600

700

800

900 1000 1100 1200

Analyze the scatterplot and provide a conclusion to her original question.

Formulate a new question that could be used to further explore the topic of relationships with internet speed.

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When Lexi got her ﬁrst paycheck, she noticed there were some deductions for things like her uniform and income tax. Her mom said that when she is older, there may be more deductions like 401(k). She is now curious about how much you need to retire. In particular, how that relates to the age someone plans to retire. a

Formulate a statistical question that Lexi could use to explore retirement age versus retirement savings.

Could she use observation, measurement, survey, experiment, or acquire secondary sources? Explain.

Explain how Lexi could collect data that could be used to answer her question from part (a).

Suppose the given data shows the age of retirement compared to the retirement savings for 20 people who plan to start their retirement with a long vacation. Analyze the data to draw a conclusion for Lexi. Age Retirement savings

54 1 200 000

56 984 530

56 854 034

60 863 844

60 672 280

63 638 002

64 622 341

Age Retirement savings

65 543 271

66 720 341

67 529 342

68 615 460

69 609 598

69 631 471

70 554 223

Age Retirement savings

71 592 318

74 578 940

77 432 093

78 605 554

78 423 404

79 349 234

Omar has a new puppy. The puppy is growing very quickly. The mother was a very large dog and the father was a medium sized dog. He has heard that smaller dogs tend to live longer and wonders if this is true and if it is true across other animals. Go through the whole data cycle at least once using a context that involves bivariate data and relationships with lifespan of different animals.

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7.02 Scatterplots and lines of fit After this lesson, you will be able to... • calculate the correlation coefficient of a set of data that follows a linear trend. • use the correlation coefficient to assess the fit of a linear function. • write the equation for a linear function that models a data set. • use linear models to make predictions. • evaluate the reasonableness of a linear model for a contextual situation.

Scatterplots and lines of fit When looking at bivariate data, a scatterplot can be used to display the relationship between the two variables. If the data has a linear trend, a line of best fit can be used to model the relationship of the data. We can use technology to find the line of best fit for a scatterplot, then use the line to help us make predictions or conclusions about the data. There are mathematical calculations we can use to measure the strength of the linear correlation between two variables.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.02 to answer these questions. 1.

Arrange these points so the value of the correlation coefficient is as large as possible. What do you notice?

Arrange these points so the value of the correlation coefficient is as close to zero as possible. What do you notice?

Move the points so they are in a straight line, then move one point so it is an outlier. What happens to the correlation coefficient value?

The correlation coeﬃcient, r, is a statistic that describes both the strength and direction of a linear correlation. Correlation Coefficient Strong −1.0

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Weak −0.5 Negative Correlation

Weak

0.0 No Correlation

Strong +0.5 Positive Correlation

+1.0

8 7

x 5 10 15 20 25 30 35 40 45

10 20 30 40 50 60 70 80 90

Perfect positive correlation, r = 1

Perfect negative correlation, r = − 1

y 260

259 258

257 2

256 255

1 x

254

x 0.2

10 20 30 40 50 60 70 80 90

Strong negative correlation, r = − 0.974 9

Weak positive correlation, r = 0.306

9 8

8 7

5 4

x 1

0.3

2 3 4 5 6 7 8 9

x 1

Moderate negative correlation, r = − 0.684

2 3 4 5 6 7 8 9

No correlation, r = 0.072

It is important to be able to distinguish between causal relationships (when changes in one variable cause changes in the other variable) and correlation where the two variables are related, but one variable does not necessarily inﬂuence the other. Correlation

Causation

A relationship between two variables

A relationship between two events where one event causes the other

Even when two variables have a strong relationship and r is close to −1 or 1, we cannot say that one variable causes change in the other variable. Causation can only be determined from an appropriately designed statistical experiment. When the correlation coeﬃcient is close to −1 or 1, we can have more conﬁdence in using the model to make predictions and draw conclusions. 7.02 Scatterplots and lines of fit mathspace.co

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A large sample can also give us more conﬁdence in our conclusions because a large sample is more likely to be representative of the population. However, some types of data can be hard to collect and we will have to do the best we can with a smaller sample, knowing that our conclusions may not be as valid.

Example 1 Data was collected on the number of concert tickets sold and the gross revenue generated by those ticket sales. The data is given in the table. Tickets Sold 75 980 71 714 66 517 63 027 74 000 68 000 72 805 70 500 73 117 65 500 69 200 71 300 76 012

Gross Revenue (in million USD) 8.7 8.3 7.9 7.7 9.1 8 8.6 8.4 9 7.6 8.2 8.5 9.2

a Formulate a question that could be answered by the data.

Create a strategy We are given the number of tickets sold and the corresponding gross revenue from selling those tickets. Think of a question that could be answered by analyzing the relationship between these two variables.

Apply the idea

Reﬂect and check

One example would be, “What is the relationship between the number of tickets sold and the gross revenue from the tickets?” or more speciﬁcally, “Does a high number of tickets sold correspond to a high gross revenue?”

We could also formulate questions about predicting the gross revenue, such as “What is the predicted gross revenue if 80 000 tickets are sold?”

b Create a scatterplot of the data.

Create a strategy We can use technology to construct the scatterplot by following these steps: 1. In the GeoGebra Statistics calculator, enter the data into two columns, one column for the independent variable and one column for the dependent variable. 2. Select all of the cells containing data and choose “Two Variable Analysis.” 3. Use the settings to adjust the scatterplot as needed. In this scenario, the number of tickets sold is the independent variable and gross revenue is the dependent variable.

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Apply the idea 1. Enter the data into two columns.

2. Select all of the cells containing data and choose “Two Variable Regression Analysis.”

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c Find the line of best ﬁt.

Create a strategy We can use technology to find the line of best fit. After following the steps in part (b), we can find the equation of the best fit line by changing the regression model at the bottom of the screen to “Linear”.

Apply the idea Find the Regression model dropdown list at the bottom part of the screen, and select “Linear”.

The equation of the line of best ﬁt is y = 0.0001x − 0.0701.

Reflect and check Although the coefficients seem like very small numbers, remember that our dependent variable is in millions. If we had written out all the place values, the coefficients in the equation would have been very large.

d Use the correlation coeﬃcient to evaluate the strength of the model.

Create a strategy We can use technology to ﬁnd the correlation coeﬃcient. After following the steps in part (c), select the Σx icon. We can use this diagram to evaluate the strength of the model. Correlation Coefficient Strong −1.0

Weak −0.5 Negative Correlation

Weak

0.0 No Correlation

Strong +0.5 Positive Correlation

+1.0

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Apply the idea Select “Show statistics”. The correlation coeﬃcient is represented by r.

The correlation coeﬃcient is r = 0.9253, which means there is strong positive correlation between the variables. This tells us that the model can make relatively accurate predictions.

Reﬂect and check Even when two variables have a strong relationship and r is close to 1 or −1, we cannot say that one variable causes change in the other variable. Causation can only be determined from an appropriately designed statistical experiment.

e Use the model to answer the question you formulated in part (a).

Create a strategy The statistical question from part (a) is, “Does a high number of tickets sold correspond to a high gross revenue?” To answer this question, we can describe the correlation in context and discuss the strength of the correlation that we found in the previous part.

Apply the idea The scatterplot shows a strong, positive linear correlation between the variables. This means that a high number of tickets sold does correspond to a high gross revenue.

Reﬂect and check Note that the term “high” is relative, so a more speciﬁc description of the number of tickets sold or the amount of gross revenue may be better when communicating the results of the analysis. For example, we might say, “When the number of tickets sold increased by about 14 000 tickets, the gross revenue from those ticket sales increased by about $1.6 million.”

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Predict the gross revenue if a concert sells 77 000 tickets.

Create a strategy We can use technology and the line of best ﬁt from part (c) to predict the gross revenue given the number of tickets sold. Remember that the number of tickets will be the input, and the output will be the gross revenue in millions of dollars. To use the graph to predict the gross revenue, we can draw a vertical line at x = 77 000 until it intersects with the line of best ﬁt. Then, we can visually trace a straight line to determine the corresponding y-value.

Apply the idea At the bottom of the graph, we can enter x = 77 000, and it will use the line of best ﬁt to calculate the y-value.

The predicted gross revenue if a concert sells 77 000 tickets is $9.17 million.

Reﬂect and check If we had used the line of best ﬁt from part (c) and calculated the gross revenue by hand, we would have gotten a very different result. y = 0.0001x − 0.0701

Equation of the line of best ﬁt

= 0.0001 (77 000) − 0.0701

Substitute x = 77 000

= 7.7 − 0.0701

Evaluate the multiplication

= 7.6299

Evaluate the subtraction

The reason this is different from the value we got when using technology is because the coefficients were rounded in our line of best fit. When calculated with the actual line of best fit, the calculator does not round the coefficients, making its result more accurate.

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Example 2 A school principal was investigating the effect of class size on the amount of time a teacher can spend with small groups of students, where each student belonged to a group of 4 or fewer students. Their statistical question was, “What size should a class be for a teacher to be able to spend at least 10 minutes with students in small groups?” 18 16 14 12 10 8 6 4 2

Time per small group (minutes)

Class size 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

a Describe a possible method that the principal could use to collect the data.

Create a strategy The data collection methods we have studied are surveys, observations, scientiﬁc experiments, polls, or questionnaires. Another possible method is researching existing data on class sizes and attention time per small group.

Apply the idea As a school principal, it is possible that the data was collected ﬁrsthand. Teachers are most likely not tracking the time they spend with small groups since they are focused on teaching the material, so the principal probably would not use a survey, poll, or questionnaire. One way the principal may have collected the data is throughout observation. They may have been able to observe various class periods of varying class sizes and tracked the amount of time each teacher was able to spend with small groups of students.

Reﬂect and check Have you ever noticed a principal or assistant principal observing one of your classes? What kind of data do you think they were collecting?

b The equation of the line of best fit shown is y = − 0.401x + 18.3, and the correlation coefficient is r = − 0.95. Could this line of best fit be used to make reasonable predictions? Explain.

Create a strategy When considering the reasonableness of a model, we want to analyze the correlation coeﬃcient. The closer the correlation coeﬃcient is to −1 or 1, the stronger the linear correlation between the variables.

Apply the idea The correlation coeﬃcient of this model is −0.95 which indicates a strong, negative linear correlation between the variables. Because the correlation is strong, the actual data values are close to the line which makes it a relatively reliable model for making predictions.

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c Describe the relationship between the variables based on the model. Include the values of the domain for which the model is appropriate.

Create a strategy To describe the relationship between the variables, we will describe the strength and direction of the correlation in context. To determine the domain for which the model is appropriate, we will look at the domain of the actual data values and the clustering of the points to the line and consider what is reasonable within the given context.

Apply the idea The model shows that as the class size increases, the time a teacher is able to spend with small groups of students decreases. This model is based on data where the class sizes range from 6 students to 26 students, so it is most appropriate to use for predictions within this domain.

Reﬂect and check The model can still be used to make predictions for class sizes outside of this domain, but the predictions become less reliable. Notice that the data on class sizes smaller than 10 are further from the line, making any prediction on smaller class sizes less reliable. And while class sizes beyond 26 are likely to be clustered near the line based on the given data, we cannot be sure that this trend will continue without collecting more data. For example, there may be a maximum class size for which this trend applies and after a class gets large enough the amount of time a teacher can spend may drop more signiﬁcantly and be better modeled by a different type of function, like an exponential function.

d Use the graph to answer the principal’s statistical question.

Create a strategy The principal’s question is, “What size should a class be for a teacher to be able to spend at least 10 minutes with students in small groups?”

Apply the idea A teacher is able to spend 10 minutes per small group when the class has about 21 students or less. 18 16 14 12 10 8 6 4 2

Time per small group (minutes)

Class size 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

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Reﬂect and check We could also have used the equation of the line that was given in part (b). y = − 0.401x + 18.3 Equation of the line of best ﬁt 10 = − 0.401x + 18.3 Substitute y = 10 − 8.3 = − 0.401x Subtract 18.3 from both sides 20.698 ≈ x Divide both sides by −0.401 This shows us that the number is actually lower than 21 so a class size of 20 or fewer students may be a better prediction.

Idea summary A line of best ﬁt for a set of data can be used to interpret a given situation and make predictions about values not represented by the data. We can use technology to perform the linear regression analysis. The correlation coeﬃcient, r, is a statistic that describes both the strength and direction of a linear correlation. Correlation Coefficient Strong −1.0

Weak −0.5 Negative Correlation

Weak

0.0 No Correlation

Strong +0.5 Positive Correlation

+1.0

Correlation does not imply causation.

Practice What do you remember? 1

548

State whether each scenario would result in data that could be organized in a scatterplot. a

A coach records the number of laps completed and the time it took to complete a single lap for students applying to join the track team.

A school collects data on the amount of time each student spends in the library.

Elora wishes to know whether there is an association between the ﬂavor of cat food and her cat’s weight.

A scientist collects data on the temperature levels and the population size of a certain species of bird.

A researcher collects data on the number of hours of sleep and academic performance of college students.

Is each statement true or false? a

The correlation coeﬃcient can be negative.

We can tell the direction of the association from the correlation coeﬃcient.

We can tell the strength of the association from the correlation coeﬃcient.

It is possible to get a correlation coeﬃcient of 10.

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For each scatterplot, determine the strength and direction of the correlation. a

200 180 160 140 120 100 80 60 40 20

y 80 70 60 50 40 30 20

20 40 60 80 100 120 140

20 40 60 80 100 120 140 160 180

22 20 18 16 14 12 10 8 6 4 2

220 200 180 160 140 120 100 80 60 40 20

x 40

80 100

Describe the strength and directions of the linear relationship between the variables with these correlation coeﬃcients: a

0.96

0.66

0.36

− 0.06

− 0.44

− 0.56

−1

The correlation coeﬃcient of two different data sets is given. i

Do the two linear relationships have the same direction?

Which data set has a stronger correlation?

Data set A: r = 0.3

10 20 30 40 50 60 70 80

Data set A: r = − 0.75

Data set B: r = 0.9

Data set B: r = − 0.5

Data set A: r = 0.8

Data set A: r = 0.4

Data set B: r = − 0.5

Data set B: r = − 0.6

Use the correlation coeﬃcient to interpret the relationship between the variables in these studies. a

A researcher plotted the life expectancy of a group of men against the number of cigarettes they smoke a day. The results were recorded and the correlation coeﬃcient was found to be −0.88.

A researcher was evaluating the relationship between the number of years in education a person completes and the number of pets they own. The results were recorded and the correlation coeﬃcient was found to be −0.3.

A study found that the correlation coeﬃcient between the population of a city and the number of speeding tickets recorded was found to be 0.83.

A study found that the correlation coeﬃcient between hair length and ﬁngernail length was found to be 0.07.

A study found that the correlation coeﬃcient between the number of laws a state has about dog breeding and the number of dogs available for adoption in shelters was found to be 0.55.

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Mae has a small herb garden with a thyme plant. She suspects that the growth of the thyme plant is linearly dependent on the amount of tea she drinks. For a month, she keeps a daily log of the amount of tea she drinks and the millimeters that the thyme plant has grown. The resultant correlation coeﬃcient was found to be 0.36. a

Is there a linear relationship between the growth of the thyme plant and the amount of tea that Mae drinks?

Can we say that there is a causal relationship between the growth of the thyme plant and the amount of tea that Mae drinks?

Let’s practice 8

For each scatterplot: i

Describe the strength and direction of the linear relationship between the variables.

Estimate the value of the correlation coeﬃcient.

5 x 5

y 15

15 10 10 5

x 5

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For each set of bivariate data: i

Find the line of best ﬁt.

Calculate the correlation coeﬃcient to two decimal places.

iii

Describe the strength of the relationship between the two variables.

x y

9 19

4 12

35 75

34 63

32 61

18 37

28 57

10 22

20 38

3 7

x y

2 88

4 83

7 60

0 95

3 84

2 88

0 104

6 72

4 85

3 84

x y

3 7

4 7.4

5 7.88

6 7.64

x y

65 25.5

72 24.2

48 16.8

84 10.4

78 24.8

56 9.6

63 6.3

64 21.4

77 13.7

93 16.3

x y

13 3.6

13 12.5

9 12.1

12 12.8

9 11.1

15 9.7

10 11.4

12 9.7

12 10.6

14 8.9

55 57

60 45

65 49

8 8.2

9 7.32

A data set for speed and time is given: Speed Time

7 7.72

20 85

25 87

30 75

35 82

40 69

45 73

50 60

Construct a scatterplot.

Find the equation of the line of best ﬁt and sketch it on the scatterplot.

Find the correlation coeﬃcient.

Describe the relationship between speed and time.

Noah is a coffee vendor. He records the maximum temperature of the day and the number of coffees sold to answer the statistical question, “How does the daily high temperature impact coffee sales?” The results are recorded in the table:

Maximum temperature (°F) Number of coffees

82 63

90 97

88 77

91 102

88 73

79 45

77 66

Maximum temperature (°F) Number of coffees

84 93

95 108

86 65

75 70

94 100

97 102

79 68

What data collection method did Noah use? A Survey

Poll

C Observation

Research

E Scientiﬁc experiment b

Construct a scatterplot for the data.

Find the line of best ﬁt.

Calculate the correlation coeﬃcient. Round your answer to two decimal places.

Describe what happens to the sales of coffee as the temperature increases.

Predict the number of coffees they will sell on a day with a high of 78 °F.

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The scatterplot shows data collected on the time taken to get to school, in minutes, and daily time spent watching TV, in hours, for 27 students. a

Formulate a question that can be answered by the scatterplot.

Which equation is most likely the line of best ﬁt?

Time watching TV (hours) 4 3

A y = 0.0265x + 3.33 B y = 0.0265x − 3.33

C y = − 0.0265x + 3.33 D y = − 0.0265x − 3.3 c

d 13

Which is closest to the correlation coeﬃcient? A −1

− 0.1

C − 0.6

− 0.5

Time to school (minutes) 25

100

125

Describe the strength of the correlation. Explain what this correlation implies in terms of the context.

The scatterplot shows the number of crashes in 2020 that involved cars of a given year of manufacture. a

If 10 000 accidents were recorded for a certain car, in which year was it likely made?

About how many crashes can we expect for a car that was made in 2006?

Identify and interpret the strength and direction of the correlation.

For what values of the domain is the model appropriate? Explain.

14000 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000

Number of crashes

Year of manufacture 1990 1995 20002005 2010 2015 2020

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Kareem is investigating the relationship between daily water levels and the height of bean plants. He places each plant in the same room, and waters them at the same time each day. The scatterplot shows the data he collected on the heights of the bean plants at 3 weeks given a certain amount of water each day. a

What method did Kareem use to collect the data?

Identify and interpret the strength and direction of the correlation.

According to the data, is it possible for a bean plant to grow if it does not receive any water for 3 weeks?

If a bean stalk’s daily water amount is increased by 1 ﬂuid ounce, by how much can we expect it to increase in height? A 0.5 in

1 in

C 2 in

3 in

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24 22 20 18 16 14 12 10 8 6 4 2

Height (in)

Daily water (fl.oz.) 1

Harleigh collected data from a random sample of people on the number of hours they exercised each week and their level of stress (on a scale from 1−10). She organized the data into the scatterplot and found the line of best ﬁt to be y = − 0.55x + 10.1. a

10 9 8 7 6 5 4 3 2 1

What data collection method did Harleigh use? A Research

Survey

C Observation

Experiment

According to the data, what is the reported stress level of someone who does not exercise?

Predict the stress level of someone who exercises for 12 hours a week.

If someone reports a stress level of 5, about how many hours might they be exercising in a week?

For what values of the domain is the model appropriate? Explain.

Stress levels

Weekly exercise (hours) 1

2 3 4 5 6 7 8 9 10

During an alcohol education program, 10 adults were offered up to 9 alcoholic drinks and were then given a simulated driving test. The driving test was scored out of 100. The results are displayed in the table: Number of drinks Driving score

2 61

3 65

5 44

3 62

4 36

2 73

7 32

6 41

9 18

Describe the data collection method used.

Find the line of best ﬁt.

Calculate the correlation coeﬃcient and round your answer to two decimal places.

Predict the driving score for someone who has had 1 drink.

If someone received a driving score of 25, how many drinks did they likely have?

Which claim is not valid?

2 54

A There is a strong correlation between the number of drinks and driving score, showing that driving gets worse as more alcohol is consumed. B This data shows that drinking and driving is very dangerous. C Although the correlation is strong, the sample size is very small, making any conclusions less reliable. D Anyone driving poorly on the road has been drinking. 17

WhichBank is researching the number of applications they receive for home loans per day based on their advertised interest rate. The results are displayed in the graph. WhichBank found the line of best ﬁt to be y = − 10.1x + 198 with a correlation coeﬃcient of r = − 0.96. Their current interest rate is 5.5%. a

Formulate a question that can be answered by the scatterplot.

For each 1% increase in the interest rate, by how much will the number of applications decrease?

According to the data, how many applications will they receive each day if the interest rate is 0%?

WhichBank makes this conclusion: “If we want people to take out a home loan, we need to lower the interest rates.” Is their claim valid? Explain.

210 Number of applications 200 190 180 170 160 150 140 130 120 110 100 90 80 Interest rate 1 2 3 4 5 6 7 8 9 10 11

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Advertisers are researching the optimal length of an advertisement so that viewers don’t click ‘Skip’. The results in the table show the length of the advertisement in minutes, and the proportion of viewers who clicked ‘Skip’: a

Formulate a question that could be answered by the data.

Calculate the correlation coeﬃcient and round your answer to two decimal places.

Predict the proportion of people who will skip an ad if it is 3 minutes long

Answer the question you formulated in part (a).

Is this claim valid: “The length of the ad is the only reason people might skip it.”?

Minutes 5.1 5.9 0.4 1.4 0.8 4 4.1 0.6 2.7 5.8 1.4 0.7 1.1 0.7 4.8

Proportion 1 1 0.6 0.4 0.7 0.8 0.8 0.3 0.9 0.8 0.7 0 0.4 0.8 0.8

Minutes 3.5 0.9 4.5 3.8 4.2 1.5 1.2 2.4 5.5 4.2 3.9 5 1 2 4.9

Proportion 0.9 0.7 0.8 0.6 0.8 0.7 0.4 0.7 1 0.7 0.7 0.9 0.3 0.5 0.9

A city started a junior soccer league in 2012 and has tracked the number of participants in the league since. Their data is shown in the table. Year Participants

2012 268

2013 240

2014 232

2015 215

2016 182

2017 175

Year Participants

2018 143

2019 122

2020 107

2021 204

2022 250

2023 277

Construct a scatterplot for the data.

The line of best ﬁt for this data is y = − 2.65x + 216, where x is the number of years since 2012, with a correlation coeﬃcient of r = − 0.17. Explain why this model does not make sense for the data.

Find the line of best ﬁt for the data from 2012 to 2020.

Find the line of best ﬁt for the data from 2020 to 2023.

Write the piecewise regression model by combining the lines in parts (c) and (d). Could this model be used to make reasonable predictions? Explain your answer.

Let’s extend our thinking 20

Create a data set which has a correlation coeﬃcient between −0.9 and −0.8.

Maeve kept track of the amount of caffeine she consumed and her pulse rate over the course of a month. Her results are shown in the table.

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Caffeine (mg) Pulse rate (bpm)

143 77

84 63

241 82

218 80

169 71

126 70

105 76

309 85

227 79

153 77

Caffeine (mg) Pulse rate (bpm)

317 89

212 75

227 83

247 82

52 70

75 68

268 86

333 92

145 80

348 93

Caffeine (mg) Pulse rate (bpm)

266 92

319 83

184 79

312 89

57 67

256 88

57 61

272 92

208 85

165 77

Predict Maeve’s pulse rate if she drinks 200 mg of caffeine.

Describe the validity of the prediction in part (a).

Predict Maeve’s pulse rate if she drinks 400 mg of caffeine.

Which prediction is more reliable, the one in part (a) or the one in part (c)? Explain.

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Use the data cycle to investigate the relationship between salary and years of teaching experience for teachers in your school division. a

Formulate a question which could be investigated using a scatterplot.

Acquire data on teacher pay for your school division.

Create a scatterplot of the data.

Find the line of best ﬁt.

Use the correlation coeﬃcient to evaluate the strength of the model.

Use the model to answer the question you formulated in part (a).

Is it reasonable to suggest that teaching experience is the sole inﬂuence on teacher salaries? If not, give an example of a different factor that affects teaching salaries.

Collect data on the teaching salaries from another district. Analyze the data and compare it to your results.

A researcher was investigating the relationship between reading speed and reading comprehension. They gave 20 people a short story to read, measured how many words they read in a minute, then gave them a quiz about the story. The quiz was scored out of 100, and the data is shown in the table. Words per minute Comprehension

150 60

155 65

172 68

178 72

183 75

205 78

206 80

214 82

228 83

238 85

Words per minute Comprehension

245 85

262 84

270 84

275 83

288 83

300 81

304 81

308 79

312 78

314 78

Predict the score someone would receive if they read 250 words per minute.

Can the line of best ﬁt be used to make reasonable predictions? Explain.

Is some subset of the data better represented by a different function? If so, use a new model to predict the score someone would receive if they read 250 words per minute.

Suppose you want to determine if there is a relationship between the amount of time you use a device (such as a phone, laptop, or tablet) and the device’s battery life. Go through the whole data cycle at least once to investigate whether a relationship between the two exists.

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7.03 Linear, quadratic, and exponential models After this lesson, you will be able to... • determine whether a data set is best represented by a linear, quadratic, exponential, or piecewise function. • write the equation for a function that models a data set. • make predictions about a data set. • evaluate the reasonableness of a mathematical model.

Curves of best fit Bivariate data can be modeled with a curve of best ﬁt, also called a regression model. If the correlation between the variables is linear, a line of best ﬁt can be used to model the data. If the correlation is not linear, the data may be modeled by a curve, such as an exponential or quadratic model. When choosing a model to represent data and make predictions, we can use what we know about the key features of the functions we have learned so far and match them to the behavior of the data. y

y 25

50000

40000

30000 20000

10000

x 10 20 30 40 50 60 70

Linear functions can represent data that increases or decreases in a constant manner. Linear functions have inﬁnite domain and range and can have both x and y-intercepts.

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0.1 0.2 0.3 0.4 0.5

Exponential functions can represent data that grows or decays rapidly and levels off around a speciﬁc value. Exponential functions have an inﬁnite domain but a restricted range, so it may not have an x-intercept.

45 40 35 30 25 20 15 10 5

x 10 20 30 40 50 60 70 80

Quadratic functions can represent data that changes direction smoothly and has a minimum or maximum value. Quadratic functions have an inﬁnite domain, and may or may not have intercepts.

Exploration Each table shown represents a different set of data. Table 1 x y

0 13

0.2 7

0.4 4

0.6 3

0.8 1

1 0

3 63

3.5 68

4 77

4.5 90

5 104

5.5 100

0 4

1 5

4 7

5 8

6 16

1.2 2

1.4 3

1.6 6

1.8 9

Table 2 x y

6 112

6.5 120

7 114

7.5 127

8 127

Table 3 x y

2 5

3 6

7 30

8 37

9 66

10 92

Without creating a scatterplot: 1.

What type of function would best model the data in Table 1? Explain your answer.

What type of function would best model the data in Table 2? Explain your answer.

What type of function would best model the data in Table 3? Explain your answer.

The correlation coefficient is used to determine the strength of a linear relationship, but it cannot describe the strength of nonlinear relationships. Instead, we can analyze the coefficient of determination (R2). The value of R2 can vary between 0 and 1. The closer the value is to 1, the more accurate the model is.

Example 1 The population of ﬁsh in a small lake over time, given in years, is shown in the table: Years 0 0.25 0.5 0.75 1 1.25 1.5 1.75

Fish Population 1000 820 650 665 500 490 425 350

Years 2 2.25 2.5 2.75 3 3.25 3.5 3.75

Fish Population 290 210 160 145 120 120 100 100

a Determine whether a linear or exponential model best ﬁts the relationship between the years and the population of ﬁsh.

Create a strategy Use technology to plot the data on a coordinate plane, then examine the shape of the data.

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To do this, enter the x- and y-values in two separate columns, then highlight the data and select Two Variable Regression Analysis:

Apply the idea

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To ﬁnd the line of best ﬁt, choose Linear under the Regression Model drop down menu:

To ﬁnd the exponential curve of best ﬁt, choose Growth under the Regression Model drop down menu:

The exponential model is a better model of the data because most of the points are tightly clustered around the curve. The linear model does not represent the smallest and largest x-values well. An exponential model would best ﬁt the data after examining the plot.

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Reflect and check We can also use the correlation coefficient and coefficient of determination to examine the fit of the linear and exponential models. To examine the fit of the exponential model, click the Σx button to show the statistics and find the coefficient of determination (R2). For this curve, R2 = 0.9844 which means this curve closely models the actual data.

To examine the fit of the linear model, find the correlation coefficient (r). For this line, r = − 0.9538 which shows a strong correlation, but it is not as strong as the exponential model.

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b Calculate the regression model for the data and use it to predict the population of ﬁsh in the lake after 4 years.

Create a strategy Use technology to calculate the exponential regression, then use the equation to determine the population when x = 4.

Apply the idea The function that ﬁts the model best is y = 1005.4784 (0.5161)x If x = 4, then y = 1005.4784 (0.5161)4 = 71.3359 According to the regression model, we can expect there to be about 71 ﬁsh remaining in the lake after 4 years.

Reflect and check Remember that the coefficients in the equation for the curve of best fit have been rounded. For that reason, the answer is not 100% accurate. If we had used technology to make this prediction, we would have gotten a slightly different answer.

The calculator’s answer is more accurate because it includes more decimal values in the coefficients and does not round them to only four place values. However, the differences between these values are small because we used four decimals in the coefficients of the equation. If we had rounded the coefficients to even fewer decimal places, the answer would have been less accurate.

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Example 2 Jocelyn plays on the basketball team and wants to improve her shot. She notices that she is really good at making shots from certain distances, but she is not as good at making shots from other distances. She decides to investigate this further using the data cycle. a Formulate a statistical question that Jocelyn can use to investigate the relationship between the shots she makes and the distance she is from the hoop.

Create a strategy We can assume that Jocelyn is investigating the relationship between the shots she makes and the distance she is from the hoop because she wants to improve her game. There are many statistical questions we can ask, but we should focus the question around the purpose of the investigation.

Apply the idea One possible statistical question is, “At what distances does Jocelyn make less than half of her shots?”

Reﬂect and check Other possible questions are: • How does the percentage of shots Jocelyn makes change with the distance from the hoop? • If Jocelyn is at the three point line, what percent of shots can we expect her to make? • How far is Jocelyn from the hoop when she makes most of her shots?

b Describe a method Jocelyn can use to collect the data.

Create a strategy Jocelyn needs to collect data on two variables: the distance she is from the hoop and the number or percent of shots she makes from that distance. Bivariate data can be collected by: • acquiring data through research • collecting data using surveys, observations, scientiﬁc experiments, polls, or questionnaires Jocelyn will need to collect the data herself, rather than acquiring it through research. She cannot collect the data through a survey, poll, or questionnaire either. The only variable Jocelyn is interested in controlling is the distance from the hoop, so she does not need to use a scientiﬁc experiment.

Apply the idea Jocelyn can use a systematic observation to collect this data. One way to do this would be to draw arcs that are various distances from the hoop (such as 2 ft from the hoop, 3 ft from the hoop, etc.). Then, she can shoot a large number of shots from each of the distances and keep track of how many she made from each distance. To get a random sample, she should not take consecutive shots from the same distance or same side of the hoop. For example, rather than taking 10 consecutive shots from the free throw line, she should take a few shots from the left or right side of the hoop that is the same distance as the free throw line. This will help the data be more representative of how she shoots in general.

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c Jocelyn collected data on the percentage of shots she made from various distances. Her data is shown in the table. Distance ( ft) Shots made (%)

2 33

3 36

4 47

5 56

6 69

7 71

8 81

9 88

10 88

Distance ( ft) Shots made (%)

12 93

13 88

14 84

15 81

16 70

17 68

18 57

19 49

20 39

11 91

Organize the data into a scatterplot.

Create a strategy We can use technology to organize this data into a scatterplot. The distance from the hoop is the independent variable, and the percentage of shots she made is the dependent variable.

Apply the idea Enter the x-values into one column and the y-values in a second column, then highlight the data and select Two Variable Regression Analysis.

d Determine the type of model that fits the data best, and find the equation of the curve of best fit.

Create a strategy We can use either the table or the scatterplot to analyze how the y-values change as x increases. Then, we can use technology to ﬁnd the equation of the curve of best ﬁt.

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Apply the idea As the x-values increase, the y-values increase, reach a maximum point, then decrease. This shows the data would be best represented by a quadratic equation. Recall that quadratic functions are polynomial functions of degree 2. Using technology, we can choose a polynomial regression model, and the degree is set to 2 by default.

The quadratic curve of best ﬁt is y = − 0.7048x2 + 16.1349x − 3.2206.

Reﬂect and check The coeﬃcient of determination, R2 = 0.9784, shows that the model is a good representation of the data.

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e Use the data to answer the statistical question from part (a).

Create a strategy Our statistical question was, “At what distances does Jocelyn make less than half of her shots?” We can use the scatterplot to ﬁnd the x-values for which the y-values are less than 50.

Apply the idea To help read the graph better, we can use the settings to adjust the axis scale and add grid lines.

The y-values are less than 50 when x is less than 4 or greater than 19. This means Jocelyn makes less than half of her shots when she is less than 4 feet from the hoop or more than 19 feet from the hoop.

Reflect and check Because the coefficient of determination is high, Jocelyn can be fairly confident in this analysis. Notice that the domain of the curve is [2, 21], so Jocelyn should not use this model to make predictions outside of this domain.

Idea summary Bivariate data can be modeled by curves of best ﬁt such as linear functions, quadratic functions, or exponential functions. We can analyze how the data changes over the domain to choose the function that is most appropriate. The correlation coefficient, r, determines of well a linear model fits the data. R2 determines how strong or weak the correlation of a nonlinear model is. An R2 value closer to 1 represents a stronger fit.

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Piecewise regression Different functions can be used to model different situations. Sometimes, it takes a combination of functions to see the full picture. A curve of best ﬁt may only be appropriate over a part of the domain, and we can create a piecewise function to model the data over different intervals of the domain. 90

For example, the data shown in the scatterplot does not have a linear pattern nor an exponential pattern.

If we use technology to find the quadratic curve of best fit, it does not fit the data well either.

70 60 50 40 30 20 10

12 16 20 24 28

Notice that the data appears to follow a linear pattern until x = 14, then it has a quadratic pattern as the x-values increase.

90 80

If we use a piecewise regression model instead, where the model is linear when x < 14 and the model is quadratic when x > 14, the model is a much better representation of the data.

70 60 50 40 30 20 10

12 16 20 24 28

Example 3 Loren posted a video on his social media account and noticed that the video was gaining lots of views. He collected data on the number of views the video had at the end of each day and organized it into a scatterplot, as shown.

566

Days

Views

Days

Views

100

16 526

132

17 538

212

19 192

485

20 537

1226

23 264

2779

23 831

6838

24 809

12 000

27 249

13 393

29 728

14 262

29 942

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Views of a video over time 30000 27000 24000 21000 18000 15000 12000 9000 6000 3000

Number of views

Days 2 4 6 8 10 12 14 16 18 20

a Is the relationship between the days and the number of views best approximated by a line, quadratic curve, exponential curve, or a combination of these functions? Explain.

Create a strategy Use the scatterplot to determine how the number of views changes over time. If the change is not consistent, consider which functions would best model the data over different subsets of the domain.

Apply the idea Over the ﬁrst 8 days, the number of views increases at an increasing rate. From 0 to 8 days, the data is best modeled by an exponential function. After 8 days, the number of views increases at a relatively constant rate. From 8 to 20 days, the data is best modeled by a linear function.

Reflect and check To check our answer, we can try to fit a linear, exponential, and quadratic model to the data. However, none of the best fit curves model the data well over the entire domain. 30000 27000 24000 21000 18000 15000 12000 9000 6000 3000

Line of best ﬁt

Number of views

Even though the correlation coefficient is high (r = 0.9877), it does not model the data well for independent values less than 10. It models the data from 10 days to 20 days fairly well, but we can get an even better fit if we remove the first several data values.

Days 2 4 6 8 10 12 14 16 18 20

30000 27000 24000 21000 18000 15000 12000 9000 6000 3000

Quadratic curve of best ﬁt

Number of views

This curve does not look like a parabola because it is very zoomed in on one section of the parabola. If we zoom out, we can see the downward facing parabola take shape. Similar to the linear model, this curve does not model the data well for independent values less than 10.

Days 2 4 6 8 10 12 14 16 18 20

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30000 27000 24000 21000 18000 15000 12000 9000 6000 3000

Exponential curve of best ﬁt

Number of views

Similar to the previous two models, this curve does not model the data very well.

Days 2 4 6 8 10 12 14 16 18 20

b Use your answer from part (a) to ﬁnd the regression model for the data.

Create a strategy In part (a), we found that the data from 0 to 8 days should be modeled by an exponential curve of best fit, and the data from 8 to 20 days should be modeled by a line of best fit. Using technology, we can enter each subset of data separately to find the two curves of best fit for our piecewise regression model.

Apply the idea To find the exponential curve of best fit for the first subset of the domain, enter the data points on the interval 0 ≤ x ≤ 8 into the calculator. Then, find the growth regression model.

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To find the line of best fit for the second subset of the domain, enter the data points on the interval 8 ≤ x ≤ 20 into the calculator. Then, find the linear regression model.

Notice that when x = 8, the exponential model represents the data better. When creating the piecewise function, we will include x = 8 in the domain for the exponential piece, but exclude x = 8 from the domain for the linear piece. The piecewise regression model for the data shown in the scatterplot is

Reﬂect and check If we graph the piecewise model, we can see that it represents the data much better than using a single function to model the data over the entire domain.

Views of a video over time 30000 27000 24000 21000 18000 15000 12000 9000 6000 3000

Number of views

Days 2 4 6 8 10 12 14 16 18 20

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Idea summary Piecewise functions can represent data that have different characteristics at different intervals. To find the curves of best fit in a piecewise function, we must first create a scatterplot and analyze the data. Then, we will need to determine an appropriate domain for each function type.

Practice What do you remember? 1

The graphs of f (x), g (x) and h (x) are shown. Identify the graphs as exponential, linear, or quadratic.

h (x) f (x)

16 14 12 g (x)

10 8 6 4 2

x 1

y = 3x

y = 3x2

For each graph, determine whether a linear, quadratic, or exponential function could represent it: a

5 x 5

570

Classify each function as linear, quadratic, or exponential. a

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x 5

5 x

x 5

Using each table: i

Create a scatterplot.

State whether each relationship is best modeled by a linear, exponential, or quadratic function.

x y

1 2

2 3

4 4

6 7

8 7

10 9

11 11

14 13

17 14

19 16

x y

1 19

2 16

4 10

5 8

9 3

14 5

16 11

17 13

19 17

20 20

x y

5 44

10 30

12 17

21 7

25 3

27 5

36 6

38 7

43 3

45 5

x y

5 3

10 30

20 55

28 68

37 70

45 78

59 64

77 39

81 11

92 9

The daily net proﬁt (in thousands of dollars) of a cafe can be modeled by the equation y = − 0.01x2 + 0.96x + 13.7 where x is the number of customers. Determine the net proﬁt when 70 customers dine in the cafe.

For each data set, use technology to ﬁt the speciﬁed regression model to the data. a

Linear x y

16 31

21 42

36 45

41 50

56 52

61 56

6 3870

9 4170

12 3999

15 3244

17 2900

19 1951

21 1678

22 1542

2 26.67

3 10.52

4 21.66

5 75.82

6 76.19

7 147.46

8 242.03

9 391.66

Quadratic x y

11 28

3 2450

24 650

Exponential x y

1 21.72

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Let’s practice SOL

Drake made a scatterplot showing the height of a student’s feet during his time in the air while performing a long jump. Height (feet) 4.5 4 3.5 3 2.5 2 1.5 1 0.5

Time (seconds) 0.2 0.4 0.6 0.8

1.2

1.4

1.6

1.8

2.2 2.4 2.6 2.8

Which is most likely the equation for the curve of best ﬁt for the relationship? A C 8

y = 0.1593x + 1.8396 2

y = − 1.7497x + 5.4083x − 0.9599

y = − 1.8396x + 0.1593

y = 5.4083x2 − 1.7497x − 0.9599

Match each scatterplot to the equation that best models the relationship. i

y = 5.41 ⋅ 0.43x

iii

y = 1.63x − 0.4

y = 0.01x2 − 0.72x + 82.64

y 90

30 80 20 70

10 x 5

y 35

30 4

20 15

10 1 x 1

572

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x 10

SOL

For each set of data, using technology: i

Create a scatterplot of the data.

Determine whether a linear, quadratic, or exponential function best models the data.

iii

Find the equation of the curve of best ﬁt.

Predict the value of y when x = 31.

x y

28 3926

x y

17 16

19 15

x y

10 860

22 732

x y

5 99

x y

30 6482

32 10 589

21 17

34 17 098

23 17

36 28 236

38 46 985

25 18

27 19

29 20

34 602

40 555

52 428

69 400

85 500

90 644

97 768

102 848

10 121

17 134

22 128

26 140

29 142

35 141

38 155

41 158

47 163

55 170

5 450

8 472

14 505

25 528

29 512

40 499

46 461

55 419

61 370

66 344

72 289

15 17 888

16 12 546

21 4750

23 1459

25 899

27 342

28 155

30 90

18 9422

19 5999

67 179

Jensen makes and sells juices and smoothies. He noticed that the sales were following a quadratic trend, as shown in the table. Juice and smoothie sales Week Profit ($)

1 50

2 100

3 180

4 425

5 500

6 405

7 350

8 370

9 290

10 240

If the sales continue to follow this trend, which is closest to the amount of proﬁt made on the 11th week? A 11

160

The table shows the population of mosquitoes (M ) versus time in days (t). Number of days (t) Mosquitoes (M )

1 12

2 24

3 29

4 63

5 86

Use the exponential regression model to approximate the number of mosquitoes on day 0.

Do you think the model can be used to make reasonable predictions after the 20th day?

100 90 80 70 60 50 40 30 20 10

210

Mosquitoes (M)

Number of days (t) 1

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For each of the following situations, assess the reasonableness of the given mathematical model. If the model is not reasonable, explain why and suggest an improvement. a

The number of people attending a concert is modeled by the equation P(t) = −50t + 400, where P(t) represents the number of people at the concert at time t hours after the gates open.

The height of a ball dropped from a building is modeled by the equation h(t) = −4t + 20, where h(t) represents the height of the ball in meters at time t seconds.

The temperature of a cup of coffee cooling down over time is modeled by the equation T(t) = 70 + 30e−0.1t, where T(t) represents the temperature of the coffee in degrees Fahrenheit after t minutes.

The scatterplot shows the population of rabbits over time (in years). a

Select the question that cannot be answered by the scatterplot.

800

A How does the population of rabbits change over time?

700

B What is the rabbit population in 2021?

600

C How many rabbits will there be after 10 years?

500

D After how many years will there be 625 rabbits?

400

Population

300

Explain why it would not be appropriate to calculate the correlation coeﬃcient for the data set shown.

200 100

A group of students formulated the question, “What is the relationship between the amount of water someone drinks in a day and their energy levels?” They asked a random group of people to track the water they drank in a day (in liters), then rate their energy level (on a scale from 0−10).

Water intake (L) Energy level

2.5 8

1.8 6

3 9

1.2 4

2.7 8

1.5 5

3.2 9

1 3

2.8 8

1.3 4

Water intake (L) Energy level

1.6 5

3.1 9

2.3 7

1.4 4

2.9 8

1.7 6

2.6 8

1.1 3

1.9 6

2.4 7

Experiment

What data collection method did the students use? A Survey

Observation

Research

Determine whether a linear, quadratic, or exponential model best ﬁts the relationship between water intake and energy level.

Find the equation of best ﬁt.

Describe the strength of the model from part (c).

Predict the energy level if one consumes 3.5 L of water.

The table shows Dillon’s credit card balance each month after he opens the line of credit. Month Debt ($)

1 512.44

2 555.78

3 578.25

4 620

Month Debt ($)

6 899.80

7 1039.93

8 1270.63

9 1589.04

5 761.88 10 1851.31

Formulate a question that can be answered by the data.

What type of function best models the relationship between the month, x, and Dillon’s debt, y? A Linear

574

Years 1 2 3 4 5 6 7 8 9

Quadratic

Exponential

Calculate the regression model for this data.

Assuming that spending continues at this rate, ﬁnd Dillon’s credit card debt after one year. Round your answer to the nearest cent.

Mathspace Virginia SOL Algebra 2 mathspace.co

The table shows the number of hours Tristan spends reading over 10 months. Month Reading hours

2 3.7

4 5.8

5 6.4

6 7.5

7 7.1

8 5.2

9 4.7

10 3.5

Describe a method Tristan may have used to collect the data.

Determine whether a linear, quadratic, or exponential model best ﬁts the relationship between the month, x, and the reading hours, y.

Calculate the regression model for this data.

Assuming that Tristan’s reading habits continue to follow this trend, ﬁnd his reading hours in the 12th month.

Ghale posted a video on her social media account about cooking tips. The application tracked the number of times the video was viewed each day as shown in the table: 1 900

2 754

3 527

4 329

5 422

6 729

7 987

Formulate a question that can be answered by the data.

Create a scatterplot and use it to determine if the data suggests an exponential, quadratic, or linear association. Explain your answer.

Determine an appropriate equation to model the data set. Round coeﬃcients to four decimal places.

Predict the number of views the video will have on the 16th day.

The table shows data on the number of pounds of litter collected each week in a national park x weeks after the park managers started an anti-littering campaign: Weeks (x) Pounds of litter collected ( y)

3 4.9

Days Views

1 3

1 6.4

2 5.5

3 3.5

4 3.1

5 1.6

6 1.2

7 0.7

Find the regression model that best ﬁts the data.

Analyze the model to determine when the campaign is no longer effective.

For what values of the domain is the model appropriate? Explain.

8 0.7

9 0.6

10 0.5

Each day, a bakery makes the same number of pastries. Any unsold pastries are donated to a local food bank at the end of the day. Chastine, the owner’s daughter, collected data on the number of unsold pastries each day and organized it into a scatterplot, as shown. Day Unsold pastries

1 20

2 30

3 36

4 38

5 32

6 28

7 21

8 15

9 9

10 6

Day Unsold pastries

11 11

12 12

13 13

14 14

15 15

16 16

17 17

18 18

19 19

20 20

Unsold pastries

36 32 28 24 20 16 12 8 4

Days 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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Select the question that Chastine may have used for her investigation. A What day of the week do we sell the most pastries? B What is the relationship between the price of a pastry and the number of unsold pastries? C What is the most amount of money that can be made by selling pastries? D How does the number of unsold pastries change over time?

What data collection method did Chastine use?

Chastine found the curve of best ﬁt to be y = 0.1372x2 − 4.0142x + 39.8667. Is some subset of the data better represented by a different function? Explain your answer.

Find the equation of the piecewise regression model.

Let’s extend our thinking 20

Imran made some syrup and then put it in the fridge to cool down. He measured the temperature of the syrup at 2-minute intervals for 10 minutes. The temperature each time is represented in the table. Minutes Temp. (°F)

0 212

2 200

4 185

6 170

10 140

12 130

14 126

16 122

20 117

Create a scatterplot and use it to determine if the data suggests an exponential, quadratic, or linear association. Explain your answer.

Determine an appropriate equation to model the data set. Round coeﬃcients to four decimal places.

After 22 minutes, the syrup has cooled enough to eat. Find the temperature that the syrup is cool enough to eat. Round your answer to one decimal place.

During a sudden bacterial outbreak, scientists must decide between two anti-bacterial treatments that are currently being trialed to try to control the outbreak. In the laboratory, they apply Treatment A and Treatment B to two samples of the bacteria, each containing 200 microbes. They keep track of the number of microbes in each sample. The table shows the results. 0 200 200

3 215 600

6 230 1800

9 245 5400

Determine whether each treatment causes the number of microbes to increase at a linear, exponential or quadratic rate.

Determine which treatment will better control the number of microbes. Explain your reasoning.

The table shown represents the revenue, in thousands of dollars, over time, of two new shoe companies: Foot Swag, F (x) and Sweet Kicks, K (x), where x is the time in weeks. a

Each company secured funding from investors to get up and running. Determine the initial revenue for each function.

576

18 119

Number of hours (t) Number of microbes using Treatment A Number of microbes using Treatment B

8 155

Determine when the revenue of Foot Swag and Sweet Kicks will be equal.

Assuming both companies stay in business for a very long time, which company would you expect to reach a revenue of $1 million ﬁrst? Foot Swag or Sweet Kicks? Explain your reasoning.

Mathspace Virginia SOL Algebra 2 mathspace.co

x 0 1 2 3 4 5

F (x) 1 2 4 8 16 32

K (x) 2.12 3.52 5.48 8 11.08 14.72

Nadia is considering two different career options. • She could accept a job with Kord Enterprise that has offered her an income of $125 000 for the ﬁrst year, and a projected income increase of $34 000 each year after that. • Alternatively, she can start her own business, where she estimates her initial annual income to be $40 000 for the ﬁrst year, increasing by 40% each year after that.

Complete the table of values:

Determine whether Nadia’s income for each situation could be modeled by a linear, quadratic, or exponential model.

In the 5th year, determine which option will result in a higher annual income for Nadia. Explain your answer.

In the 10th year, determine which option will result in a higher annual income for Nadia. Explain your answer.

1 2 3 4

Determine the domain that will result in a higher annual income at her own company versus working at Kord Enterprise.

Income in year Income in year n with her own n with Kord business Enterprise 125 000 40 000

Honorine recently got her driver’s license and bought a car that is said to get 30 miles per gallon. However, the system in her car often says that she gets fewer miles per gallon. She notices that the miles per gallon changes with her driving speed, and she wants to use the data cycle to investigate this further. a

Formulate a question which could be investigated using a scatterplot.

Describe a method which could be used to collect data.

Honorine and her brother go for a drive, and they collect the following data: Driving speed (miles per hour) Fuel efficiency (miles per gallon)

20 19

23 20

25 22

28 25

31 26

34 24

36 27

39 27

42 28

45 29

Driving speed (miles per hour) Fuel efficiency (miles per gallon)

47 30

50 29

53 31

56 30

59 30

61 30

64 30

67 28

70 29

72 26

Driving speed (miles per hour) Fuel efficiency (miles per gallon)

75 23

78 25

81 24

83 21

86 23

87 17

88 16

91 14

94 14

95 11

Organize the data using a scatterplot.

Determine an appropriate equation to model the data set.

Use the data to answer your formulated question from part (a).

Honorine concludes that if she drives 107 miles per hour, the car will get no miles per gallon. Explain why this conclusion is not valid.

The use of technology has changed in various ways over the past few decades. For example, the number of people who use a speciﬁc online company (such as Google or Amazon) or a certain application (such as Facebook or Spotify) has likely increased since the beginning of the company. Go through the whole data cycle at least once to investigate how a speciﬁc online company’s usage, revenue, or capability has changed over time.

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7.04 Univariate data, center, and spread After this lesson, you will be able to... • write investigative questions that require univariate data. • collect univariate data through research, surveys, observations, scientific experiments, polls, or questionnaires. • describe and interpret measures of center and spread of a data set. • calculate and interpret standard deviation and variance using technology. • compare the center and spread of multiple data sets.

Measures of center We can summarize data in many ways including using descriptive statistics like mean, median, mode or with data displays like histograms or boxplots. The way we summarize data can depend on the type of data. In this lesson, we will look at summarizing and formulating questions for univariate data. Univariate data Information gathered around a single characteristic. This data can be numerical or categorical. Displays include: pictographs, bar graphs, line graphs, line plots/dot plots, stem-and-leaf plots, circle graphs, and histograms. Example: Scores on assessments, time spent looking at social media, hours spent on an activity

Number of students

This histogram displays numerical data of the heights of students in a class. Notice that there is only one characteristic (or attribute), height, that is being explored. The axes are the attribute and the frequencies. We can only compare the heights of different groups/bins within the data set.

10 8 6

The ﬁrst bin is students who have heights in the interval 135− < 140 cm, the second bin is students who are 140− < 145 cm tall.

4 2 135 140 145 150 155 160 165 Height (cm)

In general, a bin contains the lower value but not the upper value.

Previously, we have seen these measures of center: Mean

Median

The point on a number line where the data distribution is balanced.

The middle value of a data set in ranked order.

This is a measure of center and summarizes the whole data set with a single number.

Mode

Example: mean

578

and μ can be used to represent the

Mathspace Virginia SOL Algebra 2 mathspace.co

This is a measure of center and summarizes the whole data set with a single number.

The piece of data that occurs most frequently. This is a measure of center and summarizes the whole data set with a single number.

Sometimes one measure may better represent the data than another. When deciding which to use we need to ask ourselves “Which measure would best represent the type of data we have?”

Exploration This histogram summarizes numerical univariate data. 6 5 4 3 2 1 0

200

400

600

800

1000

We can say the modal class is 100− < 200. What do you think that means?

One data value is 842, what could we call that value?

One of the measures of center is 199. Which measure of center would this be? Explain.

One of the measures of center is 114. Which measure of center would this be? Explain.

Depending on what the distribution looks like when graphed using a histogram, the measures of center locations can vary. • Mean and median can be the same and in the same bin as the mode 3 2.5 2 1.5 1 0.5 0

200

400

600

Mean Min Q1 Median Q3 Max

310 6 113 310 506 615

Mean and median

• Mean and median can be similar and both in the same bin as the mode 4 3.5 3 2.5 2 1.5 1 0.5 0

Mean Min Q1 Median Q3 Max 0

200

400

600

140.4444 15 306 411.5 513 812

800

Mean and median

• Mean and median can be quite different, but in the same bin as the mode. 6 5 4 3 2 1 0

0 200 Median Mean

400

600

800

Mean Min Q1 Median Q3 Max

198.7778 5 105 114 307 812

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• Mean, median, and mode can all be in different bins 7 6 5 4 3 2 1 0

Mean Median Mode

200

400 600 800 Mean Median

Beneﬁts Includes all of the data in the calculation, widely used Tells us the middle, not impacted by extreme values Quick to identify, tells us about the most frequent value(s)

Mean Min Q1 Median Q3 Max

537.5 5 406 609 712 812

Drawbacks Heavily impacted by extreme values or uneven distributions Does not include all data values Does not include all data value, is not necessarily in the middle

Example 1 The salaries of part-time employees at a company are given in the dot plot, rounded to the nearest thousand. Salaries (in $1000’s)

18 19 20 21 22 23 24 25 30 35 36 37 38

a Which measure of center best reﬂects the typical wage of a part-time employee?

Create a strategy Choose the measure that is appropriate for data sets with extreme values.

Apply the idea The median is the best measure of center that reﬂects the typical wage of a part-time employee due to presence of the three extreme values in the data set.

Reflect and check The mean is 22.9 which is not a very good estimation of the typical salary. It is pulled up by the three much larger values. This might be used to manipulate the analysis to make it look like employees are getting paid more generously than they actually are, so is not a good measure of center. The mode is 18 which is definitely not a good measure of the typical salary. It represents the minimum salary. This might be used to justify that employees need a raise as it is much lower than the typical salary. This would not accurately reflect the employee salaries.

580

Mathspace Virginia SOL Algebra 2 mathspace.co

b Calculate and interpret the chosen measure of center from part (a).

Create a strategy Remember that the median is the middle value of a data set ranked in order. Start by arranging the salaries in order from lowest to highest. If the total number of salaries is odd, then the median is the middle value. If the total number is even, then the median is the average of the two middle values.

Apply the idea 18, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 37, 38, 38 There total number is even and the two middle values are both 21. This indicates that our median for the data set is 21. The typical wage of a part-time employee at the given company is $21 000.00.

Reﬂect and check We can compare this to a mean of $22 850 and a mode of $18 000.

Example 2 A journalist wanted to report on road speed cameras being used as revenue raisers. She obtained data that showed the number of times 20 speed cameras issued a ﬁne to motorists in one month. The results were: 101, 102, 115, 115, 121, 124, 127, 128, 130, 130, 143, 143, 146, 162, 162, 163, 178, 183, 194, 977 a Determine the mean number of times a speed camera issued a ﬁne in that month. Give your answer rounded to one decimal place.

Create a strategy To ﬁnd the mean, use the formula: Mean =

Apply the idea Add all the number of times a speed camera issued a ﬁne and divide by the total number of cameras: Find the sum of the values

Divide and round your answer

b Determine the median number of times a speed camera issued a ﬁne in that month. Give your answer rounded to one decimal place.

Create a strategy The median in a data set with an even number of values is the average of the two middle data values.

Apply the idea First half of the set: 101, 102, 115, 115, 121, 124, 127, 128, 130 Second half of the set: 143, 146, 162, 162, 163, 178, 183, 194, 977

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Measures of spread and dispersion Previously, we have seen these measures of dispersion (spread): Range

Interquartile range

The difference between the upper extreme and the lower extreme.

The difference between the upper quartile and the lower quartile.

This is a measure of dispersion or spread and summarizes the variation in the data set. It can be heavily impacted by extreme values.

This is a measure of dispersion or spread and summarizes the variation in the data set. It is not usually impacted by extreme values.

There are two more measures of spread or dispersion called the variance and the standard deviation. Let’s explore why they are helpful.

Exploration Two sets of data were collected from two different samples of passengers in an airport. Passengers were asked the approximate duration of the ﬂight they had just been on. Set A Set B

1 1

2 1

3 1

4 1

5 5

6 9

7 9

8 9

9 9

10 9

11 9

12 9

13 13

14 17

15 17

16 17

17 17

Find the mean of both data sets. What does it tell us about the data?

Find the median of both data sets. What does it tell us about the data?

Find the range of both data sets. What does it tell us about the data?

Find the interquartile range of both data sets. What does it tell us about the data?

How would you describe the differences between the two sets in a way that the given summary statistics don’t show?

The variance is a way of showing how spread out numbers in a data set are. The variance looks at the square of the distances between each data value from the mean. Consider the following data set with a mean of 73.7: 100, 51, 79, 57, 60, 64, 95, 98, 56, 77 100 90 80 70 60 50 40 30 20 10

We can visualize the variance of the given data set by looking at the distance between the data value, shown on the y-axis and the mean 73.7, shown as a horizontal line.

Notice that some of these distances are positive and some are negative. To avoid the positive and negative distances ‘canceling’ each other out, we square the distances to make them all positive. Then we ﬁnd the average of these squared distances. This is the variance. x 1 2 3 4 5 6 7 8 9 10

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• Mean = 5, Median = 5, Standard deviation = 0.93 8 7 6 5 4 3 2 1 0

• Mean = 5, Median = 5, Standard deviation = 0 7 6 5 4 3 2 1 0

2 3 4 5 6 7 8 9 10

When comparing the standard deviation, we should consider the scale or order of magnitude of the data. For example, the standard deviation for human baby weights might be smaller than for whale baby weights, but that does not necessarily mean that human baby weights are more consistent.

Example 3 The number of push-ups Mario does each day is shown. 33, 32, 32, 32, 31, 32, 32, 32, 32, 32 a Calculate the variance by hand using a spreadsheet or table. Round your answer to two decimal places.

Create a strategy Use the formula No. of push-ups (x) 33 32 32 32 31 32 32 32 32 32

Apply the idea

(x − μ) ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚

(x − μ)2 ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚ ⬚

First, calculate the population mean, μ. Then, use a table like the one shown to ﬁnd the sum of (x − μ)2. Finally, divide the sum by the population size and ﬁnd the square root of the result.

1. Calculate the population mean: Use the formula for mean

Evaluate

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2. Complete the table and ﬁnd the sum of all (x − μ)2: No. of runs (x) 33 32 32 32 31 32 32 32 32 32

(x − μ) 1 0 0 0 −1 0 0 0 0 0

(x − μ)2 1 0 0 0 1 0 0 0 0 0

(x1 − μ)2 + (x2 − μ)2 + … + (xN − μ)2 = 1 + 0 + 0 + 0 + 1 + 0 + 0 + 0 + 0 + 0 =2 3. Divide by N = 10 to ﬁnd the variance. Write the formula

Substitute known values

Evaluate

The population variance is approximately σ = 0.2.

Reﬂect and check The variance is quite small, so indicates that there is not much variation or dispersion in the data set.

b Calculate his standard deviation by hand using a spreadsheet or table. Round your answer to two decimal places.

Create a strategy We have already calculated the variance, so to ﬁnd the standard deviation, we just need to take the square root.

Apply the idea Find the square root of the variance:

Use the answer from the previous part

Evaluate

The population standard deviation is approximately σ = 0.45.

c Use technology to ﬁnd his standard deviation, rounded to two decimal places.

Create a strategy Use the population standard deviation function, σ on your calculator.

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Apply the idea Using Statistics mode, enter each data point into your calculator, then choose One Variable Analysis.

Calculate the statistics by choosing the button with the Σx symbol. Then, look for the population standard deviation, σ.

σ ≈ 0.45

Reﬂect and check Notice, there was also the sample standard deviation s = 0.4714 which is close to σ = 0.4472, so would allow us to describe the spread, but not give the same precise value.

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Example 4 The given data sets show the time to get to school for 20 students at two different school, rounded to the nearest minute. Some summary statistics are given. School A 3 14

4 15

5 15

8 18

9 19

11 27

11 33

12 42

13 45

18 23

18 24

19 24

School B 15 19

15 22

17 22

17 23

18 23

School A 17.4 13.5 142.14

Mean Median Variance

School B 19.9 19 9.09

a Interpret and compare the means for both schools.

Create a strategy

Apply the idea

The mean is also called the average and describes a typical value that balances all of the data points. Check the difference between the means provided in the summary.

We can see from the given summary that School A has a lower mean of 17.4 compared to School B that has a mean of 19.9. This shows that the typical travel time for students going to School A is shorter by 2.5 minutes compared to students going to School B.

b Interpret and compare the medians for both schools.

Create a strategy Median is the middle value of a data set, which is not impacted by extreme values. Check the difference between the medians provided in the summary.

Apply the idea

Reﬂect and check

We can see from the given summary that School A has a lower median of 13.5 compared to School B that has a median of 19. This shows that the typical travel time for students going to School A is shorter by 5.5 minutes compared to students going to School B.

The two measures of center tell the same story that School B students have a longer journey to school than School A students. However, the median makes this conclusion more obvious because the difference is larger.

c Calculate the standard deviations for both schools.

Create a strategy We were given the variance, so just need to use that the standard deviation is the square root of the variance.

Apply the idea So we can square root to get:

We were given that: Variance

School A 142.14

School B 9.09

Variance Standard deviation

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School A 142.14

School B 9.09

Reﬂect and check The units for the standard deviation are the same as the original context, so we can say that the standard deviation for School A is 11.92 minutes and the standard deviation for School B is 3.01 minutes.

d Interpret and compare the standard deviations for both schools.

Create a strategy Observe the difference between the standard deviations for Schools A and B. Remember that the standard deviation quantiﬁes how spread out the values in the data set are relative to the mean value.

Apply the idea The standard deviation for school A is 11.92 while it is 3.01 for school B. This suggests that the time students take to get to school A have greater variability or the values are more spread out, compared to school B. On average, there is 11.92 minutes of difference from the mean for school A, indicating higher variability. On the other hand, there is less variability or the values are closer to each other for school B, with an average difference from the mean of 3.01 minutes.

Reﬂect and check The higher standard deviation or variability for students traveling to school A suggests that they may have encountered heavier traﬃc conditions, frequent road closures, or other factors that contribute in increased travel time.

Idea summary We can describe univariate data using measures of dispersion (spread). Measures of dispersion (spread) are a single data value that describes how varied a data set is. • • • •

Range: the difference between the upper extreme and the lower extreme. Interquartile range: the difference between the upper quartile and the lower quartile. Variance: A measure of the spread of a data set. the mean of the squares of the differences between each element and the mean of the data set. Standard deviation: A measure of the spread of a data set. The square root of the mean of the squares of the differences between each element and the mean of the data set or the square root of the variance.

Range Interquartile range Standard deviation

Beneﬁts Easy to calculate, tells about the extremes of the data Tells us about the middle half of the data, not impacted by extreme values Tells us about how far values are from the mean, widely used in other areas of statistics

Drawbacks Heavily impacted by extreme values Does not include all data values Impacted by extreme values, best to use technology to calculate

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Example 5 Determine the type of data that needs to be collected for each statistical question. a How many times have students in the school been to Washington, DC?

Create a strategy For univariate data, there is one variable of interest in the study, for bivariate data we are looking for a relationship between two variables.

Apply the idea This statistical question is looking at numerical data because it is asking for a count of how many times students have visited Washington, DC. It’s not directly comparing two variables, so it’s not bivariate.

Reﬂect and check We would be collecting discrete numerical data.

b Can we accurately predict a dog’s adult size based on their birth weight?

Create a strategy For univariate data, there is one variable of interest in the study, for bivariate data we are looking for a relationship between two variables.

Apply the idea This statistical question is bivariate because it involves analyzing the relationship between two variables: a dog’s birth weight and its adult size. It aims to determine whether there’s a predictive relationship between these two variables.

c How long do people take to run 3 mile races? Is this comparable for different age groups?

Create a strategy For univariate data, there is one variable of interest in the study, for bivariate data we are looking for a relationship between two variables.

Apply the idea This statistical question requires the collection of numerical data and compares the data over categorical variables. We can say that we are comparing sets of univariate data across different categories.

Reﬂect and check We may also say this is bivariate data, because it is looking at the relationship between two variables: the time it takes people to run a 3-mile race and their age groups. It seeks to determine whether there are differences in race times across different age groups, thus involving the comparison of two variables.

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Example 6 Atanasio’s basketball coach just had a knee replacement. Now he is interested in wait times for joint replacement surgeries. a Formulate a question that could be used to explore the scenario.

Create a strategy To formulate a question about numerical univariate data, we need consider what variable we want to explore. In this case, we want to explore the wait times for joint replacement surgeries.

Apply the idea A sample question would be “What are the waiting times for joint replacement surgeries over the past year at the local hospital?”

b Formulate a question that requires the use of a measure of center to explore the scenario.

Create a strategy To formulate a question that requires a measure of center, we need to identify which measure of center would best represent the needed data.

Apply the idea Waiting times for joint replacement surgeries may vary greatly. We should consider using the measure of center that is less impacted by extreme values, which is the median A sample question would be “What is the median wait time for joint replacement surgeries across the hospitals in our region?”

Reﬂect and check Using the mean might not be ideal for this scenario because it is more sensitive to extreme values compared to the median. If there are a few unusually long waiting times, the mean will be skewed towards these values, giving a distorted representation of the typical waiting time. The mode might not provide meaningful insight into the waiting times, because for the context of waiting times for joint replacement surgeries, it’s less likely to have repeated exact waiting times due to the variability in individual cases.

c Formulate a question that requires the use of a measure of dispersion to explore the scenario.

Create a strategy To formulate a question that requires a measure of dispersion, we need to identify which measure would best for the scenario.

Apply the idea A sample question would be “How do wait times for joint replacement surgeries at the local hospital vary?”

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Reﬂect and check We could use any measure of spread or dispersion to answer this question. The interquartile range would tell us about the middle half of the data, so would deemphasize those who got in very quickly or had to wait a very long time. Standard deviation would give a good idea of the overall variation from the average, but might be harder for those without a statistical background to understand. Using range would be straightforward and provide us a clear understanding of the spread of data by simply showing the difference between the highest and lowest values.

Example 7 Diego loves attending amusement parks, but does not like waiting in lines. This leads him to ask the question: “How long do people wait to ride the newest roller coaster at Busch Gardens Williamsburg?” a Determine which method would be the most appropriate and explain why.

Create a strategy Choose a method that is realistic, ethical and would match the question.

Apply the idea Acquiring secondary data would be the best option to know how long people wait to ride the newest roller coaster at Busch Gardens Williamsburg. Many amusement parks utilize mobile apps or online platforms to provide real-time information about ride wait times. This is not only a practical approach, but would also provide wide and accurate data easily. This method respects visitors’ privacy as participation is entirely optional.

Reﬂect and check Observation can be time-consuming and labor-intensive, especially for a popular attraction like a new roller coaster. It may not be feasible to continuously observe and record waiting times over an extended period. Survey can be challenging in a dynamic environment like an amusement park. Visitors may be unwilling to participate. Scientiﬁc experiment is clearly not applicable as it requires controlling the environment.

b Explain how a sample could be selected to get unbiased data.

Create a strategy When we select a sample, we need to make sure that it is representative of the population.

Apply the idea We can select a sample to get unbiased data by making sure the sample is randomly selected, is big enough, and has the same characteristics as the population. If we were doing an observation or survey, we would need to ensure that we were selecting people at a variety of times throughout the day. Like asking every 10th person who gets off the ride about how long they had to wait or tracking the wait time for every 10th person using observation.

Reﬂect and check When acquiring secondary data, we don’t usually have much control over the sample as demographics are not necessarily collected.

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c Explain what the standard deviation of the data set might tell us.

Create a strategy Remember that standard deviation is the measure of how far individual values in the data set are from the mean.

Apply the idea In the scenario of how long people wait to ride the newest roller coaster at Busch Gardens Williamsburg, the standard deviation tells us the variability or how dispersed the different waiting times are from the mean or average. A higher standard deviation would mean that there are waiting times that are spread out and vary a lot throughout the day or for different seats on the ride, and a lower standard deviation tells us that the waiting times are quite consistent.

d Explain what the median of the data might tell us.

Create a strategy Remember that the median is the middle value of the data set that is arranged in order.

Apply the idea In the scenario of how long people wait to ride the newest roller coaster at Busch Gardens Williamsburg, the median would tell us the usual waiting time without being affected by extremes or waiting times that are way longer or shorter than the usual.

Idea summary We can formulate questions and then collect continuous numerical data to explore univariate data with large data sets. Univariate data will have one variable or attribute collected for each member of the sample or population. A very large data set can help provide a reasonable approximation for the population. However, we need to make sure that the data is selected from a representative sample of the population that reﬂects. We can collect the data using surveys, experiments, observation, or secondary sources.

Practice What do you remember? 1

Fill in the blanks with the terms from this list: • univariate a b c d

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• categories

• numerical

Univariate data can be categorical or ⬚ data.

Histograms can be used to represent numerical ⬚ data. Univariate means one ⬚.

We can compare two sets of univariate data across different ⬚.

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• variable

Is each statement true or false? a

Two data sets can have the same mean, but different medians.

The median is always the best measure of center.

All three measures of center can be the same.

Sometimes there is no mode.

The mean is the best measure when there are extreme values.

Are the statements always, sometimes, or never true? a

If all values in a data set were increased by the same amount, the standard deviation would also increase by that amount.

If the range of a data set is large, the standard deviation will be large.

If an extreme value is added to a data set, it will not affect the standard deviation.

The standard deviation of a data set will always be positive.

The variance is larger than the standard deviation

For each question: i

Determine if it is a statistical question

Determine if it would require univariate or bivariate data to answer.

What is your blood type?

How much sleep do American teenagers get per week?

Is there a relationship between amount of food waste and age?

Is my income related to my closest size?

How fast do car drive on the freeway? Does it vary by car model?

Determine which data collection method would best be used to collect data for each of these scenarios. a

A sample of students at a university were randomly asked about their sleeping schedules and year level to see if the amount of sleep varied by year.

The seniors at a nursing home were split into two groups. Each was given a different amount of outdoor, exercise time in order to see if there’s a relationship between happiness and time spent exercising.

A researcher watched peoples’ orders and estimated if they were children, adults, or seniors in a cafe in order to determine if certain age groups order certain items more often.

For the box plot shown, ﬁnd each of the following: a

Median

Range

Interquartile range

Score

10 12 14 16 18 20

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For each data set, use technology to: i

Find the variance of each set of scores, rounded to four decimal places.

Find the standard deviation of each set of scores, rounded to two decimal places.

8, 20, 16, 9, 9, 15, 5, 17, 19, 6

− 17, 2, −6, 9, −17, −9, 3, 8, 5

81, 90, 90, 88, 73, 80, 86, 87, 75, 82, 70, 81, 71, 81, 79, 81, 80, 86, 88, 79

20, 44, 27, 25, 21, 28, 41, 24, 27, 39, 35, 43, 30, 17, 40

3, 14, 11, 17, 3, 18, 15, 6, 17, 15

Meteorologists predicted a huge variation in temperatures throughout the month of April. The temperature each day for the ﬁrst two weeks of April were recorded as follows: 61, 64, 69, 70, 70, 70, 71, 72, 72, 75, 75, 77, 79, 81

State the range of the temperatures.

Calculate the interquartile range of the temperatures.

Given that the population standard deviation is 5.2, determine whether the standard deviation or the interquartile range would be the best measure of spread to support or counter the prediction of a huge variation in temperatures.

A barista said that coffees cost less than $4. Data was recorded on how much people spend on each of their drinks: $1.2, $1.2, $1.2, $1.55, $1.55, $1.55, $1.55, $1.95, $1.95, $2.5, $5.25, $5.75, $6.2, $6.75, $6.75, $7.3, $7.3, $8.5, $8.75, $9.45

State the mean of the coffee costs.

Calculate the median of the coffee costs.

Determine whether the mean or median would be the best measure of center to support the claim of coffees cost less than $4.

The lives of two brands of batteries are tested using a sample of 10 batteries from each brand. Their battery lives (in hours) are shown below. • Brand X: 23.3, 19.7, 20.7, 25.3, 22.5, 19.1, 20.0, 20.7, 20.7, 20.9 • Brand Y: 23.2, 27.5, 25.0, 24.5, 22.7, 29.8, 22.9, 26.0, 26.4, 22.6

Find the mean for each brand of batteries.

Find the variance for each brand of batteries.

Find the standard deviation for each brand of batteries.

Compare the performance of the two brands of batteries. Be sure to use statistical ﬁndings to justify your reasoning.

Marge grows two different types of bean plants. She records the number of beans that she picks from each plant for 10 days. Her recordings are shown below: • Plant A: 10, 4, 4, 5, 7, 10, 3, 3, 9, 10 • Plant B: 8, 7, 5, 5, 9, 7, 8, 7, 5, 6 a

Determine which plant produces more beans on average. Explain.

Determine which plant has a more consistent yield of beans. Explain.

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Two machines A and B are producing chocolate bars with the following mean and standard deviation for the weight of the bars:

Machine

Mean (g)

A B

52 56

Two friends compete in 100 m sprints and the time to complete 50 sprints were recorded.

Runner

Mean (s)

The mean and standard deviation for the sprints are shown below:

Derek Sarah

13.1 14.5

Explain what the mean of the two machines tells us.

Explain what the standard deviation of the two machines tells us.

Explain what the mean of the two friends tells us.

Explain what the standard deviation of the two friends tells us.

Standard deviation (s) 1.2 0.75

A group of students record the resting heart rates for 30 students and 30 teachers. The results are represented with the histograms. Resting heart rate of teachers

Resting heart rate of teachers

40 45 50 55 60 65 70 75

Students Mean Median Range Interquartile range Standard deviation

598

Standard deviation (g) 1.5 0.65

40 45 50 55 60 65 70 75

Teachers 55.7 56 21 6 4.3

Mean Median Range Interquartile range Standard deviation

58.4 58 20 7 5.0

Compare the spread (dispersion) of the student and teacher data sets. Explain the what this means in the context.

Compare the centers of the student and teacher data sets. Explain what this means in the context.

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Let’s extend our thinking 20

Consider the test results for four Geography classes, labeled as Class W, Class X, Class Y, and Class Z:

Class W

10 9 8 7 6 5 4 3 2 1 0

Class Y

10 9 8 7 6 5 4 3 2 1 0

0 2 4 6 8 10

10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0

Class X

0 2 4 6 8 10

Class Z

0 2 4 6 8 10

Rank the classes from largest standard deviation to smallest standard deviation. Explain your reasoning. 21

A school does an analysis of the number of volunteer hours that students have completed in April of their Senior year for two different years.

Mean Median Range IQR Standard deviation Lower extreme

Formulate a question that could be answered with this data.

Which statistics support the claim that the class of 2022-2023 was more consistent with the number of hours they have?

Which statistics support the claim that the class of 2022-2023 was less consistent with the number of hours they have?

Draw a valid conclusion to answer your question. Justify your answer with data.

20222023 57.95 60 58 14 10.5 27

20232024 52.36 50 228 7 18.6 36

A travel agency wants to survey families that travel with kids under 18 to see what preferences they have when choosing destinations, hotels, transportation, etc. They gather data on the families, including the number of kids they have, and ﬁnd the standard deviation in the number of kids to be 0.1. Why might this be undesirable for the travel agency?

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The histogram and statistics for a data set are shown. n Mean σ s Σx Σx2 Min Q1 Median Q3 Max

25 22.5 10.4019 10.6164 562.5 15 361.25 2 14.5 22.5 30.5 43

4 3 2 1 0

0 5 10 15 20 25 30 35 40 45 50

What is the range of scores that fall within 1 standard deviation of the mean?

Approximately what percentage of scores lie within 1 standard deviation of the mean?

What is the range of scores that fall within 2 standard deviations of the mean?

Approximately what percentage of scores lie within 2 standard deviations of the mean?

Rodrigo collects trading cards. At a Sports Card Expo, he sells his most valuable card and is now curious what other peoples’ most valuable cards and collections are worth. a

Formulate a statistical question that Rodrigo could use to explore the value of trading cards and collections of other collectors.

Could he use observation, survey, experiment, or acquire secondary sources? Explain.

Explain how Rodrigo could collect data that could be used to answer his question from part (a).

Suppose the given data shows the value of the baseball and basketball collections of those in his collector’s club. Analyze the data to draw a conclusion for Rodrigo. Baseball Value in USD

$15

$29

$107

$228

$250

$1439

$1937

$2701

$160 316

Basketball Value in USD e 25

$18

$156

$304

$450

$557

$601

$892

$965

$1000

Formulate another statistical question that could be used to explore trading cards.

Quinn recently was told he needed to drink more water. He is curious how much water people drink and how it varies for different age groups or in different seasons. Go through the whole data cycle at least once using a context that involves water consumption.

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7.05 Histograms with smooth curves After this lesson, you will be able to... • represent univariate data in histograms. • sketch a smooth curve on a histogram. • describe the center, spread, and shape of a smooth curve. • compare the center, spread, and shape of multiple data sets.

Frequency

Histograms and smooth curves 8 7 6 5 4 3 2 1 0

Baby seal weights

We have seen that a histogram can be used to represent a set of univariate numerical data. Remember that a histogram is a data display that divides the data into bins or intervals and shows the frequency of data points within each bin with the height of each bar.

30 35 40 45 50 55 60 65 70 Weight of baby seals (lbs)

Frequency

8 7 6 5 4 3 2 1 0

Baby seal weights

Frequency

Sometimes we will see a frequency polygon on a histogram which is a line graph that follows the shape of the histogram using either the left corners, centers, or right corners of the bars.

30 35 40 45 50 55 60 65 70

8 7 6 5 4 3 2 1 0

Baby seal weights

30 35 40 45 50 55 60 65 70

Weight of baby seals (lbs)

Center of bars

Left corner of bars

Baby seal weights

30 35 40 45 50 55 60 65 70 Weight of baby seals (lbs)

Right corner of bars

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Notice that the pieces of the bars that are above the frequency polygon, could be used to ﬁll in the unﬁlled areas below the frequency polygon.

Percentage

To give us a better understanding of the overall shape of the data, we can draw a smooth curve over the histogram. This curve is sometimes called a density curve and shows where values are concentrated. The dark line is the smooth curve that can be drawn over the histogram to model the distribution.

30 25 20 15 10 5

Like the frequency polygons, the smooth curves may cross the bars in the center, left, or right. 75 80 85 90 95 100 105 110

A distribution shows the shape and how a data set is spread out or clustered within the range.

Weight

We have formal ways to describe the shape:

Frequency

This data distribution is symmetrical, or bell-shaped. It has no skew.

Score

Frequency

This data distribution shows a positive skew. Notice the “tail” at the positive end as it trails off. This is sometimes called a right skew.

Score

Frequency

This data distribution shows a negative skew. Notice the “tail” at the negative end as it trails off. This is sometimes called a left skew.

Score

Based on symmetry or skew of the distribution, we can make observations about the measures of center - mean, median, and mode.

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Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 7.05 to answer these questions. 1.

What do you notice about the mean, median and mode when the data is symmetrical?

What do you notice about the mean, median and mode when the data is positively skewed?

What do you notice about the mean, median and mode when the data is negatively skewed?

Mean median mode

For symmetrical or non-skewed data: • Roughly 50% of scores will be above the mean and 50% of scores will be below the mean. • The mean, median and mode should roughly coincide.

Mode

For a positive or right skew: • The data frequencies are much lower as you move to the right. • The mean will get pulled up by the skew, so is usually greater than the median. • The median is usually greater than the mode.

Median

Mean

Median

Mean

Mode

For a negative or left skew: • The data frequencies are much lower as you move to the left. • The mean is pulled down by the skew, so is usually less than the median. • The median is usually less than the mode.

The range is a helpful measure of spread when analyzing data distributions. In later lessons, we will look at how standard deviation can be read from a symmetrical smooth curve.

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Example 1 The given histogram represents the distribution of hours students sleep each night. Select the smooth curve that most accurately models this distribution. 200

Frequency

150 100 50 0

3 4 5 Sleep duration (hours)

200

Frequency

150 100 50 0

3 4 5 Sleep duration (hours)

6 7 8 Sleep duration (hours)

3 4 5 Sleep duration (hours)

200

Frequency

150 100 50 0

200

Frequency

150 100 50 0

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200

Frequency

150 100 50 0

Sleep duration (hours)

Create a strategy Look for the smooth curve that closely follows the shape and peak of the histogram bars, accurately representing the distribution’s center, spread, and symmetry. The correct curve will typically align well with the measures of center and spread of the data displayed in the histogram.

Apply the idea This histogram shows a negative (left) skew and mode around 6 with a frequency of 175, so we are looking for a smooth curve with the same key features. Option A: This smooth curve has a negative (left) skew, but the peak is around 5.5 with a frequency of 125, so this option is not correct. Option B: The smooth curve has a peak is around 6, but it has a frequency of 125 and is symmetric in shape, so this option is not correct. Option C: This smooth curve has a negative (left) skew and shows a peak around 6 with frequency close to 170, so this curve is correct. Option D: This is not a smooth curve, just a frequency polygon, so it is jagged. This is not the correct option.

Reﬂect and check Depending on whether we use the right corner, left corner, of middle of the bars, there can be multiple correct smooth curves.

Example 2

Frequency

The given smooth curve was created from data collected on the statistical question “What weekly pay is typical for a job while in high school?”

0 50 100 150 200 250 300 350 400 Weekly pay ($)

a Describe and interpret the shape of the distribution.

Create a strategy Is there a tail that slowly tapers off? If so, which side it is on? This will tell us the shape.

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Apply the idea There is a tail on the right side where the larger values are, so the shape is a positive (right) skew. This means that the majority of students earn a weekly play on the lower end (around $150 per week), but a few earn a high amount (around $350 per week).

b Estimate and interpret an appropriate measure of center.

Create a strategy For a symmetric distribution the mean, median, and mode occur right at the peak. Since this distribution has a positive (right) skew, the median will be pulled up (to the right) a little from the mode, and the mean will be pulled up even more from the mode.

Apply the idea

Frequency

Mode Median Mean

100

150

200

250

300

350

400

Weekly pay ($)

The most appropriate measure of center for this data is the median because the mean is more heavily impacted by the skew and the mode is not at all impacted by the skew. The median will allow us to take the higher data values into account without letting them have too much of an impact. By using the fact that the skew pulls the mean slightly higher than the mode, we can estimate the median weekly pay for high school students to be around $175 per week.

Reﬂect and check The mode would be a suitable measure at around $150 per week. The mean would likely be even higher, possibly closer to $200 per week.

c Estimate and interpret an appropriate measure of spread.

Create a strategy Measures of spread include range, interquartile range, and standard deviation.

Apply the idea Without the raw data, we cannot determine the standard deviation or interquartile range. We can estimate that the maximum pay is about $400 and the minimum pay is about $50 so the range is about 400 − 50 = 350. This means that the weekly pay for high school students varies by $350 from the lowest pay to the highest pay.

Reﬂect and check We will learn how to estimate the standard deviation from a symmetric smooth curve in future lessons.

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Shape The shape of a data set can be determined by looking at the outline of the curve and describes the distribution of data within the set. Identifying the shape of a density curve can help us understand the corresponding data set. Here are some examples of how we describe the shape of density curves: Negatively skewed (Left skewed) Mode Median

Frequency

Positively skewed (Right skewed)

Symmetrical distribution Mean median mode

Mode Median Mean

Mean

mean < median < mode

mean = median = mode

mean > median > mode

Exploration

6 5 4 3 2 1 0

Number of students

Consider the following histograms that show the height of students in two basketball teams. We know that one graph represents a team made up of Grade 12 students and the other represents a Grade 9 team. 10 8 6 4 2 0

60 62 64 66 68 70 72 74 76 Height (in)

60 62 64 66 68 70 Height (in)

Team A

Team B

Are there the same number of students in each team? Does it matter?

What are the similarities and differences in terms of measures of spread, central tendency and shape of data?

Which team do you think corresponds to the Grade 12 team, and which team do you think corresponds to the Grade 9 team?

It is important to be able to compare data sets because it helps us make conclusions or judgements about the data. For example, suppose Jim scores 50% on a geography test and 70% on a history test. Based on those grades alone, it makes sense to say that he did better in history. However, looking at the smooth curves that represent the class results, we can see they tell a different story. Number of students 12 10 History

8 6

Jim

Geography

Jim

2 10 608

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Grade (%) 80

100

Notice the geography class had a mean of 40%, while the history class had a mean of 80%. Now we know that Jim scored well above the average in geography, and well below the average in history. With this extra information, it makes more sense to say that he did better in geography.

Example 3 The following curves show the average math test results for two different classes. Curves 1 and 2 show the results for class 1 and 2 respectively. Curve 1 Curve 2

60 70 80 Test results (%)

100

a State the similarities and differences between the following pair of density curves.

Create a strategy To compare and contrast the two curves, we can look to the shapes, centers, and spreads of each curve.

Apply the idea The shape of curve 1 is skewed right and the shape of curve 2 is skewed left, so the shapes are both skewed, but in opposite directions. The mean of curve 1 will be above the peak due to the skew, so will be around 65%. The mean of curve 2 will be below the peak due to the skew, so will be around 85%. Their centers are quite different. The spread of curve 1 goes from about 40 to 90% and the spread of curve 2 goes from about 55 to 100%. Therefore we can say the spreads are over different percentages, but are about the same size.

b Interpret the test results of class 1 and class 2.

Create a strategy We can use the ﬁndings from part (a) in order to draw conclusions about the test results.

Apply the idea Class 1 has lower test results than class 2 on average and has a large spread of results. Class 2 has a much higher average test score than class 1, but also has a fairly large spread.

c If Anthony scored 60% in Class 1, and Brodie scored 80% in Class 2, who did better?

Create a strategy We can look at how their results compare to their class because the class results are very different which means that one class might have had a much harder test.

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Apply the idea Curve 1 Curve 2

Anthony Brodie

100

Test results (%)

Anthony’s grade is just above the grades of most students in the class, while Brodie’s grade is below the grades of most students in his class. This tells us that Anthony did better when we consider the results of their classmates.

Example 4 The following curves show the distributions of the race times for two different years of the Shelby Forest Loop Marathon. 2020

2021 Time (hr) 3

Describe the similarities and differences between the following pair of smooth curves.

Create a strategy We can look at the shape, measure of center or clusters, and the range as a measure of spread.

Apply the idea Both density curves have a signiﬁcant right skew. However, 2021 has another small peak around 9 hours, while 2020 tapers off smoothly. 2021 is slightly more skewed than 2020. Their centers are also fairly similar with a mean at around 5 hours. Both means are to the right of the peak. They are pulled there by the right skew of the data. The spread of 2020 is slightly lower than 2021 as there were no extreme data values with times greater than 8 hours, unlike in 2021. The range is about 8 for 2020 and about 10 for 2021, so slightly less consistent in 2021.

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Reﬂect and check Looking at the corresponding histograms and frequency polygons, we can see that the 2021 race has an extreme value which leads to the secondary bump. With real-life data, it is often not perfectly smooth. 2020 Race Results

16 14

Frequency

12 10 8 6 4 2 0

Time (hr)

2021 Race Results

16 14

Frequency

12 10 8 6 4 2 0

Time (hr)

Idea summary We can ﬁrst identify and then compare the measures of center, measures of spread, and shape of smooth curves and histograms. The context of the curves is important to consider when interpreting the comparisons.

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Practice What do you remember? 1

In your own words, describe how to identify the following from smooth curves: a

Shape

Center

Spread

Identify if the graph is positively (right) skewed, negatively (left) skewed, or symmetrical. a

Frequency

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

4 5 Score

Score

30 25 Frequency

Frequency

20 15 10 5 0

612

4 5 Score

5 6 7 8 9 10 11 12 Score

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Frequency

8 9 10 11 12 13 14 15 16 17 Score

Which line is closest to the: a

Mode?

Mean?

Median? A

B C

0 0

Fumiko formulated the question “What is a typical monthly electricity bill in my area?” She collected the data in the table from a sample of 30 households in her area. 88.35 157.17 173.22 a

115.62 160.07 176.36

116.87 160.89 177.73

Average monthly electricity bill ($) 128.40 132.64 136.28 144.64 161.05 162.29 162.60 163.60 180.77 183.15 183.83 185.95

Create a histogram of the given data set.

147.60 165.83 186.39

154.27 168.62 192.48

Describe the shape of the data set.

10 9 8 7 6 5 4 3 2 1 0

154.57 170.74 220.40

Match each histogram with its smooth curve. a

10 9 8 7 6 5 4 3 2 1 0

2 3 4 5 6 7 8

10 9 8 7 6 5 4 3 2 1 0

2 3 4 5 6 7 8

10 9 8 7 6 5 4 3 2 1 0

2 3 4 5 6 7 8

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Let’s practice For each of the given histograms: i

Sketch a smooth curve.

Describe the shape, center, and spread.

a Number of students

80 60 40 20 0

20 25 30 35 40 45 50 55 60 65 Time (seconds)

Number of days

50 40 30 20 10 0

12 14 16 18 20 22 24 26 28 30 Temperature (°F)

Number of meals

100 75 50 25 0

30 40 50 60 70 80 90 100 110 120 130 Cost of meal ($)

80 Number of flights

60 40 20 0

950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 Cost of flight to Paris, FR ($)

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For each data set: i

Create a histogram with an appropriate bin size, with or without technology.

Draw a smooth curve on the histogram. Iron levels (micromol/L)

a 17.9 35.3

19.9 35.9

22.5 38.3

23 39.7

29.4 42.7

31.2 44.7

31.9 44.8

14 42

24 44

26 45

27 48

31 52

34.9 56.3

33 56

34 67

35 77

37 83

0.78 0.97

0.81 1.02

39 91

Magnesium levels (mg/dL)

c 0.43 0.83

0.54 0.84

0.61 0.85

0.65 0.86

0.67 0.88

0.71 0.89

0.72 0.92

0.75 0.94

Describe the shape, center, and spread of each data display. a

Frequency

30 20 10 0

18 20 22 24 26 Age of babysitter (years)

50 Frequency

40 30 20 10 0 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 Time to make a batch of muffins (minutes)

Frequency

616

33.1 55.2

Vitamin D levels (mmol/L)

27.3 40.5

50 45 40 35 30 25 20 15 10 5 0

10 12 14 16 18 20 22 24 26 28 30 Hourly pay ($)

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Frequency

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Reaction time (seconds)

0.8

0.9

The height distributions of a random sample of adult females is given. a

Identify and interpret the mean of the distribution.

Identify and interpret the range of the distribution.

Identify and interpret the shape of the distribution.

Compare the mean, median, and mode. Justify your answer.

1.0

200 150 Frequency

100 90 80 70 60 50 40 30 20 10 0

100 50 0 54 56 58 60 62 64 66 68 70 72 Height (in)

Deluca formulates the question “How many messages do teenagers send per day?”. He then collects anonymized data from a telecommunications company and represents it using this smooth curve. Messages sent per day Number of messages

3000 2000 1000 0 65

85 90 95 100 105 Number of teenagers

Identify and interpret a measure of center of the distribution.

Identify and interpret the shape of the distribution.

Compare the mean, median, and mode. Justify your answer.

110

115

120

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Calla formulated the question “How does the distribution of dog ages compare to the distribution of cat ages?” Then she collected data from the local vet and represented it in the table. Number of animals (frequency)

100 90 80 70 60 50 40 30 20 10 0

Cats Dogs

20 25 Age (years)

Compare the shape of the two distributions.

Compare the centers of the two distributions and interpret them in context.

Compare the spread of the two distributions and interpret them in context.

Sergio and Peter both love to cycle. The both track the distances of all their rides for one year and organize them using these smooth curves. Number of rides

80 100 Distance (miles)

120

140

Compare the shape of the two distributions.

Compare the centers of the two distributions and interpret them in context.

Compare the spread of the two distributions and interpret them in context.

160

Number of students (frequency)

Hei formulates the question “How much time do students spend outside per month? Does it vary by age group?” She then collects some data from the middle and high school students in her community and organizes using smooth curves. 100 90 80 70 60 50 40 30 20 10 0

618

Peter Sergio

Middle school High school

30 40 50 60 70 80 Time spent outside per month (hours)

Compare the shape of the two distributions.

Compare the centers of the two distributions and interpret them in context.

Compare the spread of the two distributions and interpret them in context.

Draw at least one conclusion that answers her question.

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Curve 1 shows the times of the swimmers who competed in the ﬁnal of the women’s 50-meter freesyle at the Olympics. Curve 2 shows the times of the top swimmers at the U.S. qualiﬁers for the same event. Curve 1 Curve 2

19 20 21 22 23 24 25 26 27 28 29 30 Time in seconds

State whether the following are true or false:

The swimming times at the U.S. qualiﬁers were faster on average than the times at the Olympics.

The swimming times at the U.S. qualiﬁers were about the same on average as the times at the Olympics.

All of the Olympic ﬁnalists swam faster than the U.S. qualiﬁers.

There is a wider spread in times for the U.S. qualifiers than for the Olympics.

State the similarities and differences between the following two data distributions. Make sure to address shape, center, and spread. Distribution 1

Distribution 2 36

32 28 24 20

0 1 2 3 4 5 6 7 8

16 12 8 4

x 1

The given data display shows the number of books that people read in the past year. a

Estimate the mean, median, and mode for the given data set.

Explain which measure best represents the data.

Analyze the data representation to draw a conclusion. 500 400 Frequency

300 200 100 0 0

40 60 80 100 Number of books in the past year

120

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Let’s extend our thinking 20

The following curves represent the amount of time in seconds is takes two groups of students to complete a multiplication problem.

Curve 1 Curve 2

Compare and contrast the two groups of students, group 1 and 2 respectively, that were used to create curve 1 and 2:

1 2 3 4 5 6 7 8 9 Time in seconds

Describe two possible data sets in detail that can be described by the following curves. Explain your reasoning. Curve 1 Curve 2

3.5

4.5

5.5

height (ft)

Sketch a smooth curve that is described by the following: a

Is skewed left and has a wide spread.

Is symmetric and has a very small spread.

Sketch a smooth curve that is described by the following: a

The mean and median are just about the same.

The mean is less than the median.

The median is less than the mean.

The average salaries for people who studied in the same state as a technology company and those who studied out of state were found and displayed in the following two data displays: Salary (thousands)

200 In state study

620

200 Out of state study

Compare and contrast the two data sets.

Write a conclusion about those who study in state versus those who study out of state.

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These two histograms show the amount of sleep people got on the previous night. Sleep duration for group A

Sleep duration for group B

10 8

6 4

2 0

3 4 5 6 7 8 9 10 Hours of sleep

7 8 9 10 11 12 13 14 15 Hours of sleep

Sketch a smooth curve on the histogram for Group A.

Sketch a smooth curve on the histogram for Group B.

Sketch a histogram and smooth curve for the two groups combined.

Compare and contrast the two groups, and what happens when they are combined.

Describe what groups A and B could possibly represent.

Nellie is attending her ﬁrst live musical production this week and now wants to see more live theater. She is curious about how much it costs to see plays and musical productions. a

Formulate a statistical question that Nellie could use to explore the cost of attending plays and musicals, where a smooth curve or two smooth curves would be an appropriate data display.

Could she use observation, survey, experiment, or acquire secondary sources? Explain.

Explain how Nellie could collect data that could be used to answer her question from part (a).

Suppose the given data shows the minimum cost to attend various plays that are currently running in her area, ranging from children’s theater to professional productions. Analyze the data to draw a conclusion for her. $10 $35 $50

$41 $65 $77 e

$10 $37 $55

$45 $65 $77

$20 $38 $55

$47 $65 $80

$20 $41 $58

Plays $25 $30 $42 $42 $58 $60

$30 $42 $64

$32 $48 $64

$32 $49 $65

$35 $49 $75

$53 $65 $80

Musicals $53 $56 $70 $70 $80 $81

$57 $71 $83

$61 $72 $86

$61 $75 $89

$65 $77 $98

Formulate another statistical question that could be used to explore plays and musical productions.

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7.06 Normal distributions After this lesson, you will be able to... • describe the properties of a normal distribution. • analyze the shape, center, and spread of a set of normally distributed data. • compare the shape, center, and spread of multiple sets of normally distributed data. • solve problems about normally distributed data sets using the Empirical Rule. • solve problems about normally distributed data sets using the mean and standard deviation.

Normal distributions A data set that is symmetric and bell-shaped about the mean is said to have an approximately normal distribution. This shows how a data set that has an approximately normal distribution may appear in a histogram. A smooth, symmetrical curve can be drawn over the histogram that the data roughly follows.

Percentage

30 25 20 15

This curve is called a normal curve. The arithmetic mean (μ) is located on the line of symmetry of the curve and is approximately equivalent to the median and mode of the data set.

10 5 75 80 85 90 95 100105 110 Weight

If a data set is not symmetrical about the mean, we cannot use normal distribution to interpret it. Recall that the standard deviation, denoted by σ, describes the spread of the data. y

µ =0

σ = 0.2

σ = 0.9

x −1

A small standard deviation provides a tight cluster around the mean

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x −1

A larger standard deviation shows data that is more spread out

Exploration Consider the data sets represented by the histograms.

23 27

Histogram 1

18 20 22

Histogram 2

15 20 25 30 35 40

Histogram 3 1.

Histogram 4

Match each of the histograms to the correct mean and standard deviation. • μ = 16, σ = 3 • μ = 15, σ = 2 • μ = 19, σ = 4 • μ = 18, σ = 2

Justify your choices.

Consider this normally distributed data set with a mean of 92.5 pounds and a standard deviation of 5 pounds. The data is centered around the mean weight, and we can use the standard deviation to divide the curve into different sections. 77.5

82.5

87.5

92.5

97.5 102.5 107.5

Percentage

25 20 15 10 5 0

90 95 Weight

100

105

110

If the mean is 92.5, one standard deviation above the mean is 92.5 + 5 = 97.5. One standard deviation below the mean is 92.5 − 5 = 87.5. This means that data values between 87.5 and 92.5 pounds lie within one standard deviation of the mean. Continuing this pattern, we can say that data values between 82.5 and 102.5 pounds lie within two standard deviations of the mean, and data values between 77.5 and 107.5 pounds lie within three standard deviations of the mean. 7.06 Normal distributions mathspace.co

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Histogram

Frequency

15 10 5

6 7 8 9 10 11 12 13 14 15 Score

Apply the idea

Reﬂect and check

Most of the data is on the right side of the histogram, so the data is skewed left. Since the data is not symmetric, it does not represent a normal distribution.

Note that if the data was normally distributed, this would need to be converted to a relative frequency histogram before using the Empirical Rule to interpret it.

6 7 8 9 10 11 12 13 14 15 Sample

Apply the idea Most of the data is on the left side, so the data is skewed right. Because the data is not symmetric, it does not represent a normal distribution.

Example 2 The data on daily high temperatures for a certain town is approximately normally distributed. The mean high temperature for this city is 78 °F, and the standard deviation is 6 °F. Daily temperatures 12

Frequency

10 8 6 4 2 0

60.00 64.00 68.00 72.00 76.00 80.00 84.00 88.00 92.00 96.00 Temperature

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Example 3 Consider the normally distributed data sets shown.

Distribution A

Distribution B

a Which data set has a higher mean?

Create a strategy A normal curve is symmetric about the mean. To compare the means of the distributions, we can identify the value that lies on the line of symmetry for each curve, then compare those values.

Apply the idea The mean of Distribution A is 12 since the curve is symmetric about that value.

The mean of Distribution B is 14 since the curve is symmetric about that value. 12

Since 14 > 12, Distribution B has a higher mean value.

Reﬂect and check We do not need to consider the shape of the curve because it is not affected by the mean. The mean only affected the curve’s line of symmetry.

b Which data set has a smaller standard deviation?

Create a strategy The standard deviation affects the spread of a normal curve. To compare the standard deviations of the distributions, we can analyze and compare the spread of each curve.

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Apply the idea

Reﬂect and check

In distribution A, approximately 100% of the data lies between 10 and 14.

Because the normal curve has an area of 1, curves with a smaller standard deviation will be tall and thin. As the variation in the data increases (as the data values spread further from the center), the curve will become shorter and wider.

In distribution B, the curve stretches beyond 12 and 16. The curve is wider and less peaked, showing it has a larger spread. Distribution A has a smaller deviation.

Example 4 The grades on a recent exam are approximately normally distributed with a mean score of 72 and a standard deviation of 4. a Construct a normal curve and label the boundaries for the Empirical Rule.

Create a strategy A normal curve will have a symmetric bell-like appearance with the mean as the central value and divisions for: • Mean ±1 standard deviation • Mean ±2 standard deviations • Mean ±3 standard deviations

Apply the idea Subtract the standard deviations from the mean to ﬁnd the values on the left side of the curve: • 1 standard deviation below the mean: 72 − 4 = 68 • 2 standard deviations below the mean: 72 − 2 (4) = 64 • 3 standard deviations below the mean: 72 − 3 (4) = 60 Add the standard deviations to the mean to ﬁnd the values on the right side of the curve: • 1 standard deviation above the mean: 72 + 4 = 76 • 2 standard deviations above the mean: 72 + 2 (4) = 80 • 3 standard deviations above the mean: 72 + 3 (4) = 84

b Find the percentage of students who scored between 64 and 68 on the exam.

Create a strategy

Apply the idea

To use the Empirical Rule, we must ﬁrst determine how many standard deviations 64 and 68 are away from the mean score of 72.

64 is two standard deviations below the mean and 68 is one standard deviation below the mean.

According to the Empirical Rule, the percentage of data between 1 and 2 standard deviations below the mean is 13.5%.

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c If 32 students took the exam, determine the number of students expected to score 80 or more on the exam.

Create a strategy We ﬁrst need to determine the number of standard deviations 80 is away from the mean score of 72. Then, we can multiply the percentage found from the Empirical Rule by 32 students to determine the number of students who may have scored more than 80.

Apply the idea 80 is two standard deviations above the mean. According to the Empirical Rule, 95% of the data is within 2 standard deviations of the mean. Since all of the data lies below the curve, we know that 1 − 0.95 = 0.05 or 5% of the data lies above and below 2 standard deviations of the mean. We can divide this in half to ﬁnd the percentage of data that is only 2 standard deviations above the mean: = 0.025 or 2.5%. 2.5% of the 32 students are expected to score 80 or more. This gives us 32 (0.025) = 0.8. If the data is approximately normal, not even one student will score above an 80 in a class of 32.

Reﬂect and check We can use the Empirical Rule to check the reasonableness of this solution. According to the rule, 95% of the students will receive scores between 64 and 80 because these are 2 standard deviations from the mean. 32 ⋅ 0.95 = 30.4 Because of the rounding error, there are still 2 students that scored below 64 or above 80 on the test. A conclusion that 1 student scored 80 or above on the test would still be valid.

Example 5 Farrah is a movie buff and dreams of becoming a director. She notices that a lot of movies have similar running times and formulates the question, “How long are the most popular movies today?” She decides to investigate this further using the data cycle. a Describe a method Farrah can use to collect data.

Create a strategy First, use Farrah’s statistical question to determine the type of data that needs to be collected. Then, consider whether the data can be collected by research, a survey, an observation, or a scientiﬁc experiment.

Apply the idea “The most popular movies” is a relative term, but in this context, “most popular” is usually measured by the amount of money made while the movie was showing in theaters. Websites such as Wikipedia or IMDb generally collect and report this data. Farrah can collect data on the length of the movies from the same websites. In this case, Farrah would collect the data through research.

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b The data Farrah gathered on the running time, in minutes, of the top 30 movies is shown: 119, 126, 120, 115, 120, 133, 114, 120, 110, 105, 130, 128, 124, 129, 130, 107, 108, 119, 118, 114, 103, 124, 130, 117, 122, 113, 137, 136, 110, 119 Use technology to create a smooth curve to model the distribution and describe the shape of the curve.

Create a strategy Using technology, we can follow these steps to create a smooth curve of the data: 1. Enter the data into a single column using the GeoGebra Statistics calculator. 2. Highlight the data and select One Variable Analysis. 3. In the settings menu (represented by the gear icon), change the frequency type to Normalized. This will adjust the values on the y-axis to reﬂect a probability distribution. 4. Check the box to show the normal curve. To see the smooth curve on its own, uncheck the histogram box.

Apply the idea After entering the data and selecting One Variable Analysis, a histogram will generate. Select the gear icon to open the settings menu.

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Change the frequency type to normalized. If the frequency type is not normalized, it is not possible to create the smooth curve.

Finally, check the box for normal curve.

The curve is symmetric and bell-shaped, meaning the data is approximately normally distributed.

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Reﬂect and check To see the curve without the histogram, uncheck the histogram box.

c Answer the statistical question that Farrah formulated.

Create a strategy

Apply the idea

Farrah’s statistical question was, “How long are the most popular movies today?” We can answer this using measures of center and spread.

Because the data is normally distributed, the mean, median, and mode are approximately equal. The data distribution is symmetric about 120 minutes, which shows that most movies are 2 hours long. According to the sample data, movies range from 103 minutes to 137 minutes, showing that movie times vary by just over half an hour.

d Formulate a new question that can be answered by the normal curve that approximates the data.

Create a strategy Since the data is normally distributed, the question can be related to the mean and standard deviation of the data or require the use of the Empirical Rule.

Apply the idea

Reﬂect and check

One possible question may be, “95% of movie times are between what two running times?”

Other possible questions may be: • What percent of movies are longer than 2 hours? • How does a 2.5 hour movie compare to the lengths of other movies? • Out of 200 movies, how many are expected to be under an hour and a half long?

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e Use the data to answer the statistical question from part (d).

Create a strategy According to the Empirical Rule, 95% of movies will fall within 2 standard deviations of the mean. We can use technology to ﬁnd the standard deviation of the data, then use the standard deviation and the mean to answer the question.

Apply the idea Using technology, we can ﬁnd the standard deviation (σ ) in the summary statistics. Select the Σx icon to show the summary statistics.

The mean of the data is 120, and the standard deviation is about 9 minutes. Two standard deviations above the mean is 120 + 2 (9) = 138, and two standard deviations below the mean is 120 − 2 (9) = 102. 95% of movies are between 102 and 138 minutes long.

Reﬂect and check To visualize this better, we could have constructed a normal curve with all the standard deviation divisions labeled.

102

111

120

129

138

147

156

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Idea summary A data set that is symmetric and bell-shaped is said to have an approximately normal distribution. The mean, median, and mode are approximately equal in a normal distribution. The center of the normal distribution is at the arithmetic mean, μ. The standard deviation σ, describes the spread of the data. The normal curve represents a probability distribution, and the area under the entire curve is equal to 100%, or 1. The percentage of data between 1, 2, and 3 standard deviations can be accurately summarized using the Empirical Rule.

34%

34% 13.5%

13.5%

2.35%

68% 95% 99.7% Empirical rule

Practice What do you remember? 1

SOL

State whether the following statements are true of the normal distribution curve: a

There are the same number of scores above and below the mean.

The fewest scores lie around the mean.

The spread of the normal distribution changes depending on the standard deviation.

A normal distribution is symmetric about the mean.

The mode of a normal distribution is always greater than the mean.

The median of a normal distribution is always less than the mean.

Based on the normal distribution curve shown, what do you estimate the total area under the curve to be?

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State whether the following ﬁgures are normally distributed: a

35 30

25 20

20 15

5 80 85 90 95 100 105 110 Weight

80 85 90 95 100 105 110 Weight

State the mean and standard deviation of each data set. a

b 19

105 114 123 132 141 150 159

31 43 55 67 79 91

A sample of professional basketball players is normally distributed and gives the mean height as 78 in with a standard deviation of 4 in. a

Select the normal curve that represents the data. A

B 62 66 70 74 78 82 86

75 76 77 78 79 80 81

66 70 74 78 82 86 90

State the height of a basketball player who is 1.5 standard deviations below the mean.

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Determine if the data collected for each question is likely to be approximately normally distributed. Explain your reasoning. a

What is the average internet download speed of households within my neighborhood?

Which comic book genre is preferred by 8th grade students of School A?

How long does it take for cars to go through the drive-thru at any fast food restaurant?

What is the average daily temperature in a speciﬁc city during the month of July?

The results from an exam were approximately normally distributed. The mean score was 65, with a standard deviation of 9. The points marked on the horizontal axis are separated by 1 standard deviation:

a 8

State the value of x on the graph.

State the value of y on the graph.

In a normal distribution, state the approximate percentage of the scores that lie within: a

1 standard deviation of the mean.

3 standard deviations of the mean.

2 standard deviations of the mean.

Describe what happens to a normal curve if you: a

Increase the standard deviation but keep the mean the same?

Increase the mean but keep the standard deviation the same?

Let’s practice 10

Consider the given normal distribution:

185

636

195

200

205

210

215

Find the mean of the data set.

Find the standard deviation.

Find the range.

Find the median.

What percent of the data lies below the mean?

190

Assume the mass of sumo wrestlers is normally distributed, with a mean mass of 348 lb and a standard deviation of 22 lb. a

Construct a normal distribution curve that models the weight of sumo wrestlers and label the boundaries for the Empirical Rule.

Determine how far below the mean a sumo wrestler who weighs 326 lb is.

State the percentage of wrestler that would weigh more than 348 lb.

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In the following normal distributions, ﬁnd the approximate percentage of scores that lie in the shaded region: a

30 38 46 54 62 70 78 86 94

12 15 18 21 24 27 30 33 37

122 127 132 137 142 147 152 157 162 SOL

The shoe size of 16 year olds are normally distributed. Which is closest to the percentage of these shoe sizes that is within 3 standard deviations of the mean? A

SOL

0.3%

95%

99.7%

A normally distributed data set has a mean of 0 and a standard deviation of 0.75. Which is the closest to the percent of values between −1.5 and 1.5? A

22 34 46 58 70 82 94 106 118

34%

95%

50%

68%

The number of offspring in litters of puppies is normally distributed, as shown. a

Find the percentage of litters that have no more than 9 puppies.

Find the percentage of dogs that have at least 7 puppies.

Find the percentage of puppies that are within 2 standard deviations of the mean.

99.7% of the litters will be between what numbers of puppies?

83.85% of the litters will be between what numbers of puppies?

The distribution of grades in a class can be approximated by the normal curve shown

Find the percentage a

A grade above the average

A grade above 41

A grade above 53

A grade below 35

A grade above 29

A grade between 29 and 47

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For a particular set of scores, it was found that the mean was 61 and the standard deviation was 11. If the scores are normally distributed, ﬁnd the percentage of scores between: a

39 and 83

28 and 94

61 and 72

The grades in a test are normally distributed. The mean grade is 60 with a standard deviation of 2. State which two grades the following percentage of results lie between, if the grades lie symmetrically about the mean: a

50 and 72

68%

95%

99.7%

Naomi collects data on the times that a class of students spent talking or texting on their phones on a particular weekend. She ﬁnds the data is approximately normally distributed with mean time 173 minutes and standard deviation of 4 minutes. After drawing a normal curve, she formulates the question, “What percentage of students used their phones for at least 165 minutes on the weekend?” Answer Naomi’s statistical question.

The number of customers in the store is recorded throughout the day. The data set is approximately normally distributed with the busiest time at 10:00 am and a standard deviation of 15 minutes. The cafe owner has decided to launch a special on coffee for only 30 minutes of the morning and wants the most customers to see it. They formulated the statistical question, “For the special, what 30-minute time interval will be seen by the most customers?”

Answer the owner’s statistical question.

Determine the percentage of customers the cafe owner will reach during your selected time interval. Justify your reasoning.

The number of biscuits in a box is approximately normally distributed with mean 30 and standard deviation of 3. a

Approximately 81.5% of the scores lie between 2 standard deviations below and x standard deviation(s) above the mean. Find the value of x.

Find the range of the numbers of biscuits in 81.5% of the boxes.

The times for runners to complete a 100 m race is approximately normally distributed with mean 14 seconds and standard deviation of 1.9 seconds. Find the range of time in seconds for which 97.35% of runners completed the race. Round your answer to one decimal place.

The operating times of phone batteries are approximately normally distributed with a mean of 34 hours and a standard deviation of 4 hours. a

Construct a normal distribution curve that models the operating times of phone batteries and label the boundaries for the Empirical Rule.

Find the percentage of batteries that last between 22 and 42 hours.

Find the percentage of batteries that last between 30 hours and 42 hours.

Any battery that lasts less than 22 hours is deemed faulty. The company CEO asked one of the managers, “If we manufactured 51 000 batteries, approximately what number of batteries would we be able to sell?” Answer the CEO’s statistical question.

The heights of 600 boys are found to be approximately normal, with a mean height of 57 in and a standard deviation of 8 in. Find the number of boys with heights between: a

SOL

49 in and 65 in

41 in and 73 in

33 in and 81 in

57 in and 65 in

The times that professional divers can hold their breath are approximately normally distributed with mean 106 seconds and standard deviation 8 seconds. Which is closest to the number of divers that would be able to hold their breath for longer than 82 seconds? A

638

475

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570

680

700

The number of candies packaged in a single bag is approximately normally distributed with a mean of 38 and a standard deviation of 5 candies. If 4000 bags of this candy are produced, ﬁnd the approximate number of bags with more than 33 candies.

The coach of a sports team recorded the number of hours his players practiced during each of the two weeks. The following statistics were calculated. Week 1

10.5 12 13.5 15 16.5 18 19.5

Week 2

18 20 22 24 26

The coach concluded that there was more variation in the number of hours worked for week 2 than for week 1. The coach’s conclusion was:

valid because the mean for week 2 was greater than the mean for week 1.

valid because the standard deviation for week 2 was greater than the standard deviation for week 1.

invalid because the mean for week 1 was less than the mean for week 2.

invalid because the standard deviation for week 1 was less than the standard deviation for week 2.

The ﬁnish times of runners competing in a 10 km fun run in two different years is approximately normally distributed. The time it took one runner to ﬁnish, including the mean and standard deviation of the other runners, is shown: Year 2017 2018

Time (mins) 75 59

Mean (mins) 78 63

Standard deviation (mins) 1 4

Determine how many standard deviations below the mean the runner’s ﬁnish time was in 2017.

Determine how many standard deviations below the mean the runner’s ﬁnish time was in 2018.

Find the year that the runner performed the best relative to other runners in the race.

Let’s extend our thinking 29

The heights of adult men in the United States are approximately normally distributed with a mean of 70 in and a standard deviation of 3 in. Heights of adult women are approximately normally distributed with a mean of 64.5 in and a standard deviation of 2.5 in. a

Determine the more likely outcome: • A man has a height of 76 inches or taller (6 ft 4 in). • A woman has a height between 57 (4 ft 9 in) and 59.5 (4 ft 11.5 in) inches.

b 30

Find the range of male heights that have the same probability as a woman being between 62 and 72 in.

A machine is set for the production of cylinders of mean diameter 5.06 in, with standard deviation 0.016 in. a

Assuming a normal distribution, between what values, in inches, will 99.7% of the diameters lie?

If cylinders with diameters less than 5.012 in or more than 5.108 in are discarded, what percentage of cylinders produced are discarded?

If a cylinder, randomly selected from this production, has a diameter of 5.124 in, what conclusion could be drawn?

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In a study, the test scores from two different schools are approximately normally distributed. School A has a mean score of 75 with a standard deviation of 5. School B has a mean score of 80 with a standard deviation of 10. If a student from School A and a student from School B both receive a 70, which student performed better relative to their classmates? Explain.

Determine whether the statements regarding normally distributed data are true or false. Justify your reasoning. a

The standard deviation can be 0.

The mean can be 0.

The fewest data points lie around the mean.

A higher mean will result in a skewed curve.

The histogram displays the magnitude of earthquakes collected from a large sample. The mean of the distribution is 2.2 and the standard deviation is 0.8.

Percent

Histogram of Earthquake Magnitudes 20 18 16 14 12 10 8 6 4 2 0

Magnitude

Concetta needs to estimate the percentage of earthquakes that have a magnitude less than 3.0. Her work is shown: 1. A magnitude of 3.0 is one standard deviation above the mean magnitude of 2.2. 2. According to the Empirical Rule, 68% of the data is within one standard deviation above and below the mean, so 34% of the data is between the mean and one standard deviation above the mean. 3. Since 50% of the data is below the mean, 50 + 34 = 84% of the data is below one standard deviation above the mean. Find the error in Concetta’s reasoning. 34

Galen is a customer support intern at a tech company. Customers will call in for technological support, and the support team tries to resolve issues as eﬃciently and quickly as possible. The team manager wants them to resolve issues within 6 minutes or less. The team decides to investigate this further using the data cycle. a

Formulate an investigative question that would lead to the collection of univariate data.

Describe a method the customer support team can use to collect data.

The team collected the data shown in the table: 11.0 9.1 12.9

9.7 9.1 9.5

11.3 10.5 10.1

Call durations (minutes) 13.0 9.5 9.5 13.2 6.2 6.6 8.9 8.0 7.2 8.9 10.2 7.7

11.5 10.6 10.8

9.1 8.2 8.8

11.1 7.2 9.4

Use technology to create a smooth curve to model the distribution. Sketch the curve and describe its shape.

640

Answer the investigative question from part (a).

The team manager formulated this statistical question: “What percent of technological issues are resolved in 6 minutes or less?” Answer the manager’s statistical question.

Mathspace Virginia SOL Algebra 2 mathspace.co

The empirical rule can also be used to estimate the percentage of data within 1, 2, and 3 standard deviations on the mean in the standard normal curve.

34%

13.5%

2.35% −3

2.35% −2

−1

68% 95% 99.7%

Example 1 Brock is applying to different colleges across America and needs to decide if he should emphasize his SAT score, ACT score, or both. The test scores for both the SAT and ACT are normally distributed. The data is summarized in the table provided.

SAT ACT

Brock’s score 1450 30

Mean 1051 20.8

Standard Deviation 211 5.7

a Calculate and interpret the z-score for Brock’s SAT score.

Create a strategy The formula for ﬁnding the z-score is

where x is a test score, μ is the mean, and σ is the standard deviation.

Apply the idea For the SAT, we are given x = 1450, μ = 1051, and σ = 211: Formula for z-scores

Substitute the known values

Evaluate

Brock’s z-score for his SAT test is z = 1.89 which means Brock scored 1.89 standard deviations above the mean.

b Calculate and interpret the z-score for Brock’s ACT score.

Create a strategy We will use the formula for z-scores again, but this time with the given values for the ACT.

Apply the idea For the ACT, we are given x = 30, μ = 20.8, and σ = 5.7: Formula for z-scores

Substitute the known values

Evaluate

Brock’s z-score for his ACT test is z = 1.61 which means Brock scored 1.61 standard deviations above the mean.

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c Determine which test Brock did better on relative to all other SAT and ACT test takers.

Create a strategy Compare the z-scores found in parts (a) and (b). The higher Brock’s z-score, the better he did relative to the other test takers.

Apply the idea

Reﬂect and check

Relative to all people who took the SAT and ACT, Brock did slightly better on his SAT test than he did on his ACT test since his z-score was higher.

Both of Brock’s scores were better than average, but similar relative to the averages, so he can report either of the test scores when applying to different colleges. On college applications, only one test score is usually required.

Example 2 Three sprinters are training for a national competition. The data collected on each of their running times (in seconds) is approximately normal. Information for their mean, standard deviation, a practice 400 m sprint and its corresponding z-score are shown in the table. Lina Aurelia Mariana

μ 65 62

σ 3 2

z-score −1.27 0.85 −0.5

Practice time 65.4 59.5

a Find the 400 m sprint time Lina ran during practice.

Create a strategy To ﬁnd the practice time that had a z-score of −1.27, we can use the formula deviation, and z-score, then solve for the practice time in seconds.

with the given mean, standard

Apply the idea

Reﬂect and check

From the given information, we know μ = 65, σ = 3, and z = − 1.27.

Running 400 m in 61.2 seconds is −1.27 standard deviations below the sprinter’s average time of 65 seconds. This tells us she ran faster in that practice run than she normally does.

Formula for z-scores

Substitute known values

Multiply both sides by 3

Add 65 to both sides

Lina ran a 400 m practice time of 61.2 seconds.

b Find the standard deviation of Aurelia’s times.

Create a strategy To ﬁnd the standard deviation, we can use the z-score formula with the given mean, z-score, and practice time, then solve for the standard deviation.

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Apply the idea From the given information, we know μ = 62, z = 0.85, and x = 65.4. Formula for z-scores

Substitute known values

Evaluate the numerator

Multiply both sides by σ

Divide both sides by 0.85

Aurelia’s 400 m times have a standard deviation of 4 seconds.

c Find the average 400 m sprint time for Mariana.

Create a strategy To ﬁnd the mean, we can use the z-score formula with the given standard deviation, z-score, and practice time, then solve for the mean.

Apply the idea From the given information, we know σ = 2, z = − 0.5, and x = 59.5. Formula for z-scores

Substitute known values

Multiply both sides by 2

Subtract 59.5 from both sides

Multiply −1 to both sides

Mariana’s average 400 m time is 60.5 seconds.

Reﬂect and check Of the three sprinters, Mariana has the fastest average 400 m time, and her sprint times are more consistent.

Example 3 An extreme amusement park ride only allows riders over 60 inches tall to ride. Colette was not allowed to ride because she did not meet the height requirement, but her younger brother Gavin was able to ride because he was taller than the height requirement. This led her to ask the question, “How do the heights of men compare to the heights of women?” a Describe a method Colette can use to collect data.

Create a strategy First, use Colette’s statistical question to determine the type of data that needs to be collected. Then, consider whether the data can be collected by research, a survey, an observation, or a scientiﬁc experiment.

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Apply the idea

Reﬂect and check

Colette needs to collect data on the heights of men and women. Since most people know their heights, Colette can use a survey or poll to collect the data.

Colette could also research the average heights of men and women. While researching, she would need to make sure that the sample is representative of the population, and the data collection process did not introduce bias.

b The data Colette collected on the heights of men and women are shown. Female heights 66, 61, 62, 64, 60, 62, 64, 63, 58, 64, 60, 68, 62, 59, 64, 60, 64, 66, 62, 62

Male heights 71, 69, 71, 66, 69, 77, 74, 72, 75, 71, 68, 72, 70, 64, 73, 68, 66, 70, 67, 73

Use technology to create a smooth curve to model each distribution and describe the shape of each curve.

Apply the idea First, we will create the smooth curve that approximates the women’s heights.

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The curve is symmetric and bell-shaped, meaning the data is approximately normal. Next, we will create the smooth curve that approximates the men’s heights.

To see the full curve, we can adjust the settings by selecting the Graph tab and unchecking the automatic dimensions. Then, we can adjust the y-Max to 0.13, which will allow us to see the top of the curve.

Again, the curve is symmetric and bell-shaped, meaning the data is approximately normal.

Reﬂect and check Although both data sets are normally distributed, they have different measures of center and spread. This means the curves will have a similar shape, but one is likely taller than the other and they are centered around different values.

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c Answer the statistical question that Colette formulated.

Create a strategy Colette’s statistical question was, “How do the heights of men compare to the heights of women?”. We can answer this by analyzing the average heights of men and women, which are represented by the center of the normal curves.

Apply the idea

Reﬂect and check

Looking at the smooth curves from the previous part, we can see that the curve that approximates the women’s heights is centered at around 63 inches. The curve that approximates the men’s heights is centered at 70 inches.

Rather than using the curves to compare the means, we could have calculated the mean of each data set using technology, then compared the values.

This tells us that, on average, men are taller than women.

d Since both data sets are normally distributed, Colette wanted to further investigate men’s and women’s heights relative to the height requirement for the ride. Her new statistical question is, “How does the percentage of male riders who can ride this ride compare to the percentage of female riders who can ride?” Find and interpret the z-scores for the 60-inch height requirement relative to the average American female heights and average American male heights.

Create a strategy The average height of men and women are different, and the standard deviations of the heights are different as well. We can compare the heights of men and women by using z-scores to standardize the measurements. To ﬁnd the z-score, we can use the formula

with the given height requirement. Then, we can use technology

to ﬁnd the mean and standard deviation of each set.

Apply the idea By selecting “Show summary statistics” (Σx icon), we can ﬁnd the mean and standard deviation of women’s heights.

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From this information, we see μ ≈ 62.5 and σ ≈ 2.5, and we were given x = 60. The z-score for women’s height is

For women, the height restriction of 60 inches is only 1 standard deviation below the average female height. Next, we will ﬁnd the mean and standard deviation of men’s heights.

From this information, we see μ ≈ 70, and σ ≈ 3. The z-score is

The height restriction of 60 inches is more than three standard deviations below the average male height.

Reﬂect and check Notice that the mean and standard deviations of the sets were rounded to use “nice” values. This allows us to sketch the curves more easily. However, we should not round to “nice” values if the difference is relatively large.

Women’s heights

We can use the normal distribution curves of the data to check our answers.

57.5

62.5

67.5

The data value of 60 does lie 1 standard deviation below the mean of the women’s heights and the mean of the men’s heights.

Men’s heights

standard deviations below

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Practice What do you remember? 1

Fill in the blanks to complete each statement. a b c d

A normally distributed data set can be approximated by a ⬚ curve. A normal distribution is symmetric about the ⬚.

The mean, median, and mode of a normal distribution are approximately ⬚. The area under the curve is ⬚ which represents ⬚ of the data.

State what information a z-score tells you about a number in a data set.

For each question use the given z-score to determine:

If the value is above or below the mean.

How many standard deviation(s) the value is away from the mean.

z=4

c e g

c e

z = 2.89

State the mean.

State the standard deviation.

μ = 2, σ = 3, x = 5 μ = − 1, σ = 4, x = − 5 μ = 8, σ = 5, x = 22 μ = 16, σ = 2, x = − 7

b d f h

μ = 5, σ = 9, x = 32 μ = − 7, σ = 4, x = − 15 μ = − 15, σ = 8, x = 11 μ = 27, σ = 6, x = 54

μ = 2, σ = 3, z = 29 μ = 8, σ = 5, z = − 14 μ = 25, σ = 1.2, z = 55.7

b d f

μ = − 13, σ = 2, z = 34 μ = − 2, σ = 7, z = − 9 μ = − 18.9, σ = 4, z = 23.7

σ = 11, z = 4, x = 28 σ = 3.2, z = 2.2, x = − 7.11 σ = 5.6, z = − 4.8, x = − 34.5

b d f

σ = 4, z = − 7, x = − 40 σ = 7.8, z = − 14.7, x = − 62 σ = 3, z = 9.7, x = 27

The mean, z-score and corresponding data value of approximately normally distributed data sets are given. Find the standard deviation of each data set. a c e

650

The standard deviation, z-score and corresponding data value of approximately normally distributed data sets are given. Find the mean of each data set. a

z = − 0.48

The mean and standard deviation of approximately normally distributed data sets are given. Find the corresponding data value of each z-score. a

The mean and standard deviation of approximately normally distributed data sets are given. Find the corresponding z-score of each data value. a

z = −1

For the standard normal curve: a

μ = − 27, z = 15, x = 63 μ = − 17.5, z = − 5, x = − 22 μ = − 12.2, z = 31.9, x = 74.05

Mathspace Virginia SOL Algebra 2 mathspace.co

b d f

μ = 6.4, z = 13.6, x = 28.2 μ = 7.12, z = 3.8, x = 24.32 μ = − 7.1, z = − 10.2, x = − 82.4

In the following normal distributions, each unit on the horizontal axis indicates 1 standard deviation. Find the approximate percentage of scores that lie in the shaded region: a

−4 −3 −2 −1 0 1

Let’s practice 10

If Dave scores 96 in a test that has a mean score of 128 and a standard deviation of 16, what is his z-score?

Frank finishes a fun run in 156 minutes. If the mean time taken to finish the fun run is 120 minutes and the standard deviation is 12 minutes, find his z-score.

A particular investment fund has returned 17.2% annually on average over a period. If the mean return of all investment funds over the same period was 8% annually and the standard deviation was 2.3%, what is this fund’s z-score?

The volume of bottled water is approximately normally distributed with mean 500 mL and standard deviation 0.5 mL. a

Find the volume of water which is 2.2 standard deviations below the mean.

Find the value of the z-score that corresponds to the volume of water found in part (a).

If Harrison scores 57.6, with a z-score of 2, in a test that has a standard deviation of 5.8, what was the mean score?

An element of a data set has a value of 234, which corresponds to a z-score of −2.8. If the standard deviation of the data set is 9, which is the closest to the mean of the data set? A

259.2

− 208.8

− 259.2

208.8

The standard deviation of a normally distributed set of data is 4. If a data value of 31 has a corresponding z-score of 2.11, what is the mean of this distribution? Justify your answer.

If Luke scores 68, which corresponds to a z-score of −3, in a test that has a mean score of 93.5, what was the standard deviation of the test scores?

The mean of a normally distributed set of data is 215. A data value of 200 has a z-score of −2.7. Determine the standard deviation of this distribution. Show your work.

Statistical information for a data set is given. • The mean is 21.9. • The z-score for 14.1 is −1.6. What is the standard deviation for this data set? A

− 4.875

1.67

4.875

− 1.67

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For each of the following examples, ﬁnd x, the test score each student received: a

Iain’s z-score in a test is 1, the mean score is 62% and standard deviation is 3%.

Rochelle’s z-score in a test is −3, the mean score is 75% and standard deviation is 3%.

Aaron’s z-score in a test is 3.6, the mean score is 75% and standard deviation is 3%.

Sean’s z-score in a test is −1.1, the mean score is 76% and standard deviation is 3%.

A general ability test has a mean score of 100 and a standard deviation of 15. a

If Paul received a score of 102 in the test, what was his z-score correct to two decimal places?

If Georgia had a z-score of 3.13, what was her score in the test, correct to the nearest integer?

The mean and standard deviation of exam results in each subject are given. a

A student receives a grade of 93 in English. Determine the number of standard deviations this grade is away from the mean.

English Mathematics

Mean 75 72

Standard deviation 9 8

Find the grade in Mathematics that would have the same z-score as a grade of 93 in English.

A student receives a grade of 53.6 in Mathematics. Determine the number of standard deviations this grade is away from the mean.

Find the grade in English that would have the same z-score as a grade of 53.6 in Mathematics.

Data representing the finishing times of all runners in a recent marathon have a normal distribution. The z-score of Zenaida’s finishing time is −1.25. Explain what this z-score represents in relation to the mean time of finishing the race.

A company uses z-scores to determine employee bonuses. Employees with z-scores above 1.5 receive a bonus, where the mean performance rating is 80 and the standard deviation is 5. Given the following employees’ performance ratings, determine who will get a bonus:

Performance rating 87 89 78

The grades on a recent English exam are approximately normally distributed with mean 53 and standard deviation 4. a

Find the value of the z-score that corresponds to an English grade of 45.

What percentage of the class received a grade higher than 45?

Bags of ﬂour are each labeled as having a mass of 1 kg. The mass of these bags is normally distributed with a mean of 1.06 kg and a standard deviation of 0.03 kg. a

Copy and complete the table: Mass (kg) z-score

John Susan Mark

1.03

1.06

1.09

1.12

What percentage of bags will have a mass less than 1.06 kg?

What percentage of bags will weigh at least 1.12 kg?

A normally distributed data set has a mean of 52 and a standard deviation of 9. What percent of the data is less than z = 1.5? A

33%

50%

67%

85%

Maria scored 80% with a z-score of 4 in Physics, and 73% with a z-score of 1 in Chemistry. Determine the subject in which she scored better relative to her peers. Explain.

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Jenny scored 81% with a z-score of −2 in English, and 72% with a z-score of −4 in Mathematics. In which subject was her performance better, relative to the rest of the class?

Buzz’s best time in the half marathon is 90.2 minutes, while his best time in the full marathon is 185.6 minutes.

The mean time to complete the half marathon is 110 minutes with a standard deviation of 6 minutes among world class runners. What is the z-score of Buzz’s best time in the half marathon?

The mean time to complete the full marathon is 200 minutes with a standard deviation of 3 minutes among world class runners. What is the z-score of Buzz’s best time in the full marathon?

In which event does Buzz have a better ‘best time’, relative to world class runners?

Athletes take a lesser heart rate as a sign of better ﬁtness. Victoria and John record their heart rates in beats per minute (bpm) each time they ﬁnish a triathlon. Both their heart rates are normally distributed, with Victoria’s μ = 170 beats, σ = 9 beats and John’s μ = 161 beats, σ = 8 beats. At the end of their most recent race, Victoria’s heart rate was 192 bpm and John’s was 172 bpm. a

Calculate Victoria’s z-score in the most recent race to one decimal place.

Calculate John’s z-score in the most recent race to one decimal place.

Whose heart rate was better in the most recent race?

The table records the lengths of ﬁsh a ﬁsherman caught on a particular day: Type of fish

Length (cm)

Yellowfin Bream

20, 22, 25, 28, 26, 28, 26, 29, 32, 35

Flathead

34, 36, 33, 35, 32, 36, 38, 41, 42, 45

Find the mean and standard deviation of the lengths of the Yellowﬁsh Bream correct to two decimal places.

If the next Yellowﬁn bream he catches is 30.40 cm, calculate the z-score of this ﬁsh correct to one decimal place.

Find the mean and standard deviation of the lengths of the Flathead correct to two decimal places.

If the next Flathead he catches is 37.33 cm, calculate the z-score of this ﬁsh correct to one decimal place.

Which of these two ﬁsh is longer, relative to their cohort?

Let’s extend our thinking 33

Endo received a score of 720 points for his performance in the game “Galactic Adventure” and 850 points for his performance in the game “Fantasy Quest”. The list shows the scores of all his friends for each of the games: • Galactic Adventure: 680, 720, 600, 550, 690, 730, 710, 700, 660, 680, 710, 720, 690, 650, 670 • Fantasy Quest: 820, 850, 790, 800, 830, 820, 780, 760, 830, 840, 850, 810, 820, 830, 840 Which game is Endo better at, relative to his friends? Justify your answer.

Eileen received a grade of 55 for her History assignment and 65 for her Philosophy assignment. A grade of 60 has a z-score of 1.3143 in History and a z-score of 0.4298 in Philosophy. The list shows the grades of all her classmates for each of the assignments: • History: 51, 55, 22, 10, 34, 54, 60, 52, 31, 47, 48, 57, 35, 21, 30 • Philosophy: 54, 70, 45, 43, 70, 82, 57, 41, 43, 59, 53, 54, 40, 65, 46 In which assignment did Eileen perform the best, relative to her peers? Explain using z-scores.

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The number of days of sick leave taken by employees is approximately normally distributed. The z-scores of the number of sick days taken by four employees are shown in the table below: Employee z-score a b

Xavier Amelia −3.44 1.77

Peter −3.69

Fiona 1.89

Determine the employee who took the least amount of sick days relative to their peers. State the event that is more likely to occur: • Event 1: Randomly selecting an employee whose number of sick days were less than Xavier or greater than Amelia. • Event 2: Randomly selecting an employee whose number of sick days were less than Peter or greater than Fiona.

In Armando’s History class, his z-score for his most recent exam was 2, but he only did better than 50% of the class. Explain what this information tells Armando.

Explain the difference between a standard normal distribution curve and an approximately normal distribution curve.

Over the summer, Edmond volunteered at an elephant shelter in Asia, and Daniela volunteered at an elephant shelter in Africa. As they talked about their experiences and looked at each others’ pictures, they realized that African and Asian elephants differ in size. They decide to investigate the sizes of African and Asian elephants further using the data cycle. a

Formulate a question about African and Asian elephants that would require the collection of univariate data.

Describe a method that Edmond and Daniela could use to collect data.

Edmond and Daniela collected data on the weights of female African and Asian elephants, shown in the table. Weights of female elephants (lb) African 7558, 6740, 7087, 7845, 7621, 5914, 7070, 6409, 6438, 6746, 6586, 7373, 6957, 6573, 6766, 6700, 7396, 6377, 6688, 5988, 4968, 6892, 7019, 6055, 7862

Asian 4773, 5523, 5406, 6266, 6235, 5577, 5689, 5056, 4510, 5326, 5578, 6115, 6101, 5306, 5349, 4976, 4790, 4647, 6475, 5245, 5281, 4874, 5889, 4693, 5394

Use technology to create a smooth curve to model each distribution. Sketch each curve and describe their shapes.

654

Answer the statistical question from part (a).

Edmond and Daniela’s local zoo have a female African elephant, named Meru, and a female Asian elephant, Aiyara. Meru weighs 6842 pounds, and Aiyara weighs 5357 pounds. Which elephant is closest to the average weight of their respective species? Use z-scores to justify your answer.

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We can draw a tree diagram to represent all of the possible choices. Hot fudge Vanilla

Caramel

Hot fudge Chocolate Caramel

Hot fudge Strawberry Caramel

Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles

From the last set of branches in the tree diagram, we can count a total of 24 different ways the ice cream sundaes can be created. We can also see from the tree diagram that we could have calculated the number of possible sundaes by multiplying 3 ⋅ 2 ⋅ 4 = 24. This is an example of the fundamental counting principle. Fundamental counting principle If one decision can be made x ways and another can be made y ways, then the two decisions can be made x ⋅ y ways. This essentially means, if we know the number of ways each decision can be made, we can multiply those numbers together to ﬁnd the total number of ways all of the decisions can be made.

Example 1 Tyson’s mom is buying a brand new car, and she asked him to help her make a ﬁnal decision. She has narrowed down her options for the manufacturer, type of car, color, and type of seats to the following: • Ford or GM • Sedan, minivan, pickup truck, SUV • Black, silver, red • Leather seats or fabric seats Determine the number of unique car choices from the options above.

Create a strategy From the options that Tyson’s mom has narrowed down, there are 2 manufacturers, 4 types of cars, 3 colors, and 2 types of seats.

Apply the idea By the fundamental counting principle, we need to multiply to number of choices in each category together to ﬁnd the total number of possible cars. 2 ⋅ 4 ⋅ 3 ⋅ 2 = 48 There are 48 unique cars that could be created.

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Reﬂect and check Note that we could have changed the list of options slightly, but still ended up with the same number of possibilities. If the options were: • Ford, Chrysler-Dodge, GM • Sedan, mini-van, truck, SUV • White, black, silver, red There would still be 3 ⋅ 4 ⋅ 4 = 48 unique cars that could be created.

Example 2 Suppose you have the digits 1, 3, 4, 7, 8, 9. How many 3-digit odd numbers which are greater than 200 can be made without repeating digits?

Create a strategy To have an odd number, we know the last digit will have to be odd. To be greater than 200, we know the first digit cannot be 1. We have two cases here: one where the first number is odd in which case we lose one possibility for the last digit, and one where the first number is even.

Apply the idea Case 1: The first number is odd. For the first digit, it could be 3, 7 or 9. Now that one of those has been used, one of them cannot be included in the choices for the last digit. That means there are 2 of those could be used for the last digit, and we also have the choice of 1. In other words, there are 3 possibilities for the first digit and 3 possibilities for the second digit.

Since we have already used 2 of the 6 options for the ﬁrst and last digits, there are 4 possibilities left for the 2nd digit.

Therefore, there are = 3 ⋅ 4 ⋅ 3 = 36 possibilities for a 3-digit code that begins with an odd number greater than 2. Case 2: The first number is even. The first digit could be 4 or 8. The last digit could be any of the odd numbers 1, 3, 7, 9. Since 2 of the 6 options have already used for the first and last digits, there are 4 possibilities left for the 2nd digit.

Therefore, there are = 2 ⋅ 4 ⋅ 4 = 32 possibilities for a 3-digit code that begins with an even number. We can ﬁnd the total number of possibilities by adding the answers from both cases. There are 36 + 32 = 68 ways we can create a 3-digit odd number that is greater than 200 using the given numbers.

Idea summary To ﬁnd the number of ways multiple events can happen, we use the fundamental counting principle by multiplying together the number of ways each event can happen.

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Permutations When we’re calculating the number of possibilities for events, sometimes the order in which we make our choices makes a difference in how we calculate the possibilities. For example, consider the letters A, B, C, D, and E. We want to see how many different ways we can select and arrange a set of 3 letters.

For each position, we write down how many possible choices we have. For the 1st position, we can choose any of the 5 letters: A, B, C, D, or E.

We also write down how many possible choices we have for the 2nd position. Because we had 5 letters to start with, and we will have already selected one to go in the 1st position, there are 4 options left for the letter that goes in this next position.

For the last position, we will have already picked 2 of the 5 letters for the ﬁrst and second positions, so there are only 3 letters left to choose from.

By the fundamental counting principle, we multiply these together to determine there are 5 ⋅ 4 ⋅ 3 = 60 possible ways to arrange 3 letters. We call a situation like this, when we calculate the number of different ways a certain number of objects can be arranged from a larger set, a permutation. Permutation The number of ways in which a set of r objects can be ordered or arranged from a set of n objects. The notation is nPr. Example:

When calculating a permutation, the order the objects are chosen matters, so the number of ways the event can happen has a decreasing numerical pattern. There may be 4 ways the first event can happen, then 3 ways the second event can happen, 2 ways the third event can happen, and only 1 way for the final event. Mathematicians have created a shorthand notation to account for this pattern, called a factorial. A factorial, denoted by n!, is the product of the first n positive integers. It is calculated by multiplying n by every positive integer less than n all the way down to 1. That is:

n! = n ⋅ (n − 1) ⋅ (n − 2) ⋅ … ⋅ 2 ⋅ 1 n

a positive integer greater or equal to zero

An important property of factorial notation is 0! = 1. This property makes it possible to perform more complex calculations. Calculators and computers can evaluate factorials to help us calculate them eﬃciently. A permutation is an important application of the fundamental counting principle. We calculate a permutation using the formula:

n the total number of objects in the set r the number of objects being ordered or arranged 658

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Example 3 Suppose we need to choose a 4-digit passcode for our phones. We can use any number from 0−9, but we can only use a number once. Determine the number of passcodes possible.

Create a strategy Let’s begin by imagining the place values in the passcode.

We need to determine how many numbers are available to be chosen for each digit in the passcode. For a permutation, the order in which the objects are arranged matters. In other words, a passcode of 1234 is very different from a passcode of 2143. Although the digits in both passcodes are the same, the way in which they are arranged is different. The order in which you type the numbers into your phone’s lock screen matters.

Apply the idea For the ﬁrst digit of the code, we can use any of the 10 available numbers.

But once the number for the ﬁrst digit has been used, we only have 9 numbers left to choose for the second digit.

Following this pattern, there would be 8 numbers left to choose for the third digit, and 7 numbers left to choose for the last digit of the passcode.

By the fundamental counting principle, we multiply the options together to get a total of 10 ⋅ 9 ⋅ 8 ⋅ 7 = 5040 possibilities for a 4-digit passcode without repeated digits.

Reﬂect and check Since the digits could not be repeated, we simply removed one of the options each time. This resulted in multiplying each integer less than 10 like we would for a factorial. However, we do not want to multiply each integer all the way down to 1. We can still calculate 10! in order to use that notation, but we can remove the unwanted numbers with division.

Recall that we began with 10 digits to choose from, and we needed to choose 4 for the passcode. The denominator is just 10 − 4 = 6, so we can rewrite the expression as we use to find permutations.

. This process can be generalized into the formula

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Example 4 For each of the following scenarios, determine if a permutation would be an appropriate method for finding the total number of ways to select the objects. a A local pizza place is offering a special on a large pizza with 3 toppings of your choice. They have a total of 9 pizza toppings to choose from.

Create a strategy If we use the permutation formula in this situation, we will ﬁnd the number of different ways the three toppings could be arranged in order when selected from 9 options. We need to determine if the order that we pick the 3 toppings from the 9 options matters. Would picking sausage then pepperoni then olives be different from picking olives, then pepperoni, then sausage? If the order does matter, then we would use a permutation.

Apply the idea The order in which you pick the toppings does not matter because it is the same 3 toppings. The permutation formula would count 6 different ordered versions of olives, pepperoni, and sausage: Olives Pepperoni Sausage Olives Sausage Pepperoni Pepperoni Olives Sausage Pepperoni Sausage Olives Sausage Olives Pepperoni Sausage Pepperoni Olives But, we would count all of these as one unique type of pizza with 3 toppings. This is not a situation where we would use a permutation.

b Students need to elect a new president, vice president, and secretary for their school’s student council. There are 14 students running for any position within the student council.

Create a strategy When we think of “order” in this case, we are not necessarily looking at who we choose 1st, 2nd, or 3rd. We are determining if the students we choose for each position on the council matters. In other words, would choosing Jackie for president and Jillian for vice president be different from choosing Jillian for president and Jackie for vice president? If it is different, then the order matters.

Apply the idea When counting the outcomes, each ordered spot represents a council position:

So, the outcomes

and

indicate the students were chosen for different positions, and should be counted separately. The order that students choose the candidates or the positions the candidates receive on the council matters, so a permutation should be used.

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c Ariana has 7 bottles of nail polish, and she wants to choose 2 different colors for her nails. She wants each nail to be half one color and half the other color.

Create a strategy The way in which Ariana paints her nails is not speciﬁed. For this question, we are not worried about which color is on which half of the nail. We are only looking at the way in which she chooses the colors. So, does the order in which Ariana chooses two colors matter?

Apply the idea The order in which Ariana picks two nail polish colors does not matter because all her nails will have both colors on them. If she chooses pink ﬁrst then yellow, her nails would still have the same colors if she had chosen yellow ﬁrst then pink. A permutation would not be used in this scenario.

Reflect and check Note that the way in which Ariana paints her nails is not specified. We do not know if she is painting the bottom half of her nails with the first color and the top of her nails with the second color. All we know is that she wants to paint her nails, and she is picking two colors. If the question had specified that it was important to know which color was on top and which color was on the bottom of her nails, then it would be a different situation. In that case, pink below yellow would be different from yellow below pink, and a permutation would be used.

Example 5 Users of a website are required to create a unique PIN ID consisting of 5 characters, and they can be arranged in any order they choose. Alicia wants to use her two favorite letters, Z and S, and her three favorite digits, 7, 8 and 2. a How many unique PIN IDs can she create if the letters and digits can appear in any particular order (and no character can be repeated)?

Create a strategy We can use the permutation formula

, where n is the number of characters Alicia can choose from and r

is the number of characters in a PIN ID.

Apply the idea Since Alicia only wants to use Z, S, 7, 8, and 2, there are 5 characters she can choose from. This will be the value of n in the permutation formula. The total number of characters in a PIN ID is 5, so this will be the value of r. Using the permutation formula: Permutation formula

Substitute n = 5 and r = 5

Expand the factorial

Evaluate

There are 120 unique PIN IDs that Alicia could create.

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Example 6 Suppose that 8 people enter a room and randomly stand in a line along the back wall. Find the probability that they stand from tallest to shortest, left to right.

Create a strategy Recall probability is defined as the following: P (Event) = To find the number of total outcomes, we need to find the total possible arrangements that 8 people can stand in a line. Because the order in which they stand matters, this is a permutation.

Apply the idea We need all 8 of the people to line up along the wall, so we can ﬁnd the total by calculating 8P8. Substitute n = 8, r = 8 in the permutation formula

Evaluate the subtraction

Evaluate the factorials

Evaluate the division

There is only 1 way the people can line up from tallest to shortest, left to right. The probability of this happening is

Reﬂect and check Whenever we need to select or order all of the objects in a permutation, whenever n = r, we can simply use n! instead. This is because the denominator will always be (n − n)! = 0! = 1, and anything divided by 1 is itself.

Idea summary We use a permutation to ﬁnd the number of ways in which a set of r objects can be ordered or arranged from a set of n objects.

the total number of objects in the set

the number of objects being ordered or arranged

Permutations can also be used to calculate probabilities.

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Combinations Exploration A store has notebooks that come in 6 different colors. You need 3 new notebooks for various classes, and you want them to be different colors to easily distinguish between them. 1.

How many different color combinations can you make with the 3 notebooks you choose? Explain your process.

Why is this answer different from the answer you would get if you had used the permutation formula?

In the scenario of choosing 3 different colored notebooks from 6 color options, we can determine the total number of ways to order 3 textbooks from 6 color choices by using the permutation formula.

There are 120 different color combinations of notebooks we can create when the order matters. However, in the context of this situation, the order in which we choose the notebooks does not matter because a set of yellow, green, and blue notebooks is the same set of notebooks, regardless of order. Considering just 3 possible colors: Yellow Green Blue Yellow Blue Green Green Blue Yellow Green Yellow Blue Blue Green Yellow Blue Yellow Green We can see that the 3 colors are repeated 3 ⋅ 2 ⋅ 1 = 6 times. We can remove these repeated possibilities by dividing it from our total. In other words, we need to divide the permutation formula by r! where r is the number of objects we are choosing. This process can be generalized into the formula we use to ﬁnd combinations.

the total number of objects in the set

the number of objects that are being chosen

Combination The number of ways in which a set of r objects can be chosen from a set of n objects. The notation is nCr. Example:

For combinations, the notation

664

may also be used, and we can read it as “n choose r”.

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Example 7 For each of the following scenarios, use either a permutation, a combination, or the counting principle to solve the problem. a There are 10 parts in a school play, and 10 students auditioned for the play. How many ways can the parts be assigned to the students?

Create a strategy The order that the parts are assigned matters. We can calculate this with the permutation formula.

Apply the idea

Reﬂect and check Permutation formula

Substitute known values

Evaluate the subtraction

Evaluate the factorials

Evaluate the division

Since there are 10 parts and 10 students to hand them out to, we can also calculate this with just a factorial. 10! = 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 3 628 800

b Suppose 7 people enter a marathon. Assuming there are no ties, determine the number of ways a gold, silver, and bronze medal could be awarded.

Create a strategy This problem requires ﬁnding the number of ways to order or arrange 3 objects from 7, so it is a permutation problem. The order matters in this problem because a particular runner getting the gold medal is a different outcome than that same runner getting the silver medal. So, the order in which they get the medals matters, making it a permutation problem rather than a combination problem.

Apply the idea The formula 7 P3 can be used to represent the permutation of 7 people taking 3 different medals (gold, silver, and bronze). Substitution into formula

Expand factorials

Simplify since

Evaluate the multiplication

There are 210 ways that the gold, silver and bronze medals can be awarded in a marathon with 7 people.

Reﬂect and check Alternatively, this problem could have been solved using the counting principle. If there are three medals to win, 7 people could win gold, then there are 6 people left to win silver, then there are 5 people left to win bronze. n(gold) ⋅ n(silver) ⋅ n(bronze) = 7 ⋅ 6 ⋅ 5 = 210

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Example 8 Evaluate each expression: a

12 P8

Create a strategy Use the built-in permutation function in the GeoGebra scientific calculator. Begin by typing the letters n, p, and r. Then, select the npr (n,r) option.

Apply the idea In this expression, n = 12 and r = 8. So, type 12, comma, 8 into the parentheses of the built-in function. Then, press enter.

This shows 12P8 = 19 958 400.

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Reflect and check Instead, we could have used the permutation formula,

Permutation formula

Substitute known values

Evaluate the subtraction

Expand and divide out common factors

Evaluate the multiplication

14 C5

Create a strategy Use the combination formula

Apply the idea

Substitute known values

Evaluate the subtraction

Expand and divide out common factors

Simplify since 4 ⋅ 3 = 11 and 5 ⋅ 2 = 10

Evaluate the multiplication

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Reflect and check We can verify our answer using technology. In the GeoGebra scientific calculator, we will use the built-in function nCr (Number n, Number r). Using n = 14 and r = 5, we can verify there are 2002 combinations.

Example 9 4 letters are chosen at random from the word TRAMPOLINE. Find the number of ways to choose the letters such that the selection includes exactly 2 vowels.

Create a strategy In order for the selection to contain exactly 2 vowels, 2 letters will be vowels and the remaining 2 will be consonants. To find the number of ways this can happen, we need to: 1. Find the number of vowels in TRAMPOLINE 2. Find the number of ways to choose 2 vowels from the total number of vowels 3. Find the number of consonants in TRAMPOLINE 4. Find the number of ways to choose 2 consonants from the total number of consonants The order in which we choose the letters does not matter, so we are finding combinations. Finally, we will use the fundamental counting principle to find the total number of combinations possible.

Apply the idea The word TRAMPOLINE contains 4 vowels and 6 consonants. 4C2 = 6

Number of ways to choose two letters from the vowels

6C2 = 15

Number of ways to choose two letters from the consonants

Now, we will use the fundamental counting principle to ﬁnd the number of ways to choose 4 letters when 2 of them are vowels: 4C2 ⋅ 6C2 = 6 ⋅ 15

= 90

Evaluate the multiplication

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Reﬂect and check We can extend this question by determining the probability of choosing 4 letters such that 2 letters are vowels. To do this, we need to ﬁnd the total number of ways to choose 4 letters from the 10 letters in TRAMPOLINE: 10C4 = 210

Now, we can divide the total number of favorable outcomes by the total number of possible outcomes: Probability formula

Simplify the fraction

Therefore, the probability that the selection includes 2 vowels is .

Example 10 A box contains 6 pens of different colors: red, green, blue, yellow, black and white. Three pens are drawn at random without replacement. a Determine the total number of possible selections.

Create a strategy Because the order in which we choose the pens does not matter, we can use the combination formula.

Apply the idea There are 6 total pens, so n = 6. We need to choose 3 of them, so r = 3. State the combination formula

Substitute known values

Expand the factorials

Simplify since

Evaluate the multiplication and division

There are 20 possible ways to choose the pens.

b Determine the probability of choosing the green and black pens.

Create a strategy

Apply the idea

Because the order in which we choose the pens does not The probability of drawing a green, black, and red pen matter, drawing green then black then red is the same as together is . drawing red then black then green, and similar with the other combinations. So, drawing these 3 colors is one possible combination out of all the possible combinations found above.

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c Suppose the three pens are being placed in a specific order as they are chosen. How many different arrangements of 3 pens can be made?

Create a strategy

Apply the idea

Since the pens are being arranged in a specific order, we now use the permutation formula. The number of ways to arrange 3 pens is calculated by the permutation formula.

State the permutation formula Substitute the values

There are 120 possible arrangements. d Compare the answers to parts (a) and (c).

Create a strategy In part (a), we calculated the total number of ways to select 3 pens from 6, without considering their order, using the combination formula. In part (c), we calculated the total number of ways to arrange those 3 pens in a specific order, using the permutation formula.

Apply the idea

Reflect and check

The answer from part (a) is 20 combinations, since order does not matter. The answer from part (c) is 120 arrangements, since order matters. The number of permutations (arrangements) is greater than the number of combinations because permutations consider different orders as distinct, whereas combinations do not.

In general, the number of permutations is always greater than or equal to the number of combinations, as permutations account for more possibilities by considering different orders of the same items.

Idea summary We use combinations to ﬁnd the number of ways r objects can be chosen from n objects when the order in which we choose them does not matter.

the total number of objects in the set

the number of objects that are being chosen

Practice What do you remember? 1

Compare and contrast combinations and permutations.

Evaluate each expression: a

(11 − 8)!

3!7!

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Fill in the blank to make each equation true. a

Angie has 3 tops, 2 bottoms, and 2 types of shoes. T-shirt

Boots

Shorts

Sneakers Boots

Skirts

Sneakers Boots

Shorts OUTFITS

Sneakers Boots

Polo Shirt Skirts

Sneakers Boots

Shorts

Sneakers Boots

Sweater Skirts

Sneakers

Find the number of different outﬁts Angie can create. 5

Is each statement true or false? a

Order does not matter for combinations.

Combinations are the same as permutations.

c d

is the same as nCr. When ﬁnding the number of ways r objects can be selected from a set of n distinct objects, a permutation will result in more outcomes than a combination.

In each situation, does the order in which the objects are selected matter? a

A local pizza place is offering a special on a large pizza with 3 toppings of your choice. They have a total of 9 pizza toppings to choose from.

Members need to elect a captain and a vice-captain for their sports team. There are 14 players on the team.

There are 52 runners competing in a marathon. The top 5 runners will receive prize money based on their ﬁnishing position.

A group is selling t-shirts for a fundraiser. The t-shirts will all say the same thing, but they can choose 3 different colors for the shirts. The company that makes the t-shirts has 15 different color options to choose from.

Write using nCr or nPr notation: a

A team of ﬁve players is chosen from a squad of 12 players.

Five people are arranged in a line for an interview.

Eight chocolates are chosen from a box of 24 ﬂavors.

Locker codes are created from selecting three letters from the alphabet.

The code to open a locker is called a combination. Explain why this name is misleading.

Verify that the value of each combination or permutation is correct. a

672

5C3 = 10

Mathspace Virginia SOL Algebra 2 mathspace.co

7P2 = 42

8C4 = 70

Let’s practice 10

Use technology to evaluate each permutation: a

8P6

4P0

6P6

7C 4

9C1

10C5

7P3 ⬚ 7C3

3C3 ⬚ 3P3

12C1 ⬚ 12P12

Use technology to evaluate each combination: a

9P3

8C2

Fill in the blank with <, >, or = a

10C2 ⬚ 10 C8

What is the number of possible permutations of 12 objects taken 5 at a time?

The number of combinations of 9 objects taken 7 at a time is: A

181 440

720

For each of the following scenarios: i

Compare and contrast combinations and permutations to determine which is the best method to solve the problem.

Find the answer.

A student council election has 5 candidates running for president, vice president, and treasurer. How many different ways can the 3 positions be filled? Should you use a combination or a permutation?

A teacher wants to select 4 students from a class of 12 to participate in a group project. The order in which the students are selected doesn’t matter. How many different ways can the teacher choose the students? Should you use a combination or a permutation?

There are 10 different books on a shelf. You want to select 3 of them to read, but you care about the order in which you read them. How many different reading orders are possible? Should you use a combination or a permutation?

In a bicycle race, a quinella is a bet on the ﬁrst 2 bicycles that ﬁnish the race, but the order in which these 2 bicycles place does not matter. How many different quinella bets are possible for a bicycle race where 14 bicycles are competing?

A school baseball team has 12 players, but a coach can only choose 9 players for the batting lineup. The order in which the players bat is important. How many batting lineups are possible?

A university has 5 flagpoles and 8 different flags. If the order of the flagpoles does matter, in how many ways can the flags be chosen for the 5 flagpoles?

For each of the following problems: i

Identify it as a permutation, combination, or counting principle problem.

Solve the problem.

A book store offers a discount when you choose of 6 books from a list of 20. In how many ways can a shopper choose their books?

A car company sells a particular car in 10 different colors and in 3 different styles: hatchback, wagon or sedan. In how many ways can the car be designed?

In the qualifying round of a car race, 8 cars are given random starting positions. In how many different ways can starting positions be allocated?

A corporation has 12 members on its board of directors. In how many ways can it elect a president, vice-president, secretary and treasurer?

A popular brand of pen is available in 3 colors (red, green or blue) and 4 tips (0.3 mm, 0.5 mm and 0.7 mm). How many different choices of pens do you have with this brand?

A variety pack of chocolate consists of seven bars, each with a different ﬂavor. If three bars of chocolate are chosen at random, how many different selections are possible? 7.08 Permutations and combinations mathspace.co

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A class of 28 students visit a bowling alley and must organize themselves into groups of 8 to use a lane. How many different ways are there for a group of 8 students to be formed?

If a playlist contains 9 songs, in how many different ways could all 9 songs be played, with no song being repeated?

Five cards have different letters written on them. The letters are M, L, I, E, S. The cards are shuﬄed and laid out on a table with the letters face up next to one another. How many possible arrangements are there?

A manager wants to select 5 people from a group of 25 assistants to help with a speciﬁc project. a

In how many ways can the manager choose 5 people?

If the manager needs to ﬁll 5 speciﬁc roles within the project, in how many ways can the roles in the project be assigned?

Which way of selecting the 5 people gives the manager fewer options?

A bank requires its customers to create a 4-digit pin for their debit card. They can use the digits 0 to 9, with repetitions allowed. How many different debit cards can the bank issue if each card must have a different pin?

In a game of soccer, each team must have 1 goalkeeper and 10 other players on the ﬁeld. If a soccer team consists of 17 players, 2 of which are goalkeepers. In how many ways can players be chosen to start the game?

Motorists can create personalized license plates consisting of 6 characters, and they can be arranged in any way the motorist chooses. Tina wants to use her three favorite letters L, Y and X, and her three favorite digits, 6, 9 and 1.

How many different license plates can she create if the letters and digits can appear in any particular order (and no characters can be repeated)?

How many different license plates can she create if the three letters are followed by the three digits (and no characters can be repeated)?

A city has to accommodate for 9.96 million households requiring a telephone number. They are about to issue 8-digit telephone numbers, with 5 as the ﬁrst digit. a

How many different telephone numbers can be created if repetition of digits is not allowed?

How many different telephone numbers can be created if repetition of digits is allowed?

If each household is allowed one telephone number where digits can be repeated, will there be enough telephone numbers for the number of households?

Let’s extend our thinking 28

A newspaper editor is deciding which of 5 articles to print on the front page. a

If she can only choose 2 of them for the front page, how many different selections are possible?

If Jack wrote one of the 5 articles, ﬁnd the probability that his article is chosen for the front page.

In a horse race, a trifecta is the name given to predicting the ﬁrst three horses in their correct order. In a horse race in which there are 10 horses with equal chances of winning, what is the probability that Michael picks the trifecta?

3 letters of the word ENVIOUS are chosen at random. Find the probability that the selection includes just 1 vowel.

Amelia is playing a game of cards. She will randomly draw 5 cards from a standard deck of 52 cards, and wants only 3 of the cards to have clubs on them. How many different selections would contain the cards she wants?

A selection of 4 people are to be chosen from a group of 7 people. How many selections are possible if the youngest or oldest is included but not both?

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Mathspace Virginia SOL Algebra 2 mathspace.co

How many integers, greater than 999 but not greater than 3000, can be formed with the digits 0, 1, 2, and 3, without repetitions? Justify your answer.

In 2021, the license plates rules and populations from three different states are as follows: • California, which has a population of 39 466 917, uses the seven-character format: 1ABC234 where numbers and letters may repeat. • Delaware, which has a population of 994 735, uses the six number format: 123 456 where numbers may repeat. • New York, which has a population of 20 400 000, use the seven-character format: ABC-1234 where numbers and letters may repeat. Compare the number of possible license plates to the state’s population to determine whether the format they are using is suﬃcient. Justify your answer with work.

Show that each identity is true for any whole numbers, r and n, where 0 ≤ r ≤ n. a

nCn = 1

n Cr = n Cn − r

n + 1 Cr = n Cr + n Cr − 1

Write an inequality relating nPr and nCr where r and n are whole numbers and 0 ≤ r ≤ n. Justify your solution.

7.08 Permutations and combinations mathspace.co

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