QUADRATIC FUNCTIONS What is the name of the quadratic function shown in the graph?
Can you identify its attributes?
QUADRATIC FUNCTIONS DOMAIN – The set of possible input values. RANGE – The set of possible output values.
2 TYPES OF NOTATION DOMAIN
RANGE
INTERVAL
INEQUALITY
The height of a ball thrown up from a platform is shown versus time on the graph. What is the domain and range of this function? Domain
Range
The graph of a quadratic function is shown to the right. Complete the list of attributes: Domain Range
X-intercepts Vertex
Y-intercept Maximum/Minimum
Line of Symmetry
The graph shows the path of a punted football. What is the maximum height that the football reaches?
What is the reasonable domain?
What is the reasonable range?
Write the domain and range of the quadratic equation, y = -x2 , in inequality notation. Domain
Range
What are the domain and range for the quadratic equations shown below? Equation
Interval Notation Domain
Range
y = x2 y = x2 - 4 y = -2x2 y = -3x2 + 2
What are the domain and range of the quadratic function shown in the graph? Domain
Range
Inequality Notation Domain
Range
EFFECTS OF CHANGES
ax2 + bx + c
CONSTANT COEFFICIENT (c) The value of c determines the y-value of the y-intercept. For the graph of y = ax2 + c, the value of c determines the ycoordinate of the vertex of the parabola. If c ≠0, then the value of c determines how far up or down (translates or shifts) along the y-axis as it compares to the parent function, y = x2. When the parabola translates, the shape does not change. If c is positive, the graph shifts (translates) up.
If c is negative, the graph shifts (translates) down.
EFFECTS OF CHANGES
ax2 + bx + c
QUADRATIC COEFFICIENT (a) The value of a determines the width and direction of the parabola. Width As the absolute value of a increases (a > 1), the graph gets narrower (constricts or condenses). As the absolute value of a decreases (0 < a < 1, between 0 & 1), the graph gets wider (expands). Direction If a is positive, the graph opens upward and can be described as minimum. Minimum â&#x20AC;&#x201C; the lowest point on the graph (see the vertex) and the point with the smallest y-value. If a is negative (a < 1), the graph opens downward (reflects) and can be described as maximum. Maximum - the highest point on the graph (see the vertex) and the point with the largest y-value.
How will the graph of y = x2 â&#x20AC;&#x201C; 1 be affected if the quadratic coefficient is changed to 2?
How will the graph of y = 3x2 + 1 be affected if the quadratic coefficient is changed to 2?
How will the graph of y = -2x2 + 2 be affected if the quadratic coefficient is changed to -3? Rearrange the following equations in order from narrowest to widest; y = -2x2, y=
x 2,
y=
3x2,
y=
.5x2,
â&#x2C6;&#x2019;đ?&#x;? 2 and y = x. đ?&#x;&#x2018;
In the graph to the right, how does the graph y = -x2 - 1 compare to the graph of y = x2 - 1?
The graph of the parabola y = 3x2 + 3 is reflected over a line to obtain the graph of y = -3x2 + 3 . What is the line of reflection?
What is the equation of a parabola that is a reflection across the x-axis of the parabola y = 4x2?
If the constant term of the graph y = x2 + 3 is changed to -1, describe the effect.
How would the graph of y = x2 + 5 be changed if the function were changed to y = x2 + 10?
In the graph of the function y = x 2 â&#x20AC;&#x201C; 3, describe the shift in the vertex of the parabola if the -3 is changed to 8. Hints: For y = x2 + c, always subtract the original value of c from the new value of c to find the vertical change. When the graph of a parabola is translated vertically by changing the value of c, the graph moves as a whole. Each point is translated the same distance as the other point.
DEMONSTRATION OF LEARNING A stone was dropped from a rooftop that is 16 feet above the ground. Its height, h, in feet after t seconds can be represented by the function h = -16t2 + 16. Reasonable domain
Reasonable range
For the equation, x2 – 10x – 24, identify the following parameters: X-intercepts
Domain
Y-intercept
Range
Vertex
Minimum or Maximum
Line of Symmetry
Number of Solutions
Joe has a rectangular deck in his backyard. Its length measures 1 foot more than its width. He is planning to extend the length of the deck by 3 additional feet. If the new deck would have an area of 165 feet, what is the width of the deck.
The solutions of a quadratic equation are 3 and -2. Determine the factors and the standard form equation if a = 1.
đ?&#x;&#x2018; The sum of the solutions of a quadratic equation is . The product of its đ?&#x;? solutions is 3. What is the equation?
Kurt has a rectangular patio that is 12 feet long and 10 feet wide. He wants to increase the length and the width so the area of the new patio will be twice that of the original patio. The length will be increased by twice the amount that the width is increased. By how much should he increase each dimension?
Lucy just put a mat around her new Elvis poster. If the matted poster is 19-by-26 inches, and there is 408 ins2 of the poster showing, how wide is the mat?
Extra Graphs