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Presents
STAAR ALGEBRA I
BOOT CAMP
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Texas 1st Edition
Table of Contents Chapter 1: STAAR Algebra I Overview ....................................................... 7
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Chapter 2: Introductory Algebra .................................................................. 11 Introductory Algebra Overview .......................................................................... 12 Mini-Test One ..............................................................................................................14 Plug in Points on a Graph .......................................................................................17 Distributive Property: Show Your Work .......................................................... 19 Mini-Test Two ..............................................................................................................21 Word Problem Translation ...................................................................................23 Substitution .................................................................................................................. 24 Mini-Test Three .......................................................................................................... 25 Negative Paranoia ..................................................................................................... 27 Process of Elimination ............................................................................................ 28 Mini-Test Four ............................................................................................................30 Equations of Lines ..................................................................................................... 32 Try Numbers .................................................................................................................. 35 Mini-Test Five ..............................................................................................................36 Mini-Test Explanations ...........................................................................................38 Chapter 3: Functions ................................................................................................47 Functions Overview .................................................................................................. 48 Mini-Test One .............................................................................................................49 Domain & Range Boot Camp ................................................................................ 52 Translations & Reflections .................................................................................. 54 Mini-Test Two ............................................................................................................. 56 Process of Elimination ............................................................................................. 59 Use the Answer Choices ........................................................................................ 60
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Mini-Test Three ...........................................................................................................61 Plug It In ........................................................................................................................ 63 Don’t Overthink It .................................................................................................... 64 Mini-Test Four ............................................................................................................ 65 Read the Question .....................................................................................................67 Draw It Out ................................................................................................................. 68 Mini-Test Five ..............................................................................................................69 Create a Visual ............................................................................................................71 Mini-Test Six ................................................................................................................72 Mini-Test Explanations ........................................................................................... 74
Wrap-Up ...............................................................................................................................81
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Further Practice .............................................................................................................85 Practice Set One .......................................................................................................86 Practice Set Two ...................................................................................................... 90 Practice Set Three ................................................................................................... 94 Practice Set Explanations .................................................................................... 97
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Fourteen years ago, I made perfect scores on the ACT and the SAT. Since then, I’ve helped thousands of students improve their scores on the most popular standardized tests, including ACT, SAT, and WorkKeys. I’ve helped students who aren’t “good test takers” get the scores they need for the school, scholarship, or job they’re interested in.
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This Boot Camp was written to help you. If I were tutoring you personally, you might ask, “I’m taking my STAAR Algebra I test in two days. What do I need to know?” The information in this workshop is exactly what I would give you. This program is designed to give you the ultimate one-day cram. With this Boot Camp, you can learn the skills that you need to boost your scores on the STAAR Algebra I test. Most students I work with do best under pressure. You can call it simply “waiting until the last minute,” but I think there is a human instinct that kicks in, a survival mechanism of sorts, that clears your head and helps you do what needs to get done. This workshop is for the “final hours.” It’s here because, sometimes, the best prep comes just before the test begins. This Boot Camp will guide you through the essentials on content, test-taking strategies, and the most common question types on the STAAR Algebra I test. At MasteryPrep we’ve done research based on what students most need to do well on an STAAR Algebra I test and have distilled that information down into this workshop. It’s designed to give you everything you need—and nothing that you don’t—quickly enough that you can cover it all in a single day.
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Good luck!
Craig Gehring
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WORKKEYS OVERVIEW
A Note from the Author
Fill in the times following your instructor’s directions. This is the agenda we will follow throughout the day. There will be breaks throughout the session. Next to each section name in the schedule, you’ll find the corresponding page number where it begins in this workbook.
Section
Page Number
STAAR Algebra I Overview
7
Introductory Algebra
11
Functions
47
Boot Camp Wrap-Up
81
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Time
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WORKKEYS OVERVIEW
Schedule
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Chapter 1
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STAAR Algebra I Overview
What Is End-of-Course Testing? End-of-Course (EOC) testing measures your aptitude in a given subject after you have finished a course. Consider it a subject understanding checkup. Teachers use it to identify both your strengths and areas where improvement is needed. This helps ensure you are on track in developing the knowledge and skills needed for the next grade and, eventually, college and a career.
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Why Should You Care?
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In this Boot Camp, we’ll focus on the two most common concepts in Algebra I: Introductory Algebra and Functions. Your understanding of each of these concepts will help you pass the STAAR Algebra I test.
•
If your school uses the STAAR Algebra I test as a final exam for the course, then doing well on this test can boost your GPA.
•
A good STAAR Algebra I test score is a positive indicator that you are on track for college.
•
Mastering the foundational skills taught in this Boot Camp will help you succeed in more difficult math courses in the future.
•
Put in the effort now and save yourself from repeating a course or taking summer school.
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STAAR ALGEBRA I OVERVIEW
STAAR Algebra I Overview
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STAAR Algebra I Boot Camp
Orientation The two most common conceptual categories tested on STAAR Algebra I are Introductory Algebra and Functions. Here is the frequency breakdown from the most recent tests: Introductory Algebra makes up 44–50% of the test. Functions makes up 37–44% of the test.
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Number & Quantity, Geometry, and Statistics & Probability combined make up the remaining 13–14% of the test.
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Here is a breakdown of possible ways the two main categories will be covered on the test. Introductory Algebra tests equations, expressions, and inequalities in the following ways: •
Evaluate problems with one or two variables.
•
Create models to describe real-life situations and relationships.
•
Understand and apply basic mathematical principles.
Functions measures your ability to interpret, understand, and build functions.
The STAAR Algebra I test can be taken either on a computer or on paper. You will have up to 4 hours to complete the test. You can use a graphing calculator the entire time.
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There are 49 multiple-choice questions on the test. There are 5 grid-in questions on the test.
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STAAR ALGEBRA I OVERVIEW
STAAR Algebra I Overview
About This Boot Camp This is a one-day event, preparing you for the most important questions on the STAAR Algebra I test. This Boot Camp is not meant to be the only form of preparation for the STAAR Algebra I test. You should also find other practice tests online and ask your teachers for help and for other resources that specifically target the skills needed to do well on this test.
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This book contains key strategies for taking the test, instructional content, and mini-tests that give you practice with the type of questions you’ll see on test day (plus an explanation for how to solve every question). This book has plenty of places for you to take notes, and we highlight the most important strategies to use so you can continue practicing on your own.
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This Boot Camp will go by fast! Be ready, take notes, and stay focused!
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STAAR ALGEBRA I OVERVIEW
STAAR Algebra I Overview
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Chapter 2
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Introductory Algebra
Introductory Algebra:
Overview
Introductory Algebra Overview The Introductory Algebra conceptual category tests your proficiency over a broad range of algebra skills. The
Interpret the structure of expressions.
•
Write expressions in equivalent forms to solve problems.
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Seeing Structure in Expressions
Arithmetic with Polynomials and Rational Expressions •
Perform arithmetic operations on polynomials.
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Understand the relationship between zeros and factors of polynomials.
Creating Equations •
Create equations that describe numbers or relationships.
Reasoning with Equations and Inequalities •
Understand solving equations as a process of reasoning and explain the reasoning.
•
Solve equations and inequalities in one variable.
•
Solve systems of equations.
•
Represent and solve equations and inequalities graphically.
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INTRODUCTORY ALGEBRA
skills that will be tested on your exam include but are not limited to the following:
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STAAR Algebra I Boot Camp
Introductory Algebra:
Overview
What Are Boot Camp Mini-Tests? During this Boot Camp you will take several mini-tests, which are small segments of an Algebra I test. While taking these mini-tests, it’s important to imagine that you are in an actual testing environment. The time limits assigned
For these mini-tests, you have 8 minutes to answer 5 questions. Your instructor will signal when you are out of time. Try to get through all the questions within the time limit. Unless your instructor has provided you with an answer sheet, circle your answers directly in this book. The real test does not allow the use of cell phones,
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watches, or computers, so you shouldn’t use them on the mini-tests either.
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INTRODUCTORY ALGEBRA
as you complete the mini-tests.
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match the pace that you should try to keep during the actual test. Practice all of the skills that you have learned
Introductory Algebra - Mini-Test One 1.
A.
Which graph is a solution to 2x – 7y > 21? y
y
12 10 8
12 10 8
B.
6
6
4
4
2 2 4
6 8 10 12
–12 –10 –8 –6 –4 –2 –2
–4
2 4
6 8 10 12
–4
–6 –8
–6 –8
–10
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–10
C.
–12
–12
y
y
12 10 8
12 10 8
D.
6
6
4
4
2
–12 –10 –8 –6 –4 –2 –2
x
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–12 –10 –8 –6 –4 –2 –2
2 x
2
2 4
6 8 10 12
x
–12 –10 –8 –6 –4 –2 –2
–4
–6 –8
–6 –8
–10
–10
–12
–12
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–4
Introductory Algebra - Mini-Test One
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2 4
6 8 10 12
x
GO ON TO THE NEXT PAGE.
Use the steps in the table to answer the question. initial equation
4(x – 2)2 – 2x + x = –33x – 4x
step 1
4(x – 2)2 – x = –37x
step 2
4(x – 2)2 = –36x
step 3
(x – 2)2 = –9x
step 4
x2 – 4x + 4 = –9x
step 5
x2 + 5x + 4 = 0
The table shows the first 5 steps used to solve an equation.
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2.
Which statement is an incorrect explanation of one step in the process?
B. C. D.
3.
From step 1, apply the multiplication property of equality to –x and –37x to get 4(x – 2)2 = –36x in step 2.
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A.
From step 2, apply the division property of equality to 4(x – 2)2 and –36x to get (x – 2)2 = –9x in step 3. From step 3, apply the distributive property to (x – 2)2 to get x 2 – 4x + 4 in step 4.
From step 4, apply the subtraction property of equality to x 2 – 4x + 4 and –9x to get x 2 + 5x + 4 = 0.
A grocery store purchases crates of oranges. •
Each crate contains 75 oranges.
•
Each crate costs $60.
How much does the grocery store have to charge for each orange to make a profit of $30 per crate? A. $0.75
B. $0.83
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C. $0.90 D. $1.20
4.
Which expression is equivalent to k 2 – 16?
A. (k – 8)(k + 2)
B. (k – 8)(k – 2)
C. (k + 4)(k – 4) D. (k – 4)(k – 4)
Introductory Algebra - Mini-Test One
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GO ON TO THE NEXT PAGE.
5.
Which expression is equivalent to (x – 2)(3x 2 – 5x + 9)? A. 3x 2 – 4x + 7 B. 3x 2 – 4x + 9 C. 3x 3 – 5x 2 + 9x – 2
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D. 3x 3 – 11x 2 + 19x – 18
Introductory Algebra - Mini-Test One
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Introductory Algebra:
Plug in Points on a Graph
Plug in Points on a Graph If you’re having trouble graphing a line on the coordinate plane, try plugging in points. This is a quick and easy method for solving graph problems. If you forget what each number in the equation of a
Let’s take a look at how plugging in points can help you solve a problem on your test.
A.
Which graph is a solution to 2x – 7y > 21?
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1.
y
y
12 10 8
12 10 8
B.
6
6
4
4
2
2
–12 –10 –8 –6 –4 –2 –2
2 4
6 8 10 12
–12 –10 –8 –6 –4 –2 –2
–4
–4
–6 –8
–6 –8
–10
–10
–12
–12
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x
y
y
12 10 8
12 10 8
D.
6
6
4
4
2
2
–12 –10 –8 –6 –4 –2 –2
2 4
6 8 10 12
x
–12 –10 –8 –6 –4 –2 –2
–4
–4
–6 –8
–6 –8
–10
–10
–12
–12
2 4
6 8 10 12
2 4
6 8 10 12
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INTRODUCTORY ALGEBRA
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line represents, then plug in values for x and y to find coordinate points on the line.
Introductory Algebra:
Plug in Points on a Graph
First, simplify the inequality. 2x – 7y > 21 – 7y > –2x + 21 y<
2 x–3 7
Now, plug in a value for x to find a set of coordinates on the graph of the inequality.
2 (0) – 3 7
y < –3
Therefore, the point (0,–3) is on the graph of the inequality. Since y < –3, all the values of y below this point should
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be shaded. This makes choice B the only correct answer.
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INTRODUCTORY ALGEBRA
y<
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Let’s try x = 0.
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Introductory Algebra:
Distributive Property: Show Your Work
Distributive Property: Show Your Work Use the distributive property when you need to get rid of parentheses.
With the distributive property, you actually distribute (share) the coefficient outside of the terms in parentheses among each of the individual elements inside the parentheses. Share the wealth!
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For example: 3(2x + 5) = (3)(2x) + (3)(5) (3)(2x) + (3)(5) = 6x + 15
Be cautious about keeping track of the negative signs. –4(x – 4) = (–4)(x) – (–4)(4)
(–4)(x) – (–4)(4) = –4x – (–16) –4x – (–16) = –4x + 16
It is important to show your work throughout the entire process of solving these problems. If you try to rush and
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skip steps, you increase your chance of making a careless error.
Let’s take a look at this tricky question:
5.
Which expression is equivalent to (x – 2)(3x 2 – 5x + 9)? A. 3x 2 – 4x + 7 B. 3x 2 – 4x + 9
C. 3x 3 – 5x 2 + 9x – 2
D. 3x 3 – 11x 2 + 19x – 18
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INTRODUCTORY ALGEBRA
parentheses, this is a clue that you’ll need to use this method.
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When you are asked to simplify an expression that has parentheses and none of the answer choices have
Introductory Algebra:
Distributive Property: Show Your Work
(x – 2)(3x2 – 5x + 9) First, distribute the x to each term in the second polynomial, then distribute the –2 to every term in the second polynomial. 3x3 – 5x2 + 9x – 6x2 + 10x – 18 3x3 – 5x2 – 6x2 + 9x + 10x – 18
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By working one step at a time, you can avoid the mistakes that cause inaccuracies on your exam.
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INTRODUCTORY ALGEBRA
3x3 – 11x2 + 19x – 18
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Combine like terms.
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Introductory Algebra - Mini-Test Two 1.
Which statements about the line shown in the graph below are true? y 12 10 8 6 4 2
–4 –6 –8
6 8 10 12
x
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2 4
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–12 –10 –8 –6 –4 –2 –2
–12
Select all that apply. A. B. C. D. E. F.
The point (–30,8) is on the line. The point (0,2) is on the line.
The point (2,0) is on the line.
The graph represents the equation –x + 5y = 10. The graph represents the equation x – 5y = 10.
Seven times Hector’s age minus two times Sandra’s age equals 5. Sandra’s age is also three times Hector’s
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2.
The point (30,8) is on the line.
age. How old is Sandra?
3.
An artist spends d days expanding a mural. The existing mural is 6 feet long. Each day she adds 1.5 feet of the expansion. Which equation models the total length (L) of the mural over time? A.
d = 1.5L + 6
B.
d = 1.5L – 6
C.
L = 1.5d + 6
D.
L = 1.5d – 6
Introductory Algebra - Mini-Test Two
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5.
Jordan drove a distance represented by the equation 3x + 2. Omar drove a distance represented by the expression 18x + 12. Which of the following describes how the distance Omar drove compares to the distance Jordan drove? A.
The distance Omar drove is 3 times the distance Jordan drove.
B.
The distance Omar drove is 4 times the distance Jordan drove.
C.
The distance Omar drove is 5 times the distance Jordan drove.
D.
The distance Omar drove is 6 times the distance Jordan drove.
The formula for the area of a trapezoid is A = bases, and h is the height.
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4.
b1 + b2 h, where A is the area, b 1 and b 2 are the lengths of the 2
What is the area of the trapezoid below?
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b1 = x + 9
h=6
b2 = x – 1
A. B. C.
A = 6x + 24
A = 3x 2 + 8x – 9
A = 3x 2 + 24x – 27
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D.
A = 6x + 4
Introductory Algebra - Mini-Test Two
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Introductory Algebra:
Word Problem Translation
Word Problem Translation The secret to solving a word problem is translating it into math.
is, equal to, is the same as → =
times, product, each, per, of → •
minus, without, less, difference, change → –
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plus, together, and, combined, both → +
divided into, split between or among, divvied up → ÷
Let’s take a look at how word problem translation can help you solve a problem on your exam.
2.
Seven times Hector’s age minus two times Sandra’s age equals 5. Sandra’s age is also three times Hector’s age. How old is Sandra?
Translate the sentences into algebraic equations. Let variables represent the unknown ages and use the translations above to determine what math operations to use.
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7h – 2s = 5 s = 3h
Substitute 3h from the second equation for s in the first equation. 7h – 2(3h) = 5 7h – 6h = 5 h=5
Hector is 5 years old. Substitute 5 for h in the second equation. s = 3(5) = 15 Sandra is 15 years old. Remember to translate words into math! Only a one-two punch like this can knock out a word problem.
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operations.
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When translating word problems to algebraic equations, it is important to know which words translate to which
Introductory Algebra:
Substitution
Substitution Freezing up on a challenging question is a common problem among students. When you come across a scary-looking question and don’t know where to begin, look for a formula. If you find a formula, see if there are any values to substitute into it. Use this as a starting point for figuring out the question.
INTRODUCTORY ALGEBRA
figure into the formula provided.
5.
The formula for the area of a trapezoid is A = of the bases, and h is the height.
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For example, consider how this intimidating problem becomes manageable if you substitute values from the
b1 + b2 h, where A is the area, b 1 and b 2 are the lengths 2
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What is the area of the trapezoid below?
b1 = x + 9
h=6
b2 = x – 1
A.
A = 6x + 4
B.
A = 6x + 24
C.
A = 3x 2 + 8x – 9
D.
A = 3x 2 + 24x – 27
The formula for area is given in terms of b1, b2, and h. Scanning the problem, you’ll find these terms are also on the
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trapezoid figure. Plug their values into the formula to see if this helps you make progress on the problem. A=
A=
( b1 + b2 ) 2
h
( x + 9) + ( x − 1)
A = (6)
2
(6)
2 x + 8 2
A = 6(x + 4) A = 6x + 24
Sometimes, this method is just a starting point for understanding what a question is asking. In this case, it leads you directly to the answer to the problem!
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Introductory Algebra - Mini-Test Three 1.
What is the solution to –3(6t + 4) – 6t ≥ –11t – (17t + 4)? A. t ≤ 2 t≥2
C.
t≤4
D.
t≥4
Kedrick used the method of completing the square to solve a quadratic equation. His first two steps are shown below.
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2.
B.
Given: 3x2 + 30x + 21 = 0 Step 1: x2 + 10x + 7 = 0
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Step 2: x2 + 10x = –7
Write numbers in each box to correctly complete the square in Step 3. Step 3: x2 + 10x +
Margot, the manager of a coffee shop, is planning to order aprons for each of her employees. There are 25 employees at the shop. The baristas request aprons with pockets. Margot determines that $200 is the maximum amount that can be spent on all of the aprons. Margot finds the following prices for aprons at a restaurant supply store. Option
Features
Cost per Apron
A
2 pockets
$6.50
B
4 pockets
$8.99
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3.
=
Using a to represent the number of aprons with 2 pockets and b to represent the number of aprons with 4 pockets, which of the following inequalities represents the constraint on the number of aprons that can be purchased based on the amount of money available to spend on aprons? A. 6.50a + 8.99b ≤ 200 B. 6.50a + 8.99b ≥ 200
C. 2a + 4b > 200
D. 2a + 4b < 200
Introductory Algebra - Mini-Test Three
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4.
Tasha deposited m money into a savings account y years ago. Now she is going to use some of the money in the savings account to buy a laptop. The following expression can be used to determine the percentage of the money in savings that Tasha will use for the laptop. 299 m (1.25 )
y
× 100
What is the meaning of the denominator in the expression? The amount in Tasha’s savings account after the laptop purchase
B.
The yearly interest rate for the savings account
C.
The amount Tasha will pay for the laptop
D.
The amount in Tasha’s savings account now
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5.
A.
Which graph shows the zeros for the function g(x) = 2x 2 – 13x – 7?
y
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A.
30
B.
24
24
18
18
12
12
6
6
–30 –24 –18 –12 –6
6
12 18 24 30
x
–30 –24 –18 –12 –6
–6
–6
–12
–12
–18
–18
–24
–24
–30
–30
y
y
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D.
24
18
12
12
6
6
6
12 18 24 30
x
–30 –24 –18 –12 –6
–6
–6
–12
–12
–18
–18
–24
–24
–30
–30
Introductory Algebra - Mini-Test Three
6
12 18 24 30
x
24
18
–30 –24 –18 –12 –6
12 18 24 30
30
30
C.
6
26
x
STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Introductory Algebra:
Negative Paranoia
Negative Paranoia When solving problems loaded with negative signs, let the paranoia sink in. This test uses a lot of negative signs to trip you up. When you face a problem like this, you need to pay more attention to every calculation that involves a negative sign. It is likely that the trap answers are meant to catch you if mess up a calculation with a negative sign.
What is the solution to –3(6t + 4) – 6t ≥ –11t – (17t + 4)?
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A. t ≤ 2 B.
t≥2
C.
t≤4
D.
t≥4
Simplify both sides of the inequality and solve for t. Don’t forget to be paranoid about the negative signs! –3(6t + 4) – 6t ≥ –11t – (17t + 4) –18t – 12 – 6t ≥ –11t – 17t – 4 –24t – 12 ≥ –28t – 4 28t – 24t ≥ 12 – 4
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4t ≥ 8 t≥2
Notice that if you forget to distribute the negative sign to the 4 at the end of the inequality, you will ultimately arrive at choice D, t ≥ 4. Don’t fall into these traps! Keep a close eye on all negative signs.
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INTRODUCTORY ALGEBRA
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Let’s look at an example of when you should feel the negative paranoia:
Introductory Algebra:
Process of Elimination
Process of Elimination When you can’t find the correct answer, figuring out the wrong answers can be just as useful. Knock out answer choices that are unreasonable, that don’t answer the question, that are negative when they need to be positive, etc.
y
y
30 A.
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Which graph shows the zeros for the function g(x) = 2x 2 – 13x – 7?
30
B.
24
24 18
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12
12
6
6
–30 –24 –18 –12 –6
6
12 18 24 30
x
–30 –24 –18 –12 –6
–6
–6
–12
–12
–18
–18
–24
–24
–30
–30
y
y
D.
24
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12 18 24 30
6
12 18 24 30
x
24 18
18
12
12
6
6
–30 –24 –18 –12 –6
6
30
30
C.
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INTRODUCTORY ALGEBRA
5.
6
12 18 24 30
x
–30 –24 –18 –12 –6
–6
–6
–12
–12
–18
–18
–24
–24
–30
–30
STAAR Algebra I Boot Camp
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Introductory Algebra:
Process of Elimination
Factor the function to find the x-intercepts of the graph. g(x) = 2x2 – 13x – 7 g(x) = (2x + 1)(x – 7) x=–
1 ,7 2
Now that you know the x-intercepts of the graph you can eliminate some answer choices. Choices C and D clearly have the wrong x-intercepts. Even if you forget how to find the vertex of a parabola, you have narrowed it down to two answer choices and can make a better guess. It’s always better to guess from two answer choices instead of
To solve the question completely, use the vertex formula to find the vertex of the graph, –28.125. Choice A
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four. Making educated guesses instead of random guesses increases the odds of improving your score.
Introductory Algebra - Mini-Test Four 1.
The graph below represents which of the following system of equations? y 12 10 8 6 4 2
–4 –6 –8
2 4
6 8 10 12
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–10
x
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–12 –10 –8 –6 –4 –2 –2
–12
A. 2x – 2y = 12 x – 3y = –2 B.
2x + 2y = 12
x + 3y = 22
C. x – 7y = –18
9x – 8y = 58
D. x + 7y = 38
Select the box that correctly identifies the best description of the solution or solutions for each equation.
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2.
9x + 8y = 122
x = –3
x=1
No solution
Infinite solutions
3(3x + 5) = 4x + 5x – 8
5(x – 5) + 17 =
1 (10x – 6) – 5 2
–11x – 24 = –11x + 8(–2 – x)
–2 +
1 1 (–10x – 5) = – (9x + 18) 5 3
Introductory Algebra - Mini-Test Four
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3.
What is the largest of 3 consecutive positive integers if the product of the two smaller integers is 8 more than 2 times the largest integer?
A.
–36 meters
B.
–2 meters
C.
6 meters
D.
18 meters
E
5.
The height of a bird after s seconds is given by the expression 6 + 3(s – 2) 2. In the simplified form of the expression, the constant represents the initial height, in meters, of the bird. What is the initial height of the bird?
Kaitlyn organizes performances for a small theater company. She knows if she charges $25 per ticket, around 130 people will buy tickets. For every $1.00 she lowers the ticket price, an additional 5 people will purchase tickets. Kaitlyn wrote an expression showing the total income from ticket sales, where d is the number of dollars by which the ticket price has been reduced.
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4.
(25 – d)(130 + 5d)
Which of the following expressions is equivalent to Kaitlyn’s? A. 5d 2 – 130d + 3,250 B. –5d 2 – 5d + 3,250 C. –5d 2 + 3,125
SA
D. 4d + 155
Introductory Algebra - Mini-Test Four
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Introductory Algebra:
Equations of Lines
Equations of Lines To do well on this test you need to have a good understanding of three common equations of a line: standard form, point-slope form, and slope-intercept form.
Standard Form
E
In standard form, the variables are on the left side of the equation, and A, B, and C are integers.
Point-Slope Form
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The point-slope form of a line is y – y1 = m(x – x1)
Point-slope form is used when you already have the slope and one point on the line and you are looking for another point.
Slope-Intercept Form
The slope-intercept form of a line is y = mx + b
In slope-intercept form, m is equal to the slope of the line and b is the y-intercept.
It is important to be able to switch between these three forms. You never know if the answer choices will be given in standard form, point-slope form, or slope-intercept form.
It is easiest to use the slope-intercept form when generating the equation of a line from a graph, but if the y-intercept isn’t clearly shown, you will need to use the point-slope form.
SA
INTRODUCTORY ALGEBRA
The standard form of a line is Ax + By = C
Let’s look at an example from the test.
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Introductory Algebra:
1.
Equations of Lines
The graph below represents which of the following system of equations? y 12 10 8 6 4 2
–6 –8
6 8 10 12
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–10
2 4
–12
A. 2x – 2y = 12 x – 3y = –2 B.
2x + 2y = 12
x + 3y = 22
C. x – 7y = –18
9x – 8y = 58
D. x + 7y = 38
9x + 8y = 122
Determine the equations of the two lines in the graph and convert to standard form.
SA
The first line travels through the points (–2,0) and (10,4). The second line travels through the points (6,0) and (10,4).
Line 1: Use (–2,0) and (10,4) to find the slope. 4 −0 4 1 = = 10 − ( −2 ) 12 3 1 Use (–2,0) and m = in point-slope form. 3 1 y – 0 = (x – (–2)) 3 m=
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INTRODUCTORY ALGEBRA
–4
x
E
–12 –10 –8 –6 –4 –2 –2
Introductory Algebra: y=
Equations of Lines
1 (x + 2) 3
Convert to standard form. 1 (x + 2) 3
(3)y = (3) 3y = x + 2
m=
4 −0 4 = =1 10 − 6 4
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Use (6,0) and m = 1 in point-slope form. y – 0 = 1(x – 6) y=x–6
Convert to standard form and multiply by 2 to match the only valid answer. x–y=6
2x – 2y = 12
NOTES:
SA
INTRODUCTORY ALGEBRA
Line 2: Use (6,0) and (10,4) to find the slope.
E
x – 3y = –2
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Introductory Algebra:
Try Numbers
Try Numbers Sometimes you have to try numbers that aren’t given in the question or answer choices. You will have to read the question and determine which number should be used in that situation.
A.
–36 meters
B.
–2 meters
C.
6 meters
D.
18 meters
Instead of squaring the binomial (s – 2), you can plug in a number and solve. The question asks for the initial height of the bird, which would be found after 0 seconds have passed. Therefore, plug in 0 for s and simplify the expression.
6 + 3(s – 2)2
6 + 3(0 – 2)2 6 + 3(–2)2
SA
6 + 3(4) = 6 + 12 = 18 meters
NOTES:
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INTRODUCTORY ALGEBRA
The height of a bird after s seconds is given by the expression 6 + 3(s – 2) 2. In the simplified form of the expression, the constant represents the initial height, in meters, of the bird. What is the initial height of the bird?
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4.
E
Let’s take a look at a commonly missed word problem and see how we can unlock its answer:
Introductory Algebra - Mini-Test Five 1.
What is the solution to –3(12z + 1) – 8z < –30z – (10z + 11)? 7 2 7 B. z > – 2
A. z < –
C. z < 2
2.
z>2
E
D.
The expression (x – 2)(4x 2 – 5x + 7) is equivalent to which of the following? A. 4x 3 – 13x 2 + 17x – 14 C. D.
3.
4x 3 – 5x 2 – 3x – 2
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B.
4x 2 – 4x + 7 4x 2 – 4x + 5
Which of the following statements correctly describe the line shown on the graph below? y
10
8 6 4 2
–10 –8 –6 –4 –2
2
4
6
8
10
x
SA
–2
–4 –6 –8 –10
Select all that apply. A.
The graph represents the equation 5x – 6y = 30.
B.
The graph represents the equation 5x + 6y = 30.
C.
The point (5,0) is on the line.
D.
The point (0,5) is on the line.
E.
The point (18,10) is on the line.
F.
The point (18,–10) is on the line.
Introductory Algebra - Mini-Test Five
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4.
A runner spends w weeks training for a marathon. Her current training distance is 9 miles. Each week she adds 2.5 miles to her training distance. Which equation models the total miles (m) of her training distance over time? A. w = 2.5m + 9m B.
m = 2.5w + 9
C. w = 2.5m – 9
5.
Which graph is the solution 8x – 6y ≥ 12? y
y
10 A.
10
B.
8
8 6
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6 4
4
2
2
–10 –8 –6 –4 –2
2
4
6
8
10
x
–10 –8 –6 –4 –2
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
y
y
10
10
8 6
4
2
2
2
4
6
8
10
x
–10 –8 –6 –4 –2
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
Introductory Algebra - Mini-Test Five
37
4
6
8
10
2
4
6
8
10
x
6
4
–10 –8 –6 –4 –2
2
8
D.
SA
C.
E
D. m = 2.5w – 9
x
STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Introductory Algebra:
Mini-Test Explanations
Mini-Test Explanations MINI-TEST ONE
–7y > –2x + 21 2 x–3 7
y<
The line represented by the linear inequality has a positive slope, so choices A and C are incorrect. The shaded region should be below the line represented by the linear inequality, so choice D is incorrect. The graph in choice B is the correct representation of the inequality.
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2. The correct answer is A. In step 1, the addition property of equality, not the multiplication property of equality, is used to add x to both sides of the equation and cancel the left-side –x term. 3. The correct answer is D. To make $30 profit, a crate of oranges must be sold for 60 + 30 = $90. Divide $90 by the number of oranges in the crate.
90 = 1.2 75
Each orange must be sold for $1.20 to make a profit of $30 per crate.
4. The correct answer is C. The expression provided is a difference of squares with a2 – b2 = (a + b)(a – b). Therefore, the given expression can be factored to the following:
k2 – 16
k2 – 42
(k + 4)(k – 4)
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INTRODUCTORY ALGEBRA
2x – 7y > 21
E
1. The correct answer is B. Solve the inequality for y. Remember to flip the inequality when dividing by a negative number.
5. The correct answer is D. Distribute both terms in the first parentheses to all terms in the second parentheses and combine like terms.
(x – 2)(3x2 – 5x + 9)
3x3 – 5x2 + 9x – 6x2 + 10x – 18
3x3 – 5x2 – 6x2 + 9x + 10x – 18 3x3 – 11x2 + 19x – 18
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