Teacher Resource Sampler
Algebra 2
Common Core
Going beyond the textbook with
Pearson Algebra 2 Common Core Edition provides the teacher with a wealth
of resources to meet the needs of a diverse classroom. From extra practice, to
performance tasks, to activities, games, and puzzles, Pearson is your one-stop
shop for all teaching resources.
The wealth and flexibility of resources will enable you to easily adapt to your
classroom’s changing needs. This sampler takes one lesson from Algebra 2 and
highlights the support available for that lesson and chapter, illustrating the scope
of resources available for the program as a whole, and how they can help you
help your students achieve algebra success!
Inside this sampler you will find:
■ rigorous practice worksheets
■ extension activities
■ intervention and re-teaching resources
■ support for English Language Learners
■ leveled assessments
■ activities and projects
■ standardized test prep
■ additional problems for teaching each lesson
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2
th Pearson Algebra 2
Contents
Student Companion 4
Think About a Plan
8
Practice G9
Practice K11
Standardized Test Prep13
Solve It and Lesson Quiz
14
Additional Problems15
Reteaching17
Additional Vocabulary Support
19
Activity20
Game21
Puzzle23
Enrichment24
Teaching with TI Technology
25
Chapter Quiz29
Chapter Test31
Find the Errors!33
Performance Tasks36
Extra Practice38
Chapter Project42
Cumulative Review46
3
Polynomial Functions
5-1
Vocabulary
Review
1. Write S if the expression is in standard form. Write N if it is not.
5 1 7x 2 13x 2
47y 2 2 2y 2 1
3m2 1 4m
Vocabulary Builder
polynomial
2
polynomial (noun) pahl ah NOH mee ul
3t rt r 3
Related Words: monomial, binomial, trinomial
monomials
Definition: A polynomial is a monomial or the sum of monomials.
2. Circle the polynomial expression(s).
3
2t 4 2 5t 1 t
3x 2 2 5x 1 2
x
7g 3 1 8g 2 2 5
3. Circle the graph(s) that can be represented by a polynomial.
y
y
x
y
x
x
Write the number of terms in each polynomial.
4. 6 2 7x 2 1 3x
4
Chapter 5
5. 4b 5 2 3b4 1 7b 3 1 8b 2 2 b
118
6. 3qr 2 1 q 3r 2 2 q2r 1 7
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Use Your Vocabulary
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You can classify a polynomial by its degree or by its number of terms.
Degree
Name Using
Degree
Polynomial
Example
Number of
Terms
0
constant
5
1
monomial
1
linear
x 4
2
binomial
4x2
1
monomial
4x32x2x
2x4 5x2
3
2
trinomial
binomial
x 54x 2 2x 1
4
polynomial of 4 terms
2
quadratic
3
cubic
4
quartic
quintic
5
Name Using
Number of Terms
Problem 1 Classifying Polynomials
Got It? Write 3x 3 2 x 1 5x 4 in standard form. What is the classification of the
polynomial by degree? by number of terms?
7. Use the words in the table above to name each monomial based on its degree.
3x 3
5x 4
2x
8. The polynomial is written in standard form below. Underline each term. Then
circle the exponent with the greatest value.
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5x 4 1 3x 3 2 x
9. Classify the polynomial.
by degree
You can determine the end behavior of a
polynomial function of degree n from the
leading term ax n of the standard form.
by number of terms
End Behavior of a Polynomial Function
of Degree n with Leading Term axn
n Even
n Odd
a Positive
Up and Up
Down and Up
a Negative
Down and Down
Up and Down
Problem 2 Describing End Behavior of Polynomial Functions
Got It? Consider the leading term of y 5 24x 3 1 2x 2 1 7. What is the end
behavior of the graph?
10. Circle the leading term, ax n , in the polynomial.
y 5 24x 3 1 2x 2 1 7
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119
5
Lesson 5-1
11. Use your answer to Exercise 10 to identify a and n for the leading term.
a5
n5
12. In this polynomial, a is positive / negative , and n is even / odd .
13. Circle the graph that illustrates the end behavior of this polynomial.
The end behavior is down and up.
The end behavior is down and down.
The end behavior is up and up.
The end behavior is up and down.
Graphing Cubic Functions
Got It? What is the graph of y 5 2x 3 1 2x 2 2 x 2 2? Describe the graph.
Underline the correct word to complete each sentence.
14. The coefficient of the leading term is positive / negative .
15. The exponent of the leading term is even / odd .
16. The end behavior is down / up and down / up .
17. Circle the graph that shows y 5 2x 3 1 2x 2 2 x 2 2.
18. The end behavior of y 5 2x 3 1 2x 2 2 x 2 2 is down / up and down / up , and
there are 1 / 2 / 3 turning points.
6
Chapter 5
HSM11A2MC_0501_120 120
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Problem 3
120
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3/16/09 2:55:01 PM
HSM11A2M
Problem 4 Using Differences to Determine Degree
Got It? What is the degree of the polynomial function that generates the data
x
shown in the table at the right?
19. Complete the flowchart to find the differences of the y-values.
23
1st differences
16
39
15
1
10
13
12
29
linear
5
2nd differences
quadratic
3rd differences
cubic
4th differences
quartic
20. The degree of the polynomial is
y
3
23
2
16
1
15
0
10
1
13
2
12
3
29
.
Lesson Check • Do you UNDERSTAND?
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Vocabulary Describe the end behavior of the graph of y 5 22x7 2 8x.
21. Underline the correct word(s) to complete each sentence.
The value of a in 22x7 is positive / negative . The exponent in 22x7 is even / odd .
The end behavior is up and up / down and up / up and down / down and down .
Math Success
Check off the vocabulary words that you understand.
polynomial
polynomial function
turning point
Rate how well you can describe the graph of a polynomial function.
Need to
review
0
2
4
6
8
Now I
get it!
10
121
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2:55:01 PM
HSM11A2MC_0501_121 121
end behavior
Lesson 5-1
7
3/16/09 2:55:06 PM
Name
5-1
Class
Date
Think About a Plan
Polynomial Functions
Packaging Design The diagram at the right shows a cologne
bottle that consists of a cylindrical base and a hemispherical top.
a. Write an expression for the cylinder’s volume.
b. Write an expression for the volume of the hemispherical top.
c. Write a polynomial to represent the total volume.
1. What is the formula for the volume of a cylinder? Define any variables
you use in your formula.
_______________.
, where r is ____________________ and h is
2. Write an expression for the volume of the cylinder using the information in the
diagram.
3. What is the formula for the volume of a sphere? Define any variables you use in your
formula.
where r is________________________.
4. Write an expression for the volume of the hemisphere.
5. How can you find the total volume of the bottle?
_______________________________________________________________ .
6. Write a polynomial expression to represent the total volume of the bottle.
7. Is the polynomial expression you wrote in simplest form? Explain.
________________________________________________________________
________________________________________________________________ .
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Class
Date
Practice
Form G
Polynomial Functions
Write each polynomial in standard form. Then classify it by degree and by number of terms.
1. 4x + x + 2
2. −3 + 3x − 3x
3. 6x4 − 1
4. 1 − 2s + 5s4
5. 5m2 − 3m2
6. x2 + 3x − 4x3
7. −1 + 2x2
8. 5m2 − 3m3
9. 5x − 7x2
10. 2 + 3x3 − 2
11. 6 − 2x3 − 4 + x3
12. 6x − 7x
13. a3(a2 + a + 1)
14. x(x + 5) − 5(x + 5)
15. p(p − 5) + 6
16. (3c2)2
17. −(3 − b)
18. 6(2x − 1)
19.
20.
21.
Determine the end behavior of the graph of each polynomial function.
22. y = 3x4 + 6x3 − x2 + 12
23. y = 50 − 3x3 + 5x2
24. y = −x + x2 + 2
25. y = 4x2 + 9 − 5x4 − x3
26. y = 12x4 − x + 3x7 − 1
27. y = 2x5 + x2 − 4
28. y = 5 + 2x + 7x2 − 5x3
29. y = 20 − 5x6 + 3x − 11x3
30. y = 6x + 25 + 4x4 − x2
Describe the shape of the graph of each cubic function by determining the end
behavior and number of turning points.
31. y = x3 + 4x
32. y = −2x3 + 3x − 1
33. y = 5x3 + 6x2
Determine the degree of the polynomial function with the given data.
34.
35.
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Name
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Class
Date
Practice (continued)
Form G
Polynomial Functions
Determine the sign of the leading coefficient and the degree of the polynomial
function for each graph.
37.
36.
38.
39. Error Analysis A student claims the function y = 3x4 − x3 + 7 is a fourth-degree
polynomial with end behavior of down and down. Describe the error the student made.
What is wrong with this statement?
40. The table at the right shows data representing a polynomial function.
a. What is the degree of the polynomial function?
b. What are the second differences of the y-values?
c. What are the differences when they are constant?
Classify each polynomial by degree and by number of terms.
Simplify first if necessary.
41. 4x5 − 5x2 + 3 − 2x2
42. b(b − 3)2
43. (7x2 + 9x − 5) + (9x2 − 9x)
44. (x + 2)3
45. (4s4 − s2 − 3) − (3s − s2 − 5)
46. 13
47. Open-Ended Write a third-degree polynomial function. Make a table of values
and a graph.
48. Writing Explain why finding the degree of a polynomial is easier when the
polynomial is written in standard form.
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Name
Class
Date
Practice
5-1
Form K
Polynomial Functions
Write each polynomial in standard form. Then classify it by degree and by
number of terms.
1. 4x3 − 3 + 2x2
To start, write the terms of the polynomial
with their degrees in descending order.
4x3 + 2x2 − 3
2. 8 − x5 + 9x2 − 2x
3. 6x + 2x4 − 2
4. −6x3
5. 3 + 24x2
Determine the end behavior of the graph of each polynomial function.
6. y = 5x3 − 2x2 + 1
7. y = 5 − x + 4x2
9. y = 3x2 + 9 − x3
12. y = 1 + 2x + 4x3 − 8x4
8. y = x − x2 + 10
10. y = 8x2 − 4x4 + 5x7 − 2
11. y = 20 − x5
13. y = 15 − 5x6 + 2x − 22x3
14. y = 3x + 10 + 8x4 − x2
Describe the shape of the graph of each cubic function by determining
the end behavior and number of turning points.
15. y = x3 + 2x
To start, make a table of values to help you
sketch the middle part of the graph.
16. y = −3x3 + 4x2 − 1
x
–2
y
–12
–1
–3
0
1
0
3
2
12
17. y = 4x3 + 2x2 − x
Determine the degree of the polynomial function with the given data.
18.
x
y
–3
19.
x
y
–43
–3
65
–2
–10
–2
5
–1
1
–1
–5
0
2
0
–1
1
5
1
5
2
22
2
25
3
65
3
95
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Class
Date
Practice (continued)
Form K
Polynomial Functions
Determine the sign of the leading coefficient and the degree of the polynomial
function for each graph.
20.
21.
22.
23. Error Analysis A student claims the function y = −2x3 + 5x − 7 is a 3rd degree
polynomial with ending behavior of down and up. Describe the error the student
made. What is wrong with this statement?
24. The table to the right shows data representing a polynomial function.
a. What is the degree of the polynomial function?
b. What are the second differences of the y-values?
c. What are the differences when they are constant?
x
y
–3
–2
–1
0
98
20
6
2
1
2
2
3
48
230
Classify each polynomial by degree and by number of terms. Simplify first if necessary.
25. 3x5 − 6x2 − 5 + x2
26. a − 2a + 3a2
27. (5x2 + 2x − 8) + (5x2 − 4x)
28. c3(5 − c2)
29. (5s3 − 2s2) − (s4 + 1)
30. x(3x)(x + 2)
31. (2s − 1)(3s + 3)
32. 5
33. Open-Ended Write a fourth-degree polynomial function. Make a table of
values and a graph.
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Standardized Test Prep
Polynomial Functions
Multiple Choice
For Exercises 1–7, choose the correct letter.
1. Which expression is a binomial?
x
2
2x
3x2 1 2x 1 4
x29
2. Which polynomial function has an end behavior of up and down?
26x7 1 4x2 2 3
6x7 2 4x2 1 3
27x6 1 3x 2 2
7x6 2 3x 1 2
3. What is the degree of the polynomial 5x 1 4x2 1 3x3 2 5x?
1
2
3
4
4. What is the degree of the polynomial represented by the data in the table at
the right?
2
3
4
5
5. For the table of values at the right, if the nth differences are constant, what is
the constant value?
212
1
25
6
6. What is the standard form of the polynomial 9x2 1 5x 1 27 1 2x3 ?
27 1 5x 1 9x2 1 2x3
9x2 1 5x 1 27 1 2x3
9x2 1 5x 1 2x3 1 27
2x3 1 9x2 1 5x 1 27
x
y
3
77
2
24
1
1
0
4
1
3
2
8
3
31
7. What is the number of terms in the polynomial (2a 2 5)(a2 2 1)?
2
3
4
5
Short Response
8. Simplify (9x3 2 4x 1 2) 2 (x3 1 3x2 1 1). Then name the polynomial by
degree and the number of terms.
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5-1 Solve It!
The first column shows a sequence of numbers.
For 1st differences, subtract consecutive
numbers in the sequence:
Working
backwards unlocks
the patterns.
26 2 (24) 5 22, 4 2 (26) 5 10, and so on.
For 2nd differences, subtract consecutive 1st
differences. For 3rd differences, subtract
consecutive 2nd differences.
If the pattern suggested by the 3rd
differences continues, what is the 8th number
in the first column? Justify your reasoning.
5-1 Lesson Quiz
1. Write 2x3 2 3x2 1 x5 in standard form. What is the
classification of this polynomial by its degree? by its number
of terms?
2. Consider the leading term of y 5 22x2 2 3x 1 3.
What is the end behavior of the graph?
3. Describe the end behavior and number of turning
points in the graph of y 5
x3
1 x 1 3.
4. Do you UNDERSTAND? What is the degree of the
polynomial function that generates the data shown
in the table?
x
y
1
2
2
0
3
6
4
16
5
30
6
48
7
70
8
96
Answers
Solve It!
Lesson Quiz
1074; work backwards using
the constant third difference,
24, to find the first and second
differences and the seventh
and eighth numbers in the first
column.
1.
x5
2x3
3x2 ;
1
2
quintic
trinomial
2. down and down
3. The end behavior is down
and up. There are no turning
points.
4. 2
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Additional Problems
Polynomial Functions
Problem 1
Write 23x 1 4x3 1 7x 2 3 in standard form. What is the
classification of this polynomial by its degree? by its number
of terms?
Problem 2
Consider the leading term of y 5 3x4 2 2x3 1 x 2 1.
What is the end behavior of the graph?
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Additional Problems (continued)
5-1
Polynomial Functions
Problem 3
What is the graph of y 5 3 2 2x3 1 x? Describe the graph.
Problem 4
What is the degree of the polynomial function that generates
the data shown in the table?
x
y
2
13
1
4
0
1
1
2
2
11
3
32
4
71
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Name
Class
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Date
Reteaching
Polynomial Functions
What is the classification of the following polynomial by its degree? by its number of
terms? What is its end behavior? 5x4 − 3x + 4x6 + 9x3 − 12 − x6 + 3x4
Step 1
Write the polynomial in standard form. First, combine any like terms.
Then, place the terms of the polynomial in descending order from
greatest exponent value to least exponent value.
5x4 − 3x + 4x6 + 9x3 − 12 − x6 + 3x4
Step 2
8x4 − 3x + 3x6 + 9x3 − 12
Combine like terms.
3x6 + 8x4 + 9x3 − 3x − 12
Place terms in descending order.
The degree of the polynomial is equal to the value of the greatest
exponent. This will be the exponent of the first term when the
polynomial is written in standard form.
3x6 + 8x4 + 9x3 – 3x – 12
3x6
The first term is 3x6.
The exponent of the first term is 6.
This is a sixth-degree polynomial.
Step 3
Count the number of terms in the simplified polynomial. It has 5 terms.
Step 4
To determine the end behavior of the polynomial (the directions of the graph
to the far left and to the far right), look at the degree of the polynomial (n)
and the coefficient of the leading term (a).
If a is positive and n is even, the end behavior is up and up.
If a is positive and n is odd, the end behavior is down and up.
If a is negative and n is even, the end behavior is down and down.
If a is negative and n is odd, the end behavior is up and down.
The leading term in this polynomial is 3x6.
a (+3) is positive and n (6) is even, so the end behavior is up and up.
Exercises
What is the classification of each polynomial by its degree? by its number of
terms? What is its end behavior?
1. 8 − 6x3 + 3x + x3 − 2
2. 15x7 − 7
3. 2x − 6x2 − 9
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Date
Reteaching (continued)
Polynomial Functions
X
What is the degree of the polynomial function that generates the data
shown at the right? What are the differences when they are constant?
To find the degree of a polynomial function from a data table, you can
use the differences of the y-values.
Step 1 Determine the values of
y2 – y1, y3 – y2, y4 – y3,
y5 – y4, y6 – y5, y7 – y6.
These are called the first
differences. Make a new
column
using
these
values.
y
–3
52(y1)
–2
18(y2)
–1
2(y3)
0
–2 (y4)
Step 2 Continue determining
1
0 (y5)
differences until the y-values
2
2 (y6)
are all equal. The quantity of
3 –2 (y7)
differences is the degree of the
polynomial function.
The third differences are all equal so this is a
third degree polynomial function. The value
of the third differences is –6.
Exercises
What is the degree of the polynomial function that generates the data in the table? What
are the differences when they are constant?
4.
5.
x
y
–3
216
–2
24
–1
0
0
0
1
0
2
–24
3
–216
6.
x
y
–3
–101
–2
–37
–1
–11
0
–5
1
–1
2
19
3
73
x
y
–3
6
–2
26
–1
8
0
0
1
2
2
–34
3
–204
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ELL Support
Polynomial Functions
Match each word in Column A with the matching polynomial in Column B.
Column A
Column B
1. cubic
A. 8
2. linear
B. 3x4 + 5x2 − 1
3. quartic
C. 2x2 − 2
4. quintic
D. 7x3 + 3x2 + 4
5. constant
E. x + 10
F. 6x5 + 3x3 + 11x + 3
6. quadratic
Match each polynomial in Column A with the matching word in Column B.
Column A
Column B
7. 5x3 + 7x
A. trinomial
8. 4x5 + 6x2 + 3
B. monomial
9. 8x4
C. binomial
Use the words from the lists below to name each polynomial by its degree and its
number of terms.
Degree
linear
quadratic
cubic
quartic
quintic
Number of Terms
monomial
binomial
trinomial
10. 4x2 − 2x + 3 __________________________ .
11. 6x3___________________________________________ .
12. 3x5 + 7x3 − 4 _________________________ .
13. 8x + 3____________________________ .
14. 2x4 + 5x2 _____________________________________________ .
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Date
Activity: Researching the Factors
Dividing Polynomials
Work in small groups for this activity.
The polynomial P(x) = x4 + x3 − 28x2 + 20x + 48 can be factored into exactly four
distinct linear factors involving real numbers only. Write the polynomial in factored
form P(x) = (x − a)(x − b)(x − c)(x − d).
Notice that when the value of a polynomial changes from negative to
positive (or from positive to negative) there is a root in between, as
shown in the example at the right.
• Complete the following table to help find possible values for
the roots of the polynomial.
• P(x) = (x − a)(x − b)(x − c)(x − d). Devise a plan to find a, b, c,
and d. Describe your plan in writing. Some possible strategies
are shown at the right. Consider the advantages and
disadvantages of each approach. Explore the use of repeated
synthetic division on successive quotients.
• Write the polynomial in factored form. Show your group’s work with your plan. You may
use a combination of methods.
Wrap Up
Summarize your results in a complete logical and informative solution.
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TEACH ERS IN STRUCTION S
5-3
Game: Discovering Your Roots
Solving Polynomial Equations
Provide the host with the following equations and their solutions.
Equation
Solution
1.
(x2 – 9)(x2 + 6x + 9) = 0
–3,3
2.
(x2 – 1)(x2 + 16) = 0
±1, ±4i
3.
(x2 – 9)(2x + 9) = 0
±3i, –
4.
(x2 + 9)(x2 + 4) = 0
±3i, ±2i
5.
(x2 + 25)(x2 – 4)(x + 4) = 0
–4, ±5i, ±2
6.
(x2 + 100)(x2 – 100) = 0
±10i, ±10
7.
(x2 + 49)(3x – 5) = 0
±7i,
8.
(x2 – 81)(3x2 + 27) = 0
±9, ±3
9.
(x2 – 5x + 6)(3x2 + 27) = 0
3,2, ±3i
10.
(x2 – 6x + 9)(9x2 – 81) = 0
±3
11.
(x2 + 10x + 25)(3x2 + 27) = 0
–5, ±3i
12.
(x2 + 1)2(2x + 3)2 = 0
±i, –
13.
(x2 – 2) (2x – 3)2 = 0
14.
(x2 + 2)(2x – 4)2 = 0
± 2i, 2
15.
(x2 + 2x)(2x2 – 16) = 0
–2, 0, ±2 2
16.
(x2 + 3x)(3x2 – 24) = 0
–3, 0, ±2 2
17.
(x2 – 6x + 9)(x2 – 10x + 25) = 0
3,5
18.
(x2 + 2x + 1)(x2 + 10x + 25) = 0
–1, –5
9
2
5
3
3
2
3
± 2,
2
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Name
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5-3
Date
Game: Discovering Your Roots
Solving Polynomial Equations
This is a game for three students—a host and two players. Players alternate turns. The host will ask a
player to solve an equation below in a reasonable amount of time. Players are to write all solutions to
the given equation. Players earn 5 points for a correct answer and lose 3 points for an incorrect or
incomplete answer.
Equation
Player 1
Player 2
1. (x2 – 9)(x2 + 6x + 9) = 0
2. (x2 – 1)(x2 + 16) = 0
3. (x2 + 9)(2x + 9) = 0
4. (x2 + 9)(x2 + 4) = 0
5. (x2 + 100)(x2 – 4)(x + 4) = 0
6. (x2 + 100)(x2 – 100) = 0
7. (x2 +49)(3x – 5) = 0
8. (x2 – 81)(3x2 – 27) = 0
9. (x2 – 5x + 6)(3x2 + 27) = 0
10. (x2 – 6x + 9)(9x2 – 81) = 0
11. (x2 + 10x + 25)(3x2 + 27) = 0
12. (x2 + 1)2(2x + 3)2 = 0
13.
(x2 – 2)(2x – 3)2 = 0
14. (x2 + 2)(2x – 4)2 = 0
15. (x2 + 2x)(2x2 – 16) = 0
16. (x2 +3x)(3x2 – 24) = 0
17. (x2 – 6x + 9)(x2 – 10x) + 25 = 0
18. (x2+ 2x + 1)(x2 + 10x + 25) = 0
Total
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Puzzle: Made in the Shade
5-2
Polynomials, Linear Factors, and Zeros
Find the zeros of each polynomial below. For each corresponding row, shade in each
number that is a zero. The illustration made from shading the squares suggests the answer to
the riddle below.
A. P(x) = x(x2 − 1)
B. P(x) = x(x + 2)(x + 1)(x2 + 2x − 3)
__________________________
___________________________
D. P(x) = x(x2 − 25)(x2 + 4x + 3)
C. P(x) = x(x + 4)(x + 3)(x + 1)(x − 1)
___________________________
____________________________
E. P(x) = (x2 + x − 20)(x + 2)(x2 + 4x + 3)
F . P(x) = (x2 − 9)(x2 − 25)
_____________________________
_____________________________
G. P(x) = (x2 + 9x + 20)(x2 − 5x + 6)(x − 5)
H. P(x) = (x2 − 5x + 6)(x2 − 9x + 20)
______________________________
______________________________
I. P(x) = x2 − 6x + 9
J. P(x) = (x2 − 4x + 4)(x2 − 4x + 4)
________________________________
_______________________________
2
K. P(x) = x(x − 2x + 1)(x − 2)
_______________________________
A
–5
–4
–3
–2
–1
0
1
2
3
4
5
B
5
–5
–4
–3
–2
–1
0
1
2
3
4
C
4
5
–5
–4
–3
–2
–1
0
1
2
3
D
3
4
5
–5
–4
–3
–2
–1
0
1
2
E
2
3
4
5
–5
–4
–3
–2
–1
0
1
F
1
2
3
4
5
–5
–4
–3
–2
–1
0
G
0
1
2
3
4
5
–5
–4
–3
–2
–1
H
–1
0
1
2
3
4
5
–5
–4
–3
–2
I
–2
–1
0
1
2
3
4
5
–5
–4
–3
J
–3
–2
–1
0
1
2
3
4
5
–5
–4
K
–4
–3
–2
–1
0
1
2
3
4
5
–5
Riddle: This grows above the ground, but the solutions to the polynomials above lie beneath.
And as it grows, it provides shade to those underneath. What is it?
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Name
5-1
Class
Date
Enrichment
Polynomial Functions
Mathematicians use precise language to describe the relationships between sets. One
important relationship is described as a function. You have graphed polynomial functions.
Using this one word may not seem important, but it describes a very specific relationship
between the domain and range of a polynomial. The word function tells you that every
element of the domain corresponds with exactly one element of the range.
1. Another important relationship between two sets is described by the word onto. A
function from set A to set B is onto if every element in set B is matched with an
element in set A. Which of the following relations shows a function from set A to
set B that is onto? Explain.
2. Another relationship between two sets is described as one-to-one. A function from
set A to set B is one-to-one if no element of set B is paired with more than one
element of set A. Which of the following relations shows a function from set A to
set B that is one-to-one? Explain.
Describe each polynomial function. If it is not possible, explain why.
3. Describe a polynomial function that is onto but not one-to-one.
4. Is there a polynomial function that is one-to-one but not onto?
5. Describe a polynomial function that is both onto and one-to-one.
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Chapter 5 Quiz 1
Form G
Lessons 5-1 through 5-4
Do you know HOW?
Write each polynomial function in standard form. Then classify it by degree and by
number of terms.
1. n = 4m2 − m + 7m4
2. f(t) = 4t + 3t3 + 2t − 7
3. f(r) = 5r + 7 + 2r2
Find the zeros of each function. State the multiplicity of multiple zeros.
4. y = (x + 2)2(x − 5)4
5. y = (3x + 2)3(x − 5)5
6. y = x2(x + 4)3(x − 1)
Divide using synthetic division.
7. (x3 + 3x2 − x − 3) ÷ (x − 1)
8. (2x3 − 3x2 − 18x − 8) ÷ (x − 4)
Find all the imaginary solutions of each equation by factoring.
9. x4 + 14x2 − 32 = 0
10. x3 − 16x = 0
11. 6x3 − 2x2 + 4x = 0
Do you UNDERSTAND?
12. What is P(−4) given that P(x) = 2x4 − 3x3 + 5x2 − 1?
13. Open-Ended Write the equation of a polynomial function that has zeros at −3 and 2.
14. The product of three integers is 90. The second number is twice the first number.
The third number is two more than the first number. What are the three numbers?
15. Reasoning The volume of a box is x3 + 4x2 + 4x. Explain how you know the box is
not a cube.
16. Error Analysis For the polynomial function
1
y = x 2 + x + 6 , your friend says
3
the end behavior of the graph is down and up. What mistake did your friend
make?
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Chapter 5 Quiz 2
Form G
Lessons 5-5 through 5-9
Do you know HOW?
Expand each binomial.
1. (2a − 1)4
2. (x + 3)5
Use the Rational Root Theorem to list all possible rational roots for each
equation. Then find any actual rational roots.
3. x3 + 9x2 + 19x − 4 = 0
4. 2x3 − x2 + 10x − 5 = 0
What are all the complex roots of the following polynomial equations?
5. x4 + 3x3 − 5x2 − 12x + 4 = 0
6. 2x3 + x2 − 9x + 18 = 0
7. Describe the transformations used to change the graph of the parent function
y
y = x3 to the graph of=
1
3
( x + 4) .
6
Find a polynomial function whose graph passes through each set of points.
8. (0, 3), (−1, 0), (1, 10) and (−2, −35)
9. (−4, 215), (0, −1), (2, −1), and (3, −16)
Do you UNDERSTAND?
10. The potential energy of a spring varies directly as the square of the stretched length l.
1
The formula is PE = kl 2 , where k is the spring constant. When you stretch a
2
spring to 12 ft, it has 483 ft-lb of potential energy. What is the spring constant for
this spring? How much potential energy is created by stretching a 7 ft section?
11. In the expansion of (4r + s)7, one of the terms contains r4s3. What is the
coefficient of this term?
12. Reasoning For a set of data, you make three models. R2 for the quadratic
model is 0.825. R2 for the cubic model is 0.996. R2 for the quartic model is
0.934. Explain why the cubic model may not be the best for predicting outside
the data.
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Chapter 5 Chapter Test
Form G
Do you know HOW?
Write each polynomial in standard form. Then classify it by degree and by
number of terms.
1. 4x4 + 6x3 – 2 – x 4
2. 9x2 – 2x + 3x2
3. 4x(x – 5)(x + 6)
Find the real solutions of each equation by graphing. Where necessary, round
to the nearest hundredth.
4. x4 + 2x2 – 1 = 0
5. –x3 – 3x – 2 = 0
6. y = –x4 + 4x3 + 3 = 0
7. –x3 + 3x + 4 = 0
8. x4 + 2x – 3 = 0
9. –x3 + 2x2 + 1 = 0
Write a polynomial function with rational coefficients so that P(x) = 0 has the
given roots.
10. 2, 3, 5
11. –1, –1, 1
12.
13. 2 – i,
3 , 2i
5
Find the zeros of each function. State the multiplicity of any multiple zeros.
14. y = (x – 1)2(2x – 3)3
15. y = (3x – 2)5(x + 4)2
16. y = 4x2(x + 2)3(x + 1)
Solve each equation.
17. (x – 1)(x2 + 5x + 6) = 0
18. x3 – 10x2 + 16x = 0
19. (x + 2)(x2 + 3x – 40) = 0
20. x3 + 3x2 – 54x = 0
Divide using synthetic division.
21. (x3 – 4x2 + x – 5) ÷ (x + 2)
22. (2x3 – 4x + 3) ÷ (x – 1)
23. (x3 + 5x2 – x + 1) ÷ (x + 2)
24. (3x3 – x2 + 2x – 5) ÷ (x – 1)
Use the Rational Root Theorem to list all possible rational roots for each
equation. Then find any actual roots.
25. x3 + 2x2 + 3x + 6 = 0
26. x4 – 7x2 + 12 = 0
27. What is P(–5) if P(x) = –x3 – 4x2 + x – 2?
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Chapter 5 Chapter Test (continued)
Form G
Expand each binomial.
28. (x + y)4
29. (4 – 3x)3
30. (2r + q)5
31. (a + 4b)3
32.
a. Find a cubic function to model the data. (Let x = years after 1960.)
b. Estimate the deaths for the year 2006.
Determine the cubic function that is obtained from the parent function y = x3 after
each sequence of transformations.
33. a vertical stretch by a factor of 5, a reflection across the y-axis, and a horizontal
translation 2 units left
34. a reflection across the x-axis, a horizontal translation 3 units right, and a
vertical translation 7 units down
Do you UNDERSTAND?
35. Reasoning Would it be a good idea to use the cubic model found in
Exercise 32 to estimate the deaths for the year 2050? Why or why not?
36. Writing How do you use Pascal’s Triangle when expanding a binomial?
37. Can a function with the complex roots 5,
2 , and 3i be a fourth-degree
polynomial with rational coefficients? Explain.
38. A cubic box is 5 in. on each side. If each dimension is increased by x in., what is the
polynomial function modeling the new volume V ?
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Chapter 5 Find the Errors!
For use with Lessons 5-1 through 5-3
For each exercise, identify the error(s) in planning the solution or solving the
problem. Then write the correct solution.
1. Consider the leading term of the polynomial function. What is the end behavior of the
graph? Check your answer with a graphing calculator.
f (x) 5 23x3 1 2x2 2 x 1 1
There are 4 terms, so the function is even and the first term is negative.
The end behavior of an even negative function is down and down.
2. What are the zeros of f(x) 5 (x 2 8)2(2x 2 3)(x 1 1)? What are their multiplicities?
How does the graph behave at these zeros?
8 is a zero of multiplicity 2.
2
3
y
and 21 are zeros of multiplicity 1.
500
400
300
200
100
The graph looks close to linear at the x-intercepts
21 and 23 . It resembles a parabola at x-intercept 8.
2 O
100
200
300
x
2
4
6
8
10
3. What are the real and imaginary solutions of the equation 3x3 2 6x2 2 12x 5 0?
3x(x2 2 2x 2 4) 5 0
Use the Quadratic Formula to solve x2 2 2x 2 4 5 0.
x5
2(22) 4 "(22)2 2 4(1)(24)
2 4 !20
5
2
2(1)
x 5 1 1 !5
or
x 5 1 2 !5
The solutions are 1 1 !5 and 1 2 !5..
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Chapter 5 Find the Errors!
For use with Lessons 5-4 through 5-6
For each exercise, identify the error(s) in planning the solution or solving the
problem. Then write the correct solution.
1. Use polynomial division to divide x4 1 x3 2 7x 2 3 by x 1 3 . What is the
quotient and remainder?
x3 2 2x2 2 1 2 7x
x 1 3qx4 1 1x3 2 7x 2 3
x4 1 3x3 2 7x 2 3
22x3 2 7x 2 3
22x3 2 6x 2 3
21x 2 3
21x 2 3
0
The quotient is x3 2 2x2 2 1 with remainder 0.
2. What is a third-degree polynomial function y 5 P(x) with rational coefficients so that
P(x) 5 0 has roots 3 1 !2 and 6?
Since 3 1 !2 is a root, then 3 2 !2 is also a root.
P(x) 5 Ax 2 3 2 !2B Ax 2 3 1 !2B Ax 2 6B
P(x) 5 Ax2 1 9 2 2B Ax 2 6B
P(x) 5 Ax2 1 7B Ax 2 6B
P(x) 5 x3 2 6x2 1 7x 2 42
P(x) 5 x3 2 6x2 1 7x 2 42
3. What are all the complex roots of 2x3 1 x2 1 14x 1 7 5 0?
Find the zeros of the function.
Fin
The polynomial equation
has degree 3. There are 3
roots
Step 1
Us synthetic division and
Use
factoring.
fac
The polynomial is in standard form. The possible rational roots are
41, 47, 412 and 472 .
Step 2
Substitute 2 12 for x. The value of f (x) is 0. So, 2 12 is a root and
x 1 12 is a factor.
Step 3
1
0 2 21 14 27
2
1 2
Use synthetic division to factor out x 1 2: 0 2 21 10 27
0 2 20 14 20
Step 4
Q x 1 2 R A2x2 1 14B 5 2 Q x 1 2 R Ax2 1 7B
The complex roots are 2 12 , !7, and 2 !7.
1
1
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Chapter 5 Find the Errors!
For use with Lessons 5-7 through 5-9
For each exercise, identify the error(s) in planning the solution or solving the
problem. Then write the correct solution.
1. What is the expansion of (p 2 3q)4? Use the Binomial Theorem.
(p 2 3q)4 5 p4 1 4p3(23)q 1 6p2(23)q2 1 4p(23)q3 1 1(2 3)q4
5 p4 2 12p3q 2 18p2q2 2 12pq3 2 3q4
2. The chart shows the number, in thousands, of CDs sold
by a local band during the first 7 months. What cubic
function best models the data? Use the model to estimate
sales of CDs in the 8th month.
The n 1 1 Point Principle says that a cubic function requires
4 points. Use CUBICREG on a graphing calculator with the
first 4 points.
y 5 ax3 1 bx2 1 cx 1 d and a 5 0.75, b 5 21.5, c 5 22.25,
and d 5 5.
The function is f (x) 5 0.75x3 2 1.5x2 2 2.25x 1 5.
Month
CD sales
(thousands)
1
2
2
0.5
3
5
4
20
5
42
6
40
7
35
Use the model to estimate CD sales in the 8th month.
f (8) 5 0.75(8)3 2 1.5(8)2 2 2.25(8) 1 5 5 275
During the 8th month, about 275 thousand CDs will be sold.
3. What function do you obtain by applying the following transformations to y 5 x3?
•
vertical stretch by a factor of 6
•
vertical translation 4 units down
•
horizontal translation 5 units right
Step 1 y 5 x3
Step 2 y 5 6x3
Step 3
S y 5 6x3
S y 5 (6x 2 5)3
y 5 (6x 2 5)3
S
y 5 (6x 2 5)3 2 4
Multiply by 6 to stretch.
Replace x with x 2 5 to translate
horizontally.
Subtract 4 to translate vertically.
The transformed cubic function is y 5 (6x 2 5)3 2 4.
4. What are the real zeros of the function y 5 (x 2 4)3 1 1?
Ax 2 4B 3 1 1 5 0
Ax 2 4B 3 5 21
3
x 2 4 5 "21
x 2 4 5 21 or x 2 4 5 1
x 5 23
2 4x 5 5
The real zeros are 3 and 5.
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Chapter 5 Performance Tasks
Task 1
a. Draw the related graph of x2 2 ax 5 bx 2 ab. Determine the multiplicity of
each root.
b. Draw the related graph of (x 2 a)2(x 2 b) 5 0. Determine the multiplicity of
each root.
c. Rewrite the equations found in parts a and b in standard form.
d. Given the equation ax3 1 bx2 5 2cx, find the roots of this equation in terms
of a, b, and c.
Task 2
1
3
a. Use division to find the remaining roots of y 5 2x3 1 2x2 2 3x 2 4.
6
4
2
6
2
y
O (2, 0) x
4 6
4
6
b. Use division to find the remaining roots of y 5 x3 2 4x2 2 x 1 4.
y
(4, 0) x
42
2
6 8
6
8
c. Use the roots found in parts a and b to rewrite the functions in factored form.
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Chapter 5 Performance Tasks (continued)
Task 3
The data in the table at the right shows the times for the Men’s
500-m Speed Skating event at the Winter Olympics.
a. Find a quadratic, a cubic, and a quartic model for the data set.
Let x be the number of years since 1980.
b. Compare the models and determine which one is more
appropriate. Explain your choice.
Year
Time (sec)
1984
38.19
1988
36.45
1992
37.14
1994
36.33
1998
35.59
2002
34.42
2006
34.84
SOURCE: www.infoplease.com
Task 4
The power P generated by a circuit varies directly to the square of the current
I times the resistance R.
a. Write quadratic functions that model circuits with a power of 15 watts at
6 amps current, of 30 watts at 12 amps current, and of 60 watts at 24 amps
current.
b. Find the zeros of the functions.
c. What does each zero represent?
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Extra Practice
Chapter 5
Lesson 5-1
Write each polynomial in standard form. Then classify it by degree and by
number of terms.
1. a2 + 4a − 5a2 − a
2. 3x −
1
− 5x
3
3. 3n2+ n3– n – 3 – 3n3
5. 6c2 – 4c+ 7 – 8c2
2
4. 15 − y − 10y − 8 + 8y
6. 3x 2 − 5x − x 2 + x + 4x
Determine the end behavior of the graph of each polynomial function.
7. y = x2 – 2x + 3
10. y =
8. y = x3 – 2x
1 4
1
x + 5x 2 −
2
2
9. y = 7x5+ 3x3 – 2x
11. y = 15x9
12. y = –x12+ 6x6 – 36
Lesson 5-2
Write each polynomial in factored form. Check by multiplication.
13. x3 + 5x
14. x3 + x2 – 6x
15. 6x3 − 7x2 − 3x
Write a polynomial function in standard form with the given zeros.
16. x = 3, 2, −1
17. x= 1, 1, 2
18. x = −2, −1, 1
19. x = 1, 2, 6
20. x = −3, −1, 5
21. x = 0, 0, 2, 3
22, x = − 2 ,1, 2, 2
23. x = 2, 4, 5, 7
24. x = −2, 0,
1
,1
3
Find the zeros of each function. State the multiplicity of multiple zeros.
25. y = (x − 2)(x + 4)
26. y = (x − 7)(x − 3)
27. y = (x + 1)(x − 8)(x − 9)
28. y = x (x + 1)(x + 5)
2
2
29. y = x (x + 1)
30. y = (x − 3)(x − 4)
31. Find the relative maximum and minimum of the graph
of f(x) = x3 − 3x2 + 2. Then graph the function.
32. A jewelry store is designing a gift box. The sum of the length,
width, and height is 12 inches. If the length is one inch greater the
height, what should the dimensions of the box be to maximize
its volume? What is the maximized volume?
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Extra Practice (continued)
Chapter 5
33. Tonya wants to make a metal tray by cutting four identical
square corner pieces from a rectangular metal sheet. Then
she will bend the sides up to make an open tray.
a. Let the length of each side of the removed squares be x in.
Express the volume of the box as a polynomial function of x.
b. Find the dimensions of a tray that would have a 384-in.3
capacity.
Lesson 5-3
Find the real or imaginary solutions of each equation by factoring.
34. x3 + 27 = 0
35. 8x3 = 125
36. 9 = 4x2 − 16
37. x2 + 400 = 40x
38. 0 = 4x2 + 28x + 49
39. −9x4 = −48x2 + 64
Solve each equation.
40. t3 − 3t2 − 10t = 0
41. 4m3 + m2 − m + 5 = 0
42. t3 − 6t2 + 12t − 8 = 0
43. 2c3 − 7c2 − 4c = 0
44. w4 − 13w2 + 36 = 0
45. x 3 + 2x 2 − 13x + 10 = 0
46. The product of three consecutive integers is 210. Use N to represent the
middle integer.
a. Write the product as a polynomial function of P(N).
b. Find the three integers.
47. The product of three consecutive odd integers is 6783.
a. Write an equation to model the situation.
b. Solve the equation by graphing to find the numbers.
Lesson 5-4
Determine whether each binomial is a factor of x 3 − 5x 2 − 2x + 24.
48. x + 2
49. x − 3
50. x + 4
Divide.
51. (x3 − 3x2 + 2) ÷ (x − 1)
52. (x3 − x2 − 6x) ÷ (x − 3)
53. (2x3 + 10x2 + 8x) ÷ (x + 4)
54. (x4 + x2 − 6) ÷ (x2 + 3)
55. (x2 − 4x + 2) ÷ (x − 2)
56. (x3 + 11x + 12) ÷ (x + 3)
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Class
Date
Extra Practice (continued)
Chapter 5
Lesson 5-5
Find the roots of each polynomial equation.
57. x3 + 2x2 + 3x + 6 = 0
58. x3 − 3x2 + 4x − 12 = 0
59. 3x4 + 11x3 + 14x2 + 7x + 1 = 0
60. 3x4 − x3 − 22x2 + 24x = 0
61. 45x3 + 93x2 − 12 = 0
62. 8x4 − 66x3 + 175x2 − 132x − 45 = 0
Lesson 5-6
Find all the zeros of each function.
63. f(x) = x3 − 4x2 + x − 6
64. g(x) = 3x3 − 3x2 + x − 1
65. h(x) = x4 − 5x3 − 8x + 40
66. f(x) = 2x4 − 12x3 + 21x2 + 2x − 33
67. A block of cheese is a cube whose side is x in. long. You cut
of a 1-inch thick piece from the right side. Then you cut of a
3-inch thick piece from the top, as shown at the right. The
volume of the remaining block is 2002 in.3. What are the
dimensions of the original block of cheese?
68. You can construct triangles by connecting three vertices of a
convex polygon with n sides. The number of all possible
n 3 − 3n 2 + 2n
such triangles can be represented as f (n) =
.
6
Find the value of n such that you can construct 84 such
triangles from the polygon.
Lesson 5-7
Use the Binomial Theorem to expand each binomial.
69. (x − 1)3
70. (3x + 2)4
71. (4x + 10)3
72. (x + 2y)7
73. (5x − y)5
74. (x − 4y3)4
75. The side length of a cube is given by the expression (2x + 3y2). Write a
binomial expression for the volume of the cube.
76. What is the sixth term in the binomial expansion of (3x − 4)8?
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Date
Extra Practice (continued)
Chapter 5
Lesson 5-8
Find a polynomial function whose graph passes through each set of points.
77. (2, 5) and (8, 11)
78. (3, −3) and (7, 9)
79. (−2, 16) and (4, 13)
80. (−1, −7), (1, 1), and (2, −1)
81. (1, 5), (3, 11), and (5, 5)
82. (−4, −13), (−1, 2), (0, −1), and (1, 2)
83. The table shows the annual population of Florida for selected years.
Year
Population (millions)
1970
1980
1990
2000
6.79
9.75
12.94
15.98
a. Find a polynomial function that best models the data.
b. Use your model to estimate the population of Florida in 2020.
c. Use your model to estimate when the population of Florida will
reach 20.59 million.
Lesson 5-9
Determine the cubic function that is obtained from the parent function y = x 3 after
each sequence of transformations.
84. vertical stretch by a factor of 2;
85. vertical stretch by a factor of 3;
reflection across the x-axis;
horizontal translation 3 units left
vertical translation down 2 units;
horizontal translation 1 unit right
Find all the real zeros of each function.
(
)
(
3
86. y = 2 x − 3 + 2
)
3
87. 6 x + 3 − 6
3
1⎛
1⎞
88. − ⎜ x + ⎟ − 5
3⎝
2⎠
Find a quartic function with the given x-values as its only real zeros.
89. x = −3 and x = 3
90. x = 1 and x = 3
91. x = 0 and x = 4
92. x = −8 and x = −6
93. x = −2 and x = 8
94. x = −3 and x = 5
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Name
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Date
Chapter 5 Project: Curves by Design
Beginning the Chapter Project
A continuous curve can be approximated by the graph of a polynomial. This fact is
central to modern car design. Scale models are first produced by a designer. Even such
apparently minor parts of the design such as door handles are included in models.
When the modeling process is complete, every curve in the design becomes an
equation that is adjusted by the designer on a computer. Minor changes can
be made through slight changes in an equation. Although in many programs
the computer adjusts the equations, you can do the same thing on a graphing
calculator. When the design has been finalized, the information is used to produce
dies and molds to manufacture the car.
List of Materials
• Graphing calculator
• Graph paper
Activities
Activity 1: Graphing
A hood section of a new car is modeled by the equation
y 5 0.00143x4 1 0.00166x3 2 0.236x2 1 1.53x 1 0.739. The graph of this
polynomial equation is shown at the right. Use a graphing calculator to
fine-tune the equation. Keep the same window but change the equation.
Pretend you are the designer and produce a curve with a shape more
pleasing to your eye!
Activity 2: Analyzing
Research the design of a car or another object that has curved parts.
• On graph paper, sketch a curve that models all or part of the object you chose
to research. Label four points that you think would help identify the curve.
Find the cubic function that fits these four points.
• Use the equation y 5 ax3 1 bx2 1 cx 1 d. Solve for the variables
a, b, c, and d using a 4 3 4 inverse matrix.
Activity 3: Graphing
Identify and label ten points on the sketch you made in Activity 2. Do you think
the function that best fits these points will be more accurate than the function you
found using four points? Explain your reasoning. Then find the new function using
a graphing calculator and the CubicReg feature.
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Chapter 5 Project: Curves by Design (continued)
Finishing the Project
The activities should help you to complete your project. Make a poster to display
the sketch and graphs you have completed for the object you have chosen. On the
poster, include your research about the object.
Reflect and Revise
Before completing your poster, check your equations for accuracy, your graph
designs for neatness, and your written work for clarity. Is your poster eye-catching,
exciting, and appealing, as well as accurate? Show your work to at least one adult
and one classmate. Discuss improvements you could make.
Extending the Project
Interview someone who uses a computer-assisted design (CAD) program at
work. If possible, arrange to have a demonstration of the program. Find out what
skills, education, or experience helped the person successfully enter the field of
computer-assisted design.
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Name
Class
Date
Chapter 5 Project Manager: Curves by Design
Getting Started
Read the project. As you work on the project, you will need a calculator, materials
on which you can record your calculations, and materials to make accurate and
attractive graphs. Keep all of your work for the project in a folder.
Checklist
Suggestions
☐ Activity 1: modeling a curve
☐ Make small changes in the equation at first.
☐ Activity 2: finding a cubic model
☐ Label the turning points.
☐ Activity 3: finding a better fit
☐ Use the regression feature of your graphing
calculator.
☐ object model
☐ Is a cubic function the best model for the
object you chose? Why or why not? How
can you determine the curve that best
models the shape of your object using a
graphing calculator?
Scoring Rubric
4
Your equations and solutions are correct. Graphs are neat and accurate. All
written work, including the poster, is neat, correct, and pleasing to the eye.
Explanations show careful reasoning.
3
Your equations are fairly close to the graph designs, with some minor errors.
Graphs, written work, and the poster are neat and mostly accurate with
minor errors. Most explanations are clear.
2
Your equations and solutions contain errors. Graphs, written work, and the
poster could be more accurate and neater. Explanations are not clear.
1
Major concepts are misunderstood. Project satisfies few of the requirements
and shows poor organization and effort.
0
Major elements of the project are incomplete or missing.
Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.
Teacher’s Evaluation of Project
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TEACHER INSTRUCTIONS
Chapter 5 Project Teacher Notes: Curves by Design
About the Project
The Chapter Project gives students an opportunity to adjust a polynomial
equation to fit the curve for their designs of the hood section of a car. They
also write cubic equations for curves of objects of their choice by using inverse
matrices and by using their calculator’s regression feature.
Introducing the Project
• Encourage students to keep all project-related materials in a separate folder.
• Ask students if they have ever wondered how car designers change the shapes
of a car’s parts. Ask students what they think an equation for a curved section
of a car would look like.
Activity 1: Graphing
Students graph the given polynomial and fine-tune the equation to make the
graph a pleasing shape for a car hood.
Activity 2: Analyzing
Students research the designs of cars or other objects that have curved parts and
use inverse matrices to write equations for one of their curves.
Activity 3: Graphing
Students use their calculators to find more accurate equations to model the curves
for their projects.
Finishing the Project
You may wish to plan a project day on which students share their completed
projects. Encourage students to explain their processes as well as their results. Ask
students to review their project work and update their folders.
• Have students review their methods for finding and recording curves and
equations used for the project.
• Ask groups to share their insights that resulted from completing the project,
such as techniques they found to make fitting the equations to the curves
easier or more accurate.
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Date
Chapter 5 Cumulative Review
Multiple Choice
For Exercises 1–14, choose the correct letter.
1. Which relation is not a function?
y50
y 5 2x
y5x12
x52
2. For which of the following sets of data is a linear model reasonable?
{(0, 11), (2, 8), (3, 7), (7, 2), (8, 0)}
{(215, 8), (28, 27), (23, 0), (0, 5), (7, 23)}
{(210, 3.5), (25.5, 6.5), (20.1, 24), (3.5, 27.5), (12, 25)}
{(21, 3.5), (0, 2.5), (2, 6.5), (23, 11.5), (5, 27.5)}
3. Which is a solution of the system of inequalities e
(3, 3)
(21, 2)
y14.0
?
y#x11
(1, 5)
(0, 2)
4. Which of the following is the equation of a parabola?
y 5 ux 1 3 u
y5x21
y 5 x2 1 1
x5y12
y 5 8x
y 5 8x2
5. Which of these is a direct variation?
x58
y58
6. Which of these quadratic equations has the factors (x 2 2) and (x 2 3)?
x2 2 x 2 6
x2 1 x 2 6
x2 2 5x 1 6
x2 1 5x 1 6
4x 2 5x2
6x3 2 x 1 7
(4, 3)
(24, 25)
(24, 3)
x,3
x.3
x53
7. Which polynomial is written in standard form?
1 1 3x 2 5x2
8. Solve the system e
3x2 1 2 1 x3
x1450
.
y5x11
(24, 23)
9. Solve 8x , 12 1 4x.
x,1
10. What is the axis of symmetry of y 5 2(x 2 3)2 1 5?
y53
x 5 2.5
x53
x55
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Chapter 5 Cumulative Review
Date
(continued)
11. A sixth-degree polynomial function with rational coefficients has complex
roots 6, !2, and 25i. Which of the following cannot be another complex root
of this polynomial?
5i
12. Solve (x 1 3)(x 1 4) 5 0
i !3
2!2
x 5 3 or x 5 4
x 5 23 or x 5 24
x50
none of the above
0
13. Which relation is a function?
{(2, 3), (3, 5), (1, 4), (2, 21)}
{(3, 1), (3, 3), (3, 2), (3, 0)}
{(1, 0), (0, 2), (3, 9), (21, 8)}
{(1, 4), (2, 4), (4, 3), (4, 4)}
14. Find the roots of x3 1 x2 2 17x 1 15 5 0.
1, 3, 5
21, 3, 5
25, 23, 21
25, 1, 3
Short Response
15. Open-Ended Write the equation of a direct variation in slope-intercept form.
Write the x-and y-intercepts.
16. Writing Explain how to write a polynomial equation in standard form with
roots x = a, b, c.
17. Evaluate 2a2 2 5a 1 4 for a 5 3.
18. Graph the inequality: 2x 2 3y , 6.
19. Use Pascal’s Triangle or the Binomial Theorem to expand (x 2 y2)3.
20. Determine the equation of the graph of y 5 x3 under a vertical stretch by a
factor of 8, a reflection across the x-axis, a horizontal translation 3 units left,
and a vertical translation 5 units up.
Extended Response
21. An arrow is shot upward. Its height h, in feet, is given by the equation
h 5 216t2 1 32t 1 5, where t is the time in seconds. The arrow is released at t 5 0 s.
a. How many seconds does it take until the arrow hits the ground?
b. How high is the arrow after 2 seconds?
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