teacher resource samplerassets.pearsonschool.com/asset_mgr/current/201135/... · solve it and...

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Algebra 2Common Core

ContentsStudent Companion 4

Think About a Plan 8

Practice G 9

Practice K 11

Standardized Test Prep 13

Solve It and Lesson Quiz 14

Additional Problems 15

Reteaching 17

Additional Vocabulary Support 19

Activity 20

Game 21

Puzzle 23

Enrichment 24

Teaching with TI Technology 25

Chapter Quiz 29

Chapter Test 31

Find the Errors! 33

Performance Tasks 36

Extra Practice 38

Chapter Project 42

Cumulative Review 46

Going beyond the textbook with Pearson Algebra 2

Log On and LearnIntroducing PowerAlgebra.com and PowerGeometry.com—the gateway for students and teachers to all the digital components available for the series. Go to PowerAlgebra.com or PowerGeometry.com and fill in the information below.

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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User Name: CommonCore2012*

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For review purposes only. Not for sale or resale. For review purposes only. Not for sale or resale.

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119 Lesson 5-1

Classifying Polynomials

Got It? Write 3x3 2 x 1 5x4 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.

3x3 2x 5x4

8. The polynomial is written in standard form below. Underline each term. Then circle the exponent with the greatest value.

5x4 1 3x3 2 x

9. Classify the polynomial. by degree by number of terms

Problem 1

You can classify a polynomial by its degree or by its number of terms.

Describing End Behavior of Polynomial Functions

Got It? Consider the leading term of y 5 24x3 1 2x2 1 7. What is the end behavior of the graph?

10. Circle the leading term, axn, in the polynomial.

y 5 24x3 1 2x2 1 7

Problem 2

Name UsingDegree

PolynomialExample

Number ofTerms

Name UsingNumber of TermsDegree

3

4

5

0

1

2 1

3

2

4

1

2

monomial

trinomial

binomial

polynomial of 4 terms

monomial

binomial

cubic

quartic

quintic

constant

linear

quadratic

4x3 2x2 x

2x4 5x2

x5 4x2 2x 1

5

x 4

4x2

n Even

Up and Up

n Odd

Down and Up

Up and Down

End Behavior of a Polynomial Functionof Degree n with Leading Term axn

a Positive

Down and Downa Negative

You can determine the end behavior of a polynomial function of degree n from the leading term axn of the standard form.

HSM11A2MC_0501_119 119 3/16/09 2:54:58 PM

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Vocabulary

5-1 Polynomial Functions

Review

1. Write S if the expression is in standard form. Write N if it is not.

5 1 7x 2 13x2

47y2 2 2y 2 1

3m2 1 4m

Vocabulary Builder

polynomial (noun) pahl ah NOH mee ul

Related Words: monomial, binomial, trinomial

Definition: A polynomial is a monomial or the sum of monomials.

Use Your Vocabulary

2. Circle the polynomial expression(s).

2t 4 2 5t 1 3t 7g 3 1 8g 2 2 5 3x2 2 5x 1 2

x

3. Circle the graph(s) that can be represented by a polynomial.

Chapter 5 118

y

x

y

x

y

x

polynomial

3t rt r3

monomials

2

Write the number of terms in each polynomial.

4. 6 2 7x2 1 3x 5. 4b5 2 3b4 1 7b3 1 8b2 2 b 6. 3qr2 1 q3r 2 2 q2r 1 7

HSM11A2MC_0501_118 118 3/16/09 2:54:51 PM

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Lesson Check

121 Lesson 5-1

• Do you UNDERSTAND?

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 end behavior

Rate how well you can describe the graph of a polynomial function.

Using Differences to Determine Degree

Got It? What is the degree of the polynomial function that generates the data shown in the table at the right?

19. Complete the flowchart to find the differences of the y-values.

20. The degree of the polynomial is .

Problem 4

x y

3

2

1

0

1

2

3

23

16

15

10

13

12

29

linear

quadratic

cubic

quartic

23

39 1 5

16 15 10 13 12 29

1st differences

2nd differences

3rd differences

4th differences

Now Iget it!

Need toreview

0 2 4 6 8 10

HSM11A2MC_0501_121 121 3/16/09 2:55:06 PM

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Chapter 5 120

Graphing Cubic Functions

Got It? What is the graph of y 5 2x3 1 2x2 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 2x2 2 x 2 2.

18. The end behavior of y 5 2x 3 1 2x2 2 x 2 2 is down / up and down / up , and

there are 1 / 2 / 3 turning points.

Problem 3

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.

11. Use your answer to Exercise 10 to identify a and n for the leading term.

a 5 n 5

12. In this polynomial, a is positive / negative , and n is even / odd .

HSM11A2MC_0501_120 120 3/16/09 2:55:01 PM

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Name Class Date

5-1 Think About a PlanPolynomial 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 variablesyou use in your formula.

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.

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.

________________________________________________________________

________________________________________________________________ .

Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

2

, where r is ____________________ _______________. and h is

________________________.where r is

Name Class Date

5-1 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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38 9


Name Class Date

5-1 Practice Form K



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


6. y = 5x3 − 2x2 + 1 7. y = 5 − x + 4x2 8. y = x − x2 + 10

9. y = 3x2 + 9 − x3 10. y = 8x2 − 4x4 + 5x7 − 2 11. y = 20 − x5

12. y = 1 + 2x + 4x3 − 8x4 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 + 2xTo start, make a table of values to help you sketch the middle part of the graph.

x y –2 –12 –1 –3

0 0 1 3 2 12

16. y = −3x3 + 4x2 − 1 17. y = 4x3 + 2x2 − x

Determine the degree of the polynomial function with the given data.

Prentice Hall Foundations Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

5

18. 19.x y

–3 –43 –2 –10 –1 1

0 21 52 223 65

x y –3 65 –2 5 –1 –5

0 –11 5 2 253 95

Name Class Date

5-1 Practice Form G(continued)


Determine the sign of the leading coefficient and the degree of the polynomial function for each graph.

37.

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.


4

38.36.

10 11


Name Class Date

Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

7

Multiple Choice

For Exercises 1–7, choose the correct letter.

1. Which expression is a binomial?

2x x2 3x2 1 2x 1 4 x 2 9

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 diff erences are constant, what is the constant value?

212 25 1 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

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). Th en name the polynomial by degree and the number of terms.

5-1 Standardized Test Prep Polynomial Functions

x

�3 77

24�2

�1

0

1

2

3

1

�4

�3

�8

�31

y

Name Class Date

5-1 Practice (continued) Form K


Determine the sign of the leading coefficient and the degree of the polynomial function for each graph.

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?

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 ofvalues and a graph.

Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

6

20. 21. 22.

x y –3 98 –2 20 –1 6 0 2

1 2 2 48 3 230

12 13


Prentice Hall Algebra 2 • Solve It/Lesson Quiz on TransparenciesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

30

AnswersSolve It!1074; work backwards using the constant third difference, 24, to fi nd the fi rst and second differences and the seventh and eighth numbers in the fi rst column.

Lesson Quiz 1. x5 1 2x3 2 3x2; quintic

trinomial 2. down and down

3. The end behavior is down and up. There are no turning points.

4. 2

1. Write 2x3 2 3x2 1 x5 in standard form. What is the classifi cation 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?

5-1 Lesson Quiz

5-1 Solve It!

The first column shows a sequence of numbers. For 1st differences, subtract consecutivenumbers in the sequence:

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 3rddifferences continues, what is the 8th numberin the first column? Justify your reasoning.

Working backwards unlocks the patterns.

x y

1

2

3

4

5

6

7

8

2

0

6

16

30

48

70

96

Prentice Hall Algebra 2 • Additional ProblemsCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

59

Name Class Date

5-1 Additional ProblemsPolynomial Functions

Problem 2

Consider the leading term of y 5 3x4 2 2x3 1 x 2 1. What is the end behavior of the graph?

Problem 1

Write 23x 1 4x3 1 7x 2 3 in standard form. What is the classifi cation of this polynomial by its degree? by its number of terms?

14 15


Prentice Hall Algebra 2 • Additional ProblemsCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

60

Name Class Date

Problem 4

What is the degree of the polynomial function that generates the data shown in the table?

Problem 3

What is the graph of y 5 3 2 2x3 1 x? Describe the graph.

5-1 Additional Problems (continued)


x y

2

1

0

1

2

3

4

13

4

1

2

11

32

71

Name Class Date

5-1 Reteaching

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

8x4 − 3x + 3x6 + 9x3 − 12 Combine like terms.

3x6 + 8x4 + 9x3 − 3x − 12 Place terms in descending order.

Step 2 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 The first term is 3x6.

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.

ExercisesWhat 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


9


16 17


Name Class Date

5-1ELL Support


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

6. quadratic F. 6x5 + 3x3 + 11x + 3

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 _____________________________________________ .


1

Name Class Date

5-1

Reteaching (continued)


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.

Step 2 Continue determiningdifferences until the y-valuesare all equal. The quantity ofdifferences is the degree of the

X y–3 52(y1) –2 18(y2) –1 2(y3) 0 –2 (y4) 1 0 (y5)

2 2 (y6)

3 –2 (y7)

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.

polynomial function.The third differences are all equal so this is athird degree polynomial function. The valueof the third differences is –6.

ExercisesWhat is the degree of the polynomial function that generates the data in the table? What are the differences when they are constant?

4. 5. 6.


10

x y –3 6 –2 26 –1 8 0 0 1 2

2 –34

3 –204

x y –3 216 –2 24 –1 0 0 0 1 0 2 –24 3 –216

x y

–3 –101 –2 –37 –1 –11 0 –5 1 –1 2 19 3 73

18 19


TEACHERS INSTRUCTIONS

5-3 Game: Discovering Your RootsSolving Polynomial Equations

Provide the host with the following equations and their solutions.

Prentice Hall Algebra 2 • Activities, Games, and Puzzles Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

39

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 93 , –2

i±

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 57 ,3

i±

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 3, –2

i±

13. (x2 – 2) (2x – 3)2 = 0 32,2

±

14. (x2 + 2)(2x – 4)2 = 0 2 ,2i±

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

Name Class Date

5-4 Activity: Researching the FactorsDividing 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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4120 21


Name Class Date

5-2 Puzzle: Made in the ShadePolynomials, 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) __________________________ ___________________________

C. P(x) = x(x + 4)(x + 3)(x + 1)(x − 1) D. P(x) = x(x2 − 25)(x2 + 4x + 3)

___________________________ ____________________________

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)

________________________________ _______________________________

K. P(x) = x(x2 − 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?


38

Name Class Date

5-3Game: Discovering Your RootsSolving 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


4022 23

For review purposes only. Not for sale or resale.For review purposes only. Not for sale or resale.

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5-1 EnrichmentPolynomial 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 toset 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 toset 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.


8

24 25


26 27


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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 21 63

y x x= + + , your friend says

the end behavior of the graph is down and up. What mistake did your friend make?

Prentice Hall Gold Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Chapter 5 Chapter Test Form G

Do you know HOW?


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. 3 , 2i 13. 2 – i, 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?


93

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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 = x3 to the graph of ( )31 4y x= +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.

The formula is 212

PE kl= , where k is the spring constant. When you stretch a

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.


92 30 31


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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 aftereach 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 avertical 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-degreepolynomial 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 ?


94Prentice Hall Algebra 2 • Find the Errors!

Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

25

Name Class Date

For each exercise, identify the error(s) in planning the solution or solving the problem. Th en 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

Th ere are 4 terms, so the function is even and the fi rst term is negative.

Th e 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.

23 and 21 are zeros of multiplicity 1.

Th e graph looks close to linear at the x-intercepts

21 and 23. It resembles a parabola at x-intercept 8.

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.

x 52(22) 4 "(22)2 2 4(1)(24)

2(1)5

2 4 !202

x 5 1 1 !5 or x 5 1 2 !5

Th e solutions are 1 1 !5 and 1 2 !5..

�2�100

100200300400500

�200�300

O

y

x

2 4 6 8 10

Chapter 5 Find the Errors!For use with Lessons 5-1 through 5-3

32 33


Prentice Hall Algebra 2 • Find the Errors!Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Name Class Date


1. What is the expansion of (p 2 3q)4? Use the Binomial Th eorem.

(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. Th e chart shows the number, in thousands, of CDs sold by a local band during the fi rst 7 months. What cubic function best models the data? Use the model to estimate sales of CDs in the 8th month.

Th e n 1 1 Point Principle says that a cubic function requires 4 points. Use CUBICREG on a graphing calculator with the fi rst 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. Th e function is f (x) 5 0.75x3 2 1.5x2 2 2.25x 1 5.

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 275During 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 S y 5 6x3 Multiply by 6 to stretch.Step 2 y 5 6x3 S y 5 (6x 2 5)3 Replace x with x 2 5 to translate

horizontally. Step 3 y 5 (6x 2 5)3 S y 5 (6x 2 5)3 2 4 Subtract 4 to translate vertically.

Th e 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 4B3 1 1 5 0

Ax 2 4B3 5 21

x 2 4 53"21

x 2 4 5 21

x 5 23

or

x 2 4 5 1

2 4x 5 5

Th e real zeros are 3 and 5.


Month

3

2

1 2

4

0.5

5

5

6

20

7

42

40

35

CD sales(thousands)

Prentice Hall Algebra 2 • Find the Errors!Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Name Class Date


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 3x4 1 3x3 2 7x 2 3

22x3 2 7x 2 322x3 2 6x 2 3

21x 2 321x 2 3

0

Th e quotient is x3 2 2x2 2 1 with remainder 0.

2. What is a third-degree polynomial function y 5 P(x) with rational coeffi cients 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 6BP(x) 5 Ax2 1 9 2 2B Ax 2 6BP(x) 5 Ax2 1 7B Ax 2 6BP(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?

Use synthetic division and factoring.Usfac

Find the zeros of the function.FinThe polynomial equation has degree 3. There are 3 roots

Step 1 Th e polynomial is in standard form. Th e possible rational roots are

41, 47, 412 and 47

2.

Step 2 Substitute 212 for x. Th e value of f (x) is 0. So, 21

2 is a root and

x 1 12 is a factor.

Step 3 Use synthetic division to factor out x 1 12: 0 2 21 14 27

0 2 21 10 27

0 2 20 14 20

Qx 1 12R A2x2 1 14B 5 2Qx 1 1

2R Ax2 1 7BStep 4 Th e complex roots are 21

2, !7, and 2!7.


212

34 35



100

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Chapter 5 Performance Tasks (continued)

Task 3

Th e 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.

Task 4

Th e 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?

Year

1984 38.19

36.451988

1992

1994

1998

2002

2006

37.14

36.33

35.59

34.42

34.84

Time (sec)

SOURCE: www.infoplease.com


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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 , fi nd the roots of this equation in terms of a, b, and c.

Task 2

a. Use division to fi nd the remaining roots of y 5 12x

3 1 32x

2 2 3x 2 4.

b. Use division to fi nd the remaining roots of y 5 x3 2 4x2 2 x 1 4.

c. Use the roots found in parts a and b to rewrite the functions in factored form.

Chapter 5 Performance Tasks

x

y

O4 6�2�6

246

�4�6

(2, 0)

x

y

2 6�2�4

�6

8

�8

(4, 0)

36 37

For review purposes only. Not for sale or resale.

Name Class Date

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 themiddle 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)

Prentice Hall Algebra 2 • Extra PracticeCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Extra PracticeChapter 5

Lesson 5-1


1.a 2.2 + 4a − 5a2 − a 3x −13− 5x 3. 3n2+ n3– n – 3 – 3n3

4.15 5. 6c− y 2 −10y − 8+8y 2 – 4c+ 7 – 8c2 6. 3x 2 − 5x − x 2 + x + 4x


7. y = x2 – 2x + 3 8. y = x3 – 2x 9. y = 7x5+ 3x3 – 2x

10. y =12

x 4 + 5x 2 −12

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, 13

, 1

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)

29. y = x2(x + 1) 30. y = (x − 3)(x − 4)2

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?


17 For review purposes only. Not for sale or resale.38 39


Name Class Date


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 1970 1980 1990 2000

Population (millions) 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 willreach 20.59 million.

Lesson 5-9

Determine the cubic function that is obtained from the parent function y = x 3 aftereach sequence of transformations.

84. vertical stretch by a factor of 2; 85. vertical stretch by a factor of 3;reflection across the x-axis; vertical translation down 2 units;horizontal translation 3 units left horizontal translation 1 unit right

Find all the real zeros of each function.

86. 87. 6y = 2 x − 3( )3 + 2 x + 3( )3 − 6 88. −13

x +12

⎛⎝⎜

⎞⎠⎟

3

− 5

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

Prsentice Hall Algebra 2 • Extra PracticeCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Name Class Date


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 cutof 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

such triangles can be represented as f (n) =n3 − 3n2 + 2n

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 abinomial expression for the volume of the cube.

76. What is the sixth term in the binomial expansion of (3x − 4)8?


1940 41



105

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Chapter 5 Project: Curves by Design (continued)

Finishing the Project

Th e 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 fi eld of computer-assisted design.


104

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Chapter 5 Project: Curves by Design

Beginning the Chapter Project

A continuous curve can be approximated by the graph of a polynomial. Th is fact is central to modern car design. Scale models are fi rst 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 fi nalized, 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. Th e graph of this polynomial equation is shown at the right. Use a graphing calculator to fi ne-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 fi ts 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: GraphingIdentify and label ten points on the sketch you made in Activity 2. Do you think the function that best fi ts these points will be more accurate than the function you found using four points? Explain your reasoning. Th en fi nd the new function using a graphing calculator and the CubicReg feature.

42 43



103

T E A C H E R I N S T R U C T I O N S

Chapter 5 Project Teacher Notes: Curves by Design

About the Project

Th e Chapter Project gives students an opportunity to adjust a polynomial equation to fi t the curve for their designs of the hood section of a car. Th ey 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.


Students graph the given polynomial and fi ne-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.


Students use their calculators to fi nd 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 fi nding 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 fi tting the equations to the curves easier or more accurate.


106

Name Class Date

Chapter 5 Project Manager: Curves by Design

Getting StartedRead 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 fi rst.

☐ Activity 2: fi nding a cubic model ☐ Label the turning points.

☐ Activity 3: fi nding a better fi t ☐ 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 Rubric4 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 satisfi es few of the requirements and shows poor organization and eff ort.

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

44 45



102

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Chapter 5 Cumulative Review (continued)

11. A sixth-degree polynomial function with rational coeffi cients has complex roots 6, !2, and 25i. Which of the following cannot be another complex root of this polynomial?

5i i!3 2!2 0

12. Solve (x 1 3)(x 1 4) 5 0

x 5 3 or x 5 4 x 5 23 or x 5 24

x 5 0 none of the above

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 25, 23, 21 21, 3, 5 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 Th eorem 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 refl ection 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. Th e 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?


101

Name Class Date

Chapter 5 Cumulative Review

Multiple Choice

For Exercises 1–14, choose the correct letter.

1. Which relation is not a function?

y 5 0 y 5 2x y 5 x 1 2 x 5 2

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 y 1 4 . 0

y # x 1 1 ?

(3, 3) (21, 2) (1, 5) (0, 2)

4. Which of the following is the equation of a parabola?

y 5 x 2 1 y 5 ux 1 3 u y 5 x2 1 1 x 5 y 1 2

5. Which of these is a direct variation?

x 5 8 y 5 8 y 5 8x y 5 8x2

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

7. Which polynomial is written in standard form?

1 1 3x 2 5x2 3x2 1 2 1 x3 4x 2 5x2 6x3 2 x 1 7

8. Solve the system e x 1 4 5 0

y 5 x 1 1 .

(24, 23) (4, 3) (24, 25) (24, 3)

9. Solve 8x , 12 1 4x.

x , 1 x , 3 x . 3 x 5 3

10. What is the axis of symmetry of y 5 2(x 2 3)2 1 5?

y 5 3 x 5 2.5 x 5 3 x 5 5

46 47

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