Additional Vocabulary Support

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

1

Name Class Date

6-1 Additional Vocabulary Support Roots and Radical Expressions

Complete the vocabulary chart by fi lling in the missing information.

Word or Word Phrase

Defi nition Example

nth root Given the equation an 5 b, a is the nth root of b.

1.

radicand 2. Th e radicand in the expression !3 64 is 64.

index Th e number that gives the degree of the root.

3.

cube root Th e third root of a number. 4.

principal root 5. Th e principal square root of 4 is 2.

Choose the word or phrase from the list that best completes each sentence.

cube root nth root radicand index principal root

6. Th e is the number under the radical sign in a radical expression.

7. Th e of 27 is 3.

8. Given the equation an 5 b, a is the of b.

9. In a radical expression, the indicates the degree of the root.

10. When a number has both a positive and a negative root, the positive root is

considered the .

34 5 81; 3 is the 4th root of 81.

The cube root of 8 is 2.

The index in the

expression 5!32 is 5.

The number under the radical sign.

The positive root when a number has both a positive and a negative root.

radicand

cube root

nth root

index

principal root


2

Name Class Date

6-1 Think About a PlanRoots and Radical Expressions

Boat Building Boat builders share an old rule of thumb for sailboats. Th e maximum speed K in knots is 1.35 times the square root of the length L in feet of the boat’s waterline. a. A customer is planning to order a sailboat with a maximum speed

of 12 knots. How long should the waterline be? b. How much longer would the waterline have to be to achieve a maximum

speed of 15 knots?

1. Write an equation to relate the maximum speed K in knots to the length L in feet of a boat’s waterline.

2. How can you fi nd the length of a sailboat’s waterline if you know its maximum speed?

.

3. A customer is planning to order a sailboat with a maximum speed of 12 knots. How long should the waterline be?

4. How can you fi nd how much longer the waterline would have to be to achieve a maximum speed of 15 knots, compared to a maximum speed of 12 knots?

.

5. If a customer wants a sailboat with a maximum speed of 15 knots, how long should the waterline be?

6. How much longer would the waterline have to be to achieve a maximum speed of 15 knots?

K 5 1.35!L

Substitute the maximum speed for K and solve the resulting equation for L

Subtract the waterline length needed for a 12-knot maximum speed from the

waterline length needed for a 15-knot maximum speed

about 123 ft

about 44 ft

about 79 ft

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

3

Name Class Date

6-1 Practice Form G

Roots and Radical Expressions

Find all the real square roots of each number.

1. 400 2. 2196 3. 10,000 4. 0.0625

Find all the real cube roots of each number.

5. 216 6. 2343 7. 20.064 8. 100027

Find all the real fourth roots of each number.

9. 281 10. 256 11. 0.0001 12. 625

Find each real root.

13. !144 14. 2!25 15. !20.01 16. !3 0.001

17. !4 0.0081 18. !3 27 19. !3 227 20. !0.09

Simplify each radical expression. Use absolute value symbols when needed.

21. "81x4 22. "121y10 23. "3 8g6

24. "3 125x9 25. "5 243x5y15 26. "3 (x 2 9)3

27. "25(x 1 2)4 28. %3 64x9

343 29. !3 20.008

30. %4 x4

81 31. "36x2y6 32. "4 (m 2 n)4

33. A cube has volume V 5 s3, where s is the length of a side. Find the side length for a cube with volume 8000 cm3.

34. Th e temperature T in degrees Celsius (8C) of a liquid t minutes after heating is given by the formula T 5 8!t . When is the temperature 488C?

12

0.3

9x2

3xy3

6

220, 20

no real fourth roots

25

3

11 »y5…

x 2 9

20.2

6»x…y3»x…3

»m 2 n…

36 min

20 cm

27

no real square roots

24, 4

not a real number

23

2g2

5(x 1 2)2

20.4

2100, 100

20.1, 0.1

0.1

0.3

5x3

4x3

7

103

20.25, 0.25

25, 5


4

Name Class Date

Find the two real solutions of each equation.

35. x2 5 4 36. x4 5 81

37. x2 5 0.16 38. x2 51649

39. x4 516

625 40. x2 5121625

41. x2 5 0.000009 42. x4 5 0.0001

43. Th e number of new customers n that visit a dry cleaning shop in one year is directly related to the amount a (in dollars) spent on advertising. Th is relationship is represented by n3 5 13,824a. To attract 480 new customers, how much should the owners spend on advertising during the year?

44. Geometry Th e volume V of a sphere with radius r is given by the formula V 5

43 pr3.

a. What is the radius of a sphere with volume 36p cubic inches? b. If the volume increases by a factor of 8, what is the new radius?

45. A clothing manufacturer fi nds the number of defective blouses d is a function of the total number of blouses n produced at her factory. Th is function is d 5 0.000005n2.

a. What is the total number of blouses produced if 45 are defective? b. If the number of defective blouses increases by a factor of 9, how does the

total number of blouses change?

46. Th e velocity of a falling object can be found using the formula v2 5 64h, where v is the velocity (in feet per second) and h is the distance the object has already fallen.

a. What is the velocity of the object after a 10-foot fall? b. How much does the velocity increase if the object falls 20 feet

rather than 10 feet?

6-1 Practice (continued) Form G


22, 2

$8000

3 in.

6 in.

3000

about 25.30 ft/sec

It has tripled.

about 10.48 ft/sec

20.003, 0.003

225, 25

23, 3

20.1, 0.1

20.4, 0.4

21125,

1125

247, 47

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

5

Name Class Date

Find all the real square roots of each number.

1. 625 2. 21.44 3. 1681

Find all the real cube roots of each number.

4. 2216 5. 164 6. 0.027

Find all the real fourth roots of each number.

7. 0.2401 8. 1 9. 21296

Find each real root. To start, fi nd a number whose square, cube, or fourth is equal to the radicand.

10. !400 11. 2!4 256 12. !3 2729

5 "(20)2

Simplify each radical expression. Use absolute value symbols when needed. To start, write the factors of the radicand as perfect squares, cubes, or fourths.

13. "25x6 14. "3 343x9y12 15. "4 16x16y20

5 "(5)2(x3)2

6-1 Practice Form K


625

26

60.7

20

5»x 3…

no real roots

14

61

24

7x 3y

4

649

0.3

no real fourth roots

29

2x 4»y

5…


6

Name Class Date

16. Th e formula for the volume of a sphere is V 543 pr3. Solving for r, the radius of

a sphere is r 5 Å3 3V4p . If the volume of a sphere is 20 ft3, what is the radius of the

sphere to the nearest hundredth?


17. x4 5 81 18. x2 5 144 19. x4 52401625

20. Writing Explain how you know whether or not to include the absolute value symbol on your root.

21. Arrange the numbers !3 264, 2!3 264, !64, and !6 64, in order from least to greatest.

22. Open-Ended Write a radical that has no real values.

23. Reasoning Th ere are no real nth roots of a number b. What can you conclude about the index n and the number b?

6-1 Practice (continued) Form K


6 3 6 12 w 75

1.68 ft

If the index is odd, then you do not use the absolute value symbol on your root. If the index is even, then you need the absolute value symbol on those variable terms with an odd power.

Answers may vary. Sample: any even index radical with a negative radicand

The index n is even and the number b is negative.

3!264 , 6!64, 2 3!264 , !64


7

Name Class Date

Multiple Choice

For Exercises 1−6, choose the correct letter.

1. What is the real square root of 0.0064?

0.4 0.04

0.08 no real square root

2. What is the real cube root of 264?

4 28

24 no real cube root

3. What is the real fourth root of 2 1681?

23 2

49

223 no real fourth root

4. What is the value of !3 20.027?

20.3 0.3 20.03 0.03

5. What is the simplifi ed form of the expression "4x2y4?

2xy2 2 u x uy2 4xy2 2 u xy u

6. What are the real solutions of the equation x4 5 81?

29, 9 3 23, 3 23

Short Response

7. Th e volume V of a cube with side length s is V 5 s3. A cubical storage bin has volume 5832 cubic inches. What is the length of the side of the cube? Show your work.

6-1 Standardized Test Prep Roots and Radical Expressions

B

G

D

F

B

H

[2] V 5 s3, 5832 5 s3, s 53!5832 5 18; 18 in.

[1] incorrect side length OR no work shown[0] incorrect answer and no work shown OR no answer given


8

Name Class Date

Rounding Roots and RadicalsComputers treat radicals such as !2 as if they were rounded to a preassigned number of decimal places. Most computers round numbers according to an algorithm that uses the largest integer less than or equal to a given number. Th is function is called the greatest integer function and is written as y 5 fxg .

As you can see, the graph of the greatest integer function is not continuous. Th e open circles indicate that the endpoint is not included as part of the graph.

Th e command INT in most popular spreadsheet programs serves the same purpose as the greatest integer function. For instance, INT(3.84) 5 3; INT(21.99) 5 22; INT(7) 5 7.

To round a number x to r decimal places, a computer performs the following procedure:

Step 1 Multiply x by 10r.

Step 2 Add 0.5 to the result.

Step 3 Find INT of the result.

Step 4 Multiply the result by 102r.

Fill in the table below to see how this procedure works.

x r Step 1 Step 2 Step 3 Step 4

11.4825 3 11482.5 11483

132.718 2

34.999 1

A computer that rounds numbers after each operation may introduce rounding errors into calculations. To see the eff ects of rounding errors, perform each of the following computations for x 5 2 and diff erent r values. First fi nd the given root and write the answer to r 1 1 digits after the decimal. Carry out the four steps to get the answer and then raise the result to the given power. Write the answer again to r 1 1 digits after the decimal and carry out the four steps to get the fi nal answer.

x r Q!xR2 Q!3 x R3

2 6

2 3

2 1

6-1 EnrichmentRoots and Radical Expressions

O

2

�2 2

x

y

2.000001

1.999

2.0

2.000000

2.000

2.2

11483

13271.8

349.99

13272

350

13272.3

350.49

132.72

35.0

11.483


9

Name Class Date

For any real numbers a and b and any positive integer n, if a raised to the nth power equals b, then a is an nth root of b. Use the radical sign to write a root. Th e following expressions are equivalent:

an 5 b g !n b 5 a

Problem

What are the real-number roots of each radical expression?

a. !3 343 Because (7)3 5 343, 7 is a third (cube) root of 343. Therefore, 3!343 5 7. (Notice that (27)3 5 2343, so 27 is not a cube root of 343.)

b. 4 Å 1

625 Because Q15R4 51

625 and Q215R4 5

1625, both 15 and 2

15 are real-number fourth roots of 1

625.

c. !3 20.064 Because (20.4)3 5 20.064,20.4 is a cube root of 20.064 and is, in fact, the only one. So, 3!20.064 5 20.4.

d. !225 Because (5)2 5 (25)2 5 25, neither 5 nor 25 are second (square) roots of 225. There are no real-number square roots of 225.

Exercises

Find the real-number roots of each radical expression.

1. !169 2. !3 729 3. !4 0.0016

4. Å3 2 18 5. Å 4

121 6. Å3 125216

7. Å2 4

25 8. !4 0.1296 9. !3 20.343

10. !4 20.0001 11. Å5 1243 12. Å3 8

125

6-1 Reteaching Roots and Radical Expressions

an 5 g !n!!b!b 5 a!power index radicand

radical sign

213, 13

212

no real sq root

no real 4th root

9

2 211, 2

11

20.6, 0.6

13

20.2, 0.2

56

20.7

25


10

Name Class Date

You cannot assume that "n an 5 a. For example, "(26)2 5 !36 5 6, not 26. Th is leads to the following property for any real number a:

If n is odd "n an 5 a

If n is even "n an 5 u a u

Problem

What is the simplifi ed form of each radical expression?

a. "3 1000x3y9

"3 1000x3y9 5 "3 103x3(y3)3 Write each factor as a cube.

5 "3 (10xy3)3 Write as the cube of a product.

5 10xy3 Simplify.

b. Å4 256g8

h4k16

Write each factor as a power of 4.

5 Å4 a4g2

hk4b4 Write as the fourth power of a quotient.

54g2

u h u k4 Simplify.

Th e absolute value symbols are needed to ensure the root is positive when h is negative. Note that 4g2 and k4 are never negative.

Exercises


13. "36x2 14. "3 216y3 15. Å 1

100x2

16. "x20

"y8 17. Å3 (x 1 3)3

(x 2 4)6 18. "5 x10y15z5

19. Å3 27z3

(z 1 12)6 20. "4 2401x12 21. Å3 1331

x3

22. Å4 (y 2 4)8

(z 1 9)4 23. Å3 a6b6

c3 24. "3 2x3y6

6-1 Reteaching (continued) Roots and Radical Expressions

Å4 44(g2)4

h4(k4)45Å4 256g8

h4k16

6»x…

x10

y4

3z(z 1 12)2

(y 2 4)2

»z 1 9…

6y

x 1 3(x 2 4)2

7»x3…

a2b2

c

110»x…

x2y3z

11x

2xy2


11

Name Class Date

6-2 Additional Vocabulary Support Multiplying and Dividing Radical Expressions

Combining Radicals: Products

If !n a and !n b are real numbers, then !n a ? !n b 5 !n ab.

Sample !3 8 ? !3 27 5 !3 8 ? 27 5 !3 216 5 6

Solve.

1. !3 16 ? !3 4 5

2. Which of the following products can be simplifi ed? Circle the correct answer.

!3 12 ? !6 !4 16 ? !4 24 !4 35 ? !3 10

3. Write the radical expression !3 32x4 in simplest form.

4. Which of the following products cannot be simplifi ed? Circle the correct answer.

!4 15 ? !4 4 !4 ? !12 !4 10 ? !3 5

5. "4x2y3 ? "27x2y2 5

Combining Radicals: Quotients

If !n a and !n b are real numbers and b 2 0, then !n a!n b5 nÅa

b .

Sample !8!25 Å8

2 5 !4 5 2

Solve.

6. Which of the following quotients can be simplifi ed? Circle the correct answer.

3 !12

3!4

!3 6"3

!4 20!3 15

7. Write the radical expression "64x4

"4x2 in simplest form.

8. Rewriting an expression so that there are no radicals in any denominator and no

denominators in any radical is called .

3!16 ? 4 53!64 5 4

3!4x

6x2y2"3y

l4xl

rationalizing the denominator

2x


12

Name Class Date

6-2 Think About a Plan Multiplying and Dividing Radical Expressions

Satellites Th e circular velocity v, in miles per hour of a satellite orbiting Earth is

given by the formula v 5 Å1.24 3 1012

r , where r is the distance in miles from the satellite to the center of the Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi? (Th e radius of the Earth is 3950 mi.)

Know

1. Th e fi rst satellite orbits at an altitude of z z.

2. Th e second satellite orbits at an altitude of z z.

3. Th e distance from the surface of the Earth to its center is z z.

Need

4. To solve the problem I need to fi nd:

.

Plan

5. Rewrite the formula for the circular velocity of a satellite using a for the altitude of the satellite.

6. Use your formula to fi nd the velocity of a satellite orbiting at an altitude of 100 mi.

7. Use your formula to fi nd the velocity of a satellite orbiting at an altitude of 200 mi.

8. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than one orbiting at an altitude of 200 mi?

100 mi

200 mi

the difference in the velocities of a satellite orbiting at an altitude of 100 mi

and one orbiting at an altitude of 200 mi

v 5 Å1.24 3 1012

a 1 3950

about 17,498 mi/h

about 17,286 mi/h

about 212 mi/h

3950 mi


13

Name Class Date

Multiply, if possible. Th en simplify.

1. !4 ? !25 2. !81 ? !36 3. !3 ? !3 27

4. "3 45 ? "3 75 5. !18 ? !50 6. !3 216 ? !3 4

Simplify. Assume that all variables are positive.

7. "36x3 8. "3 125y2z4 9. "18k6

10. "3 216a12 11. "x2y10z 12. "4 256s7t12

13. "3 216x4y3 14. "75r3 15. "4 625u5v8

Multiply and simplify. Assume that all variables are positive.

16. !4 ? !6 17. "9x2 ? "9y5 18. "3 50x2z5 ? "3 15y3z

19. 4!2x ? 3!8x 20. !xy ? !4xy 21. 9!2 ? 3!y

22. "12x2y ? "3xy4 23. "3 29x2y4 ? "3 12xy 24. 7"3y2 ? 2"6x3y

Divide and simplify. Assume that all variables are positive.

25. "75"3

26. "63xy3

"7y 27.

"54x5y3

"2x2y

28. "6x"3x

29. 3"4x2

3"x 30. 4Ä243k3

3k7

31. "(2x)2

"(5y)4 32.

3"18y2

3"12y 33. Ä162a

6a3

6-2 Practice Form G

Multiplying and Dividing Radical Expressions

15 30 24

10

2"6

6xy2"xy 23xy 3"4y2 42xy"2xy

54

9xy2!y

9

5yz2 3"6x2

6x!x 5z 3"y2z 3k3"2

22a4 3!2 xy5"z 4st3 4"s3

6xy 3"x 5r"3r 5uv2 4!u

48x 2xy 27!2y

5 3y!x 3xy!3x

3"3a

"2 3"4x 3k

2x25y2

3"12y2


14

Name Class Date

Rationalize the denominator of each expression. Assume that all variables are positive.

34. !y!5

35. "18x2y"2y3

36. "3 7xy2

"3 4x2

37. Å9x2 38.

!xy!3x 39. Ä3 x2

3y

40. !4 2x"4 3x2

41. Å x8y 42. Ä3 3a

4b2c

43. What is the area of a rectangle with length !175 in. and width !63 in.?

44. Th e area of a rectangle is 30 m2. If the length is !75 m, what is the width?

45. Th e volume of a right circular cone is V 513pr2h, where r is the radius of the base and h

is the height of the cone. Solve the formula for r. Rationalize the denominator.

46. Th e volume of a sphere of radius r isV 543pr3.

a. Use the formula to fi nd r in terms of V. Rationalize the denominator. b. Use your answer to part (a) to fi nd the radius of a sphere with volume

100 cubic inches. Round to the nearest hundredth.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive.

47. !14 ? !21 48. !3 150 ? !3 20 49. !3Q!12 2 !6R 50.

6!2x5!3

51. 8"3 2x2 52.

5!3 xy4

"3 25xy2



"5y5

3xy

3"14x2y2

2x

3"2x2

4"54x3

3x

"3y3

"2xy4y

3"9x2y2

3y

3"6abc2

2bc

105 in.2

2"3 m

r 5"3πhVπh

2.88 in.

7"6 6 2 3"2

2"6x5

43"4xx

3"5y2

103"3

r 53"3V

4π; r 5

3"6π2V2π


15

Name Class Date

6-2 Practice Form K


Multiply, if possible. Th en simplify. To start, identify the index of each radical.

1. !3 4 ? !3 6 2. !5 ? !8 3. !3 6 ? !4 9

index of both radicals is 3!3 4 ? 6

Simplify. Assume all variables are positive. To start, change the radicand to factors with the necessary exponent.

4. "3 27x6 5. "48x3y4 6. "5 128x2y25

5 "3 33 ? (x2)3

Multiply and simplify. Assume all variables are positive.

7. !12 ? !3 8. "4 7x6 ? "4 32x2 9. 2"3 6x4y ? 3"3 9x5y2

Simplify each expression. Assume all variables are positive.

10. !3 4 ? !3 80 11. 5"2xy6 ? 2"2x3y 12. !5Q!5 1 !15R

13. Error Analysis Your classmate simplifi ed "5x3 ? "3 5xy2 to 5x2y. What mistake did she make? What is the correct answer?

14. A square rug has sides measuring !3 16 ft by !3 16 ft. What is the area of the rug?

2

3 !3

3x 2

6

4 3!5

They are different, so you cannot multiply the radicands.

4 3!4 ft 2

2!10

4xy 2!3x

2x 2 4!14

20x 2y

3!y

The indexes are different, so you cannot multiply.

2y 5 5"4x

2

18x 3y 3!2

5 1 5!3

She thought the indexes were the same.


16

Name Class Date

Divide and simplify. Assume all variables are positive. To start, write the quotient of roots as a root of a quotient.

15. "36x6

"9x4 16.

"4 405x8y2

"4 5x3y2 17.

"3 75x7y2

"3 25x4

5 Å36x6

9x4

Rationalize the denominator of each quotient. Assume all variables are positive. To start, multiply the numerator and denominator by the appropriate radical expression to eliminate the radical.

18. !26!3

19. !3 x!3 2

20. "7x4y!5xy

5 !26!3?!3!3

21. Einstein’s famous formula E 5 mc2 relates energy E, mass m, and the speed of light c. Solve the formula for c. Rationalize the denominator.

22. Th e formula h 5 16t2 is used to measure the time t it takes for an object to free fall from height h. If an object falls from a height of h 5 18a5 ft, how long did it take for the object to fall in terms of a?



2x

!783

c 5 ÅEm; c 5

!Emm

3a 2!2a4 seconds

3x 4!x

3!4x2

x 3"3y

2

x !35x5


17

Name Class Date

6-2 Standardized Test Prep Multiplying and Dividing Radical Expressions

Multiple Choice

For Exercises 1−5, choose the correct letter. Assume that all variables are positive.

1. What is the simplest form of !3 249x ? "3 7x2?

7x!7x 27x 7x 27"3 x2

2. What is the simplest form of "80x7y6?

2x3y3!20x 4x6y6"5x3 4"5x7y6 4x3y3!5x

3. What is the simplest form of "3 25xy2 ? "3 15x2?

5x"3 3y2 5x!3 3y 15xy!3 y 5xy!15x

4. What is the simplest form of "75x5

"12xy2?

5"3x4

2"3y2 5x2

2y 5x!x2y

5x2y2

5. What is the simplest form of 2"3 x2y"3 4xy2

?

"3 x2y

2y x!3 2y

y "3 2xy2

y !3 2yxy

Short Response

6. Th e volume V of a wooden beam is V 5 ls2, where l is the length of the beam and s is the length of one side of its square cross section. If the volume of the beam is 1200 in.3 and its length is 96 in., what is the side length? Show your work.

B

I

A

G

C

[2] V 5 ls2; s 5 ÅVl 5 Å1200

96 5 !12.5 N 3.5 in.[1] appropriate methods but with computational errors OR correct answer without

work shown[0] incorrect answer and no work shown OR no answer given


18

Name Class Date

6-2 EnrichmentMultiplying and Dividing Radical Expressions

To simplify the radical !n a, you look for a perfect nth power among the factors of the radicand a. When this factor is not obvious, it is helpful to factor the number into primes. Prime numbers are important in many aspects of mathematics. Several mathematicians throughout history have unsuccessfully tried to fi nd a pattern that would generate the nth prime number. Other mathematicians have off ered conjectures about primes that remain unresolved.

1. Goldbach’s Conjecture states that every even number n . 2 can be written as the sum of two primes. For example, 4 5 2 1 2 and 10 5 3 1 7. Choose three even numbers larger than 50 and write them as a sum of two primes.

2. Th e Odd Goldbach’s Conjecture states that every odd number n . 5 can be written as the sum of three primes. For example, 7 5 2 1 2 1 3. Choose three odd numbers larger than 50 and write them as the sum of three primes.

3. Another interesting pattern emerges when you examine a subset of the prime numbers. Make a list of the primes less than 50.

4. Make this list smaller by eliminating 2 and all primes that are 1 less than a multiple of 4.

5. Th e remaining primes in the list above are related in an interesting way. You can write each prime as the sum of two squares. Express each of these primes as a sum of two squares.

6. A Cullen number, named after an Irish mathematician James Cullen, is a natural number of the form n 3 2n 1 1. Determine the fi rst four Cullen numbers. Th at is, let n 5 1, 2, 3, 4.

7. What is the smallest Cullen number that is a prime number? (Th e next Cullen number that is a prime occurs when n 5 141!)

8. A palindrome is a number that reads the same forward and backward. For example, 121 is a palindromic number. List the seven palindromic primes that are less than 140.

Answers may vary. Sample: 52 5 47 1 5.

Answers may vary. Sample: 51 5 37 1 11 1 3.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

5, 13, 17, 29, 37, 41

5 5 1 1 4, 13 5 4 1 9, 17 5 1 1 16, 29 5 4 1 25, 37 5 1 1 36, 41 5 16 1 25

3, 9, 25, 65

3 when n 5 1

2, 3, 5, 7, 11, 101, 131


19

Name Class Date

6-2 ReteachingMultiplying and Dividing Radical Expressions

You can simplify a radical if the radicand has a factor that is a perfect nth power and n is the index of the radical. For example:

!n xynz 5 y!n xz

Problem

What is the simplest form of each product?

a. !3 12 ? !3 10

!3 12 ? !3 10 5 !3 12 ? 10 Use

5 "3 22 ? 3 ? 2 ? 5 Write as a product of factors.

5 "3 23 ? 3 ? 5 Find perfect third powers.

5 "3 23 ? "3 3 ? 5 Use n!ab 5n !a ?

n!b.

5 2!3 15 Use n"an 5 a to simplify.

b. "7xy3 ? "21xy2

"7xy3 ? "21xy2 5 "7xy3 ? 21xy2 Use n!a ?n!b 5

n !ab.

5 "7xy2y ? 3 ? 7xy2 Write as a product of factors.

5 "72x2(y2)2 ? 3y Find perfect second powers.

5 7xy2"3y Use n!an 5 a to simplify.

Exercises

Simplify each product.

1. !15x ? !35x 2. "3 50y2 ? "3 20y 3. "3 36x2y5 ? "3 26x2y

4. 5"7x3y ? "28y2 5. 2"3 9x5y2 ? "3 2x2y5 6. !3 Q!12 2 !21 R

n!a ?n!b 5

n !ab.

5x"21 10y 26xy2 3!x

70xy"xy 2x2y2 3!18xy 6 2 3"7


20

Name Class Date

Rationalizing the denominator means that you are rewriting the expression so that no radicals appear in the denominator and there are no fractions inside the radical.

Problem

What is the simplest form of !9y!2x

?

Rationalize the denominator and simplify. Assume that all variables are positive.

!9y!2x

5 Å 9y

2x Rewrite as a square root of a fraction.

5 Å 9y ? 2x

2x ? 2x Make the denominator a perfect square.

5 Å18xy

4x2 Simplify.

5!18xy

"22 ? x2 Write the denominator as a product of perfect squares.

5"18xy

2x Simplify the denominator.

5"32 ? 2 ? x ? y

2x Simplify the numerator.

53"2xy

2x Use n!an 5 a to simplify.

Exercises

Rationalize the denominator of each expression. Assume that all variables are positive.

7. !5!x

8. "3 6ab2

"3 2a4b 9.

"4 9y"4 x 10.

"10xy3

"12y2

11. 4"3 k9

16"3 k5 12. Ä3x5

5y 13. "4 10"4 z2

14. Ä3 19a2babc4

6-2 Reteaching (continued)


"5xx

3"3ba

4"9x3yx

"30xy6

k 3"k4

x2"15xy5y

4"10z2

z

3"19ac2

c2


21

Name Class Date

Th e column on the left shows the steps used to rationalize a denominator. Use the column on the left to answer each question in the column on the right.

Problem Rationalizing the Denominator

Write the expression 4!3!7 1 !3 with a

rationalized denominator.

1. What does it mean to rationalize a denominator?

Multiply the numerator and the denominator by the conjugate of the denominator.

4!3!7 1 !3?!7 2 !3!7 2 !3

2. What are conjugates?

The radicals in the denominator cancel out.4!3A!7 2 !3B

7 2 3

3. Write and solve an equation to show why the radicals in the denominator cancel out.

Distribute !3 in the numerator.4(!3 ? !7 2 !3 ? !3)

7 2 3

4. What property allows you to distribute the !3?

Simplify.4(!21 2 3)

4

5. Why do the fours in the numerator and the denominator cancel out?

Simplify.

!21 2 3

6. What number multiplied by !21 would produce a product of 21?

6-3 Additional Vocabulary Support Binomial Radical Expressions

Sample answer: It means to write

an expression so that there are no

radicals in any denominators and no

denominators in any radicals.

Conjugates are expressions that differ

only in the signs of the fi rst or second

terms.

(!7 1 !3)(!7 2 !3) 5

(!7 ? !7) 2 (!7 ? !3) 1

(!7 ? !3) 2 (!3 ? !3) 5 7 2 3

The Distributive Property

Sample answer: Because 4 divided by

4 equals 1.

!21


22

Name Class Date

6-3 Think About a PlanBinomial Radical Expressions

Geometry Show that the right triangle with legs of length !2 2 1 and !2 1 1 is similar to the right triangle with legs of length 6 2 !32 and 2.

Understanding the Problem

1. What is the length of the shortest leg of the fi rst triangle? Explain.

2. What is the length of the shortest leg of the second triangle? Explain.

3. Which legs in the two triangles are corresponding legs?

Planning the Solution

4. Write a proportion that can be used to show that the two triangles are similar.

Getting an Answer

5. Simplify your proportion to show that the two triangles are similar.

!2 2 1 ; because !2 5 !2, !2 2 1 must be less than !2 1 1

6 2 !32; because !32 is between 5 and 6, 6 2 !32 must be between 0 and 1,

which is less than 2.

The smaller leg in the fi rst triangle corresponds to the smaller leg in the second

triangle. The larger leg in the fi rst triangle corresponds to the larger leg in the

second triangle.

!2 2 1!2 1 10

6 2 !322

!2 2 1!2 1 10

6 2 !322

2(!2 2 1) 0 (!2 1 1)(6 2 !32)

2!2 2 2 0 6!2 2 !64 1 6 2 !32

2!2 2 2 0 6!2 2 8 1 6 2 4!2

2!2 2 2 5 2!2 2 2 �


23

Name Class Date

6-3 Practice Form G

Binomial Radical Expressions

Add or subtract if possible.

1. 9!3 1 2!3 2. 5!2 1 2!3 3. 3!7 2 7 3!x

4. 14!3 xy 2 3!3 xy 5. 8!3 x 1 2!3 y 6. 5!3 xy 1 !3 xy

7. !3x 2 2!3x 8. 6!2 2 5!3 2 9. 7!x 1 x!7

Simplify.

10. 3!32 1 2!50 11. !200 2 !72 12. !3 81 2 3!3 3

13. 2!4 48 1 3!4 243 14. 3!75 1 2!12 15. !3 250 2 !3 54

16. !28 2 !63 17. 3!4 32 2 2!4 162 18. !125 2 2!20

Multiply.

19. A1 2 !5B A2 2 !5B 20. A1 1 4!10B A2 2 !10B 21. A1 2 3!7B A4 2 3!7B 22. (4 2 2!3)2 23. (!2 1 !7)2 24. A2!3 2 3!2B2 25. A4 2 !3B A2 1 !3B 26. A3 1 !11B A4 2 !11B 27. A3!2 2 2!3B2Multiply each pair of conjugates.

28. (3!2 2 9)(3!2 1 9) 29. (1 2 !7)(1 1 !7)

30. (5!3 1 !2)(5!3 2 !2) 31. (3!2 2 2!3)(3!2 1 2!3)

32. (!11 1 5)(!11 2 5) 33. (2!7 1 3!3)(2!7 2 3!3)

8 3!x 1 2 3!y

6!2 2 5 3!2

4!2

19!3

0

238 1 7"10

5!2 1 2!3

9 1 2!14

1 1 !11

2!3x

22!2

13 4!3

2!7

7 2 3!5

11!3

28 2 16!3

11 3!xy

5 1 2!3

263

73

214

26

6

1

6 3!xy

7!x 1 x!7

0

2 3!2

!5

67 2 15!7

3!7 2 7 3!x

30 2 12!6

30 2 12!6


24

Name Class Date



Rationalize each denominator. Simplify the answer.

34. 3 2 !10!5 2 !2

35. 2 1 !14!7 1 !2

36. 2 1 !3 x!3 x

Simplify. Assume that all the variables are positive.

37. !28 1 4!63 2 2!7 38. 6!40 2 2!90 2 3!160

39. 3!12 1 7!75 2 !54 40. 4!3 81 1 2!3 72 2 3!3 24

41. 3!225x 1 5!144x 42. 6"45y2 1 4"20y2

43. A3!y 2 !5B A2!y 1 5!5B 44. A!x 2 !3B A!x 1 !3B 45. A park in the shape of a triangle has a sidewalk dividing it into two parts.

a. If a man walks around the perimeter of the park, how far will he walk? b. What is the area of the park?

46. Th e area of a rectangle is 10 in.2. Th e length is A2 1 !2B in. What is the width?

47. One solution to the equation x2 1 2x 2 2 5 0 is 21 1 !3. To show this,

let x 5 21 1 !3 and answer each of the following questions. a. What is x2? b. What is 2x? c. Using your answers to parts (a) and (b), what is the sum x2 1 2x 2 2?

600 ft

300 ft

300 6 ft

side

wal

k V

300 3 ftV

300

3

ftV

5(2 2 !2) in.

(900 1 300!3 1 300!6) ft or about 2154 ft270,000 1 90,000!3

2 ft2 or about 212,942 ft2

4 2 2!322 1 2!3

0

!5 2 2!23

12!7

41!3 2 3!6

105!x

6y 1 13!5y 2 25 x 2 3

26y!5

6 3!3 1 4 3!9

26!10

!2 x 1 2 3"x2

x


25

Name Class Date

Simplify if possible. To start, determine if the expressions contain like radicals.

1. 3!5 1 4!5 2. 8!3 4 2 6!3 4 3. 2!xy 1 2!y

both radicals

4. A fl oor tile is made up of smaller squares. Each square measures 3 in. on each side. Find the perimeter of the fl oor tile.

Simplify. To start, factor each radicand.

5. !18 1 !32 6. !4 324 2 !4 2500 7. !3 192 1 !3 24

5 !9 ? 2 1 !16 ? 2

Multiply.

8. A3 2 !6B A2 2 !6B 9. A5 1 !5B A1 2 !5B 10. A4 1 !7B2

Multiply each pair of conjugates.

11. A7 2 !2B A7 1 !2B 12. A1 1 3!3B A1 2 3!3B 13. A6 1 4!7B A6 2 4!7B

6-3 Practice Form K


7!5

7!2

12 2 5!6

47

2 3!4

22 4!4 or 22!2

24!5

226

no; cannot simplify

6 3!3

23 1 8"7

276

24!2 in.


26

Name Class Date

Rationalize each denominator. Simplify the answer.

14. 32 1 !6

15. 7 1 !56 2 !5

16. 1 2 2!104 1 !10

5 32 1 !6

?2 2 !62 2 !6

17. A section of mosaic tile wall has the design shown at the right. Th e design is made up of equilateral triangles. Each side of the large triangle is 4 in. and each side of a small triangle is 2 in. Find the total area of the design to the nearest tenth of an inch.


18. !45 2 !80 1 !245 19. A2 2 !98B A3 1 !18B 20. 6"192xy2 1 4"3xy2

21. Error Analysis A classmate simplifi ed the

expression 11 2 !2

using the steps shown.

What mistake did your classmate make?

What is the correct answer?

22. Writing Explain the fi rst step in simplifying !405 1 !80 2 !5.



11 2 !2

?1 2 !21 2 !2

5 1 2 !21 2 2 5

1 2 !221 5 21 1 !2

23 1 32!6

6!5

47 1 13!531

236 2 15!2

4 2 32!10

52y!3x

A N 17.3 in.2

The student multiplied the denominator by itself instead of by its conjugate; 21 2 !2

First, factor each radicand so you can combine like radicals.


27

Name Class Date

Multiple Choice


1. What is the simplest form of 2!72 2 3!32?

2!72 2 3!32 24!2 22!2 0

2. What is the simplest form of A2 2 !7B A1 1 2!7B? 212 1 3!7 16 1 5!7

212 2 3!7 3 1 !7

3. What is the simplest form of A!2 1 !7B A!2 2 !7B? 9 1 2!14 9 2 2!14 25 9

4. What is the simplest form of 72 1 !5

?

214 1 7!5 214 2 7!5

14 1 7!5 14 2 7!5

5. What is the simplest form of 8!3 5 2 !3 40 2 2!3 135?

16!3 5 12!3 5 4!3 5 0

Short Response

6. A hiker drops a rock from the rim of the Grand Canyon. Th e distance it falls d in feet after t seconds is given by the function d 5 16t2. How far has the rock fallen after (3 1 !2) seconds? Show your work.

6-3 Standardized Test PrepBinomial Radical Expressions

D

F

C

F

D

[2] d 5 16t2 5 16(3 1 !2)2 5 16(11 1 6!2) 5 176 1 96!2 ft[1] appropriate method but with computational errors[0] incorrect answer and no work shown OR no answer given


28

Name Class Date

Consider how you might use a calculator to fi nd the square of negative three. If you enter the expression 232, your calculator produces an answer of 29. However, the square of negative three is (23)2 5 (23)(23) 5 9. Calculators follow the order of operations. Th erefore, a calculator will compute 232 as the opposite of 32. Th e correct input is (23)2, which is correctly evaluated as 9. Be sure to follow the order of operations when expanding binomial radical expressions.

1. Consider the algebraic expression (a 1 b)2. Is (a 1 b)2 equivalent to a2 1 b2? If yes, explain. If not, explain why it is not mathematically logical and give a counterexample.

2. Are there values of a and b for which (a 1 b)2 5 a2 1 b2?

Consider each pair of expressions below for nonnegative values of the variables. State whether they are equivalent expressions. If yes, explain. If not, give a counterexample.

3. "x2 1 y2, "x2 1 "y2

4. "ab , Åa

b

5. Q!aR2, a

6. Q"x2 1 y2R2, x 1 y

6-3 EnrichmentBinomial Radical Expressions

Answer may vary. Sample: (a 1 b)2 means (a 1 b)(a 1 b) which, when expanded, is a2 1 2ab 1 b2, which is not equivalent to a2 1 b2.

Answers may vary. Sample: a 5 1, b 5 0

Answers may vary. Sample: These expressions are not equivalent. Let x 5 2 and y 5 3 then "22 1 32 5 "13 u"4 1 "9

Answers may vary. Sample: These expressions are not equivalent. Let a = 6 and

b 5 2 then !62 N 1.22 and Å6

2 5 !3 N 1.73

Answers may vary. Sample: These expressions are equivalent. A!aB2 5 A!aB A!aB 5 "a2 5 a for all a L 0.

Answers may vary. Sample: These expressions are not equivalent. Q"x2 1 y2RQ"x2 1 y2R 5 "(x2 1 y2)2 5 x2 1 y2


29

Name Class Date

Two radical expressions are like radicals if they have the same index and the same radicand.

Compare radical expressions to the terms in a polynomial expression.

Like terms: 4x3 11x3 Th e power and the variable are the same

Unlike terms: 4y3 11x3 4y2 Either the power or the variable are not the same.

Like radicals: 4!3 6 11!3 6 Th e index and the radicand are the same

Unlike radicals: 4!3 5 11!3 6 4 2"6 Either the index or the radicand are not the same.

When adding or subtracting radical expressions, simplify each radical so that you can fi nd like radicals.

Problem

What is the sum? !63 1 !28

!63 1 !28 5 !9 ? 7 1 !4 ? 7 Factor each radicand.

5 "32 ? 7 1 "22 ? 7 Find perfect squares.

5 "32"7 1 "22"7 Use n!ab 5n !a ?

n!b.

5 3!7 1 2!7 Use n"an 5 a to simplify.

5 5!7 Add like radicals.

Th e sum is 5!7.

Exercises

Simplify.

1. !150 2 !24 2. !3 135 1 !3 40 3. 6!3 2 !75

4. 5!3 2 2 !3 54 5. 2!48 1 !147 2 !27 6. 8!3 3x 2 !3 24x 1 !3 192x

6-3 ReteachingBinomial Radical Expressions

3!6 !3

10 3!3x

2 3!2

5 3!5

0


30

Name Class Date

• Conjugates, such as !a 1 !b and !a 2 !b, diff er only in the sign of the second term. If a and b are rational numbers, then the product of conjugates produce a rational number:

Q!a 1 !bRQ!a 2 !b R 5 Q!a R22 Q!b R2

5 a 2 b.

• You can use the conjugate of a radical denominator to rationalize the denominator.

Problem

What is the product? Q2!7 2 !5RQ2!7 1 !5R Q2!7 2 !5RQ2!7 1 !5R These are conjugates.

5 Q2!7R22 Q!5R2

Use the difference of squares formula.

5 28 2 5 5 23 Simplify.

Problem

How can you write the expression with a rationalized denominator? 4!21 1 !3

4!2

1 1 !3

54!2

1 1 !3?

1 2 !31 2 !3

Use the conjugate of 1 1 !3 to rationalize the denominator.

54!2 2 4!6

1 2 3 Multiply.

54!2 2 4!6

22 5 2A4!2 2 4!6B

2 Simplify.

524!2 1 4!6

2 5 22!2 1 2!6

Exercises

Simplify. Rationalize all denominators.

7. A3 1 !6B A3 2 !6B 8. 2!3 1 15 2 !3

9. Q4!6 2 1RQ!6 1 4R 10.

2 2 !72 1 !7

11. A2!8 2 6B A!8 2 4B 12. !5

2 1 !3



3 "3 1 12

20 1 15!6

40 2 28!2 2!5 2 !15211 1 4"73


31

Name Class Date

6-4 Additional Vocabulary Support Rational Exponents

Choose the word or phrase from the list that best matches each sentence.

rational exponent radical form exponential form

1. Th e expression "4 y3 is written in .

2. A is an exponent written in fractional form.

3. Th e expression x 35 is written in .

Write each expression in exponential form.

4. "4 y7 5

5. (!3 x)4 5

6. (!5 a)3 5

7. !8 r 5

Write each expression in radical form.

8. w34 5

9. b52 5

10. h12 5

11. g 37 5

Multiple Choice

12. What is "6 y4

!3 y in simplest terms?

y12 !3 y "3 y4 y

23

radical form

rational exponent

exponential form

y

74

x

43

a35

4"w3

7"g3

"b5

"h

r

18

B


32

Name Class Date

6-4 Think About a PlanRational Exponents

Science A desktop world globe has a volume of about 1386 cubic inches. Th e radius of the Earth is approximately equal to the radius of the globe raised to the 10th power. Find the radius of the Earth. (Hint: Use the formula V 5

43pr3 for the

volume of a sphere.)

Know

1. Th e volume of the globe is z z.

2. Th e radius of the Earth is equal to .

Need

3. To solve the problem I need to fi nd .

Plan

4. Write an equation relating the radius of the globe rG to the radius of the Earth rE.

5. How can you represent the radius of the globe in terms of the radius of the Earth?

6. Write an equation to represent the volume of the globe.

7. Use your previous equation and your equation from Exercise 5 to write an equation to fi nd the radius of the Earth.

8. Solve your equation to fi nd the radius of the Earth.

1386 in.3

the radius of the globe raised to the 10th power

the radius of the Earth

about 251,000,000 in. or 3961 mi

rE 5 rG10

rG 5 r1

10E

1386 5 43πr 3

G

1386 5 43π ¢r

110

E ≤3

Name Class Date


33

6-4 Practice Form G

Rational Exponents

Simplify each expression.

1. 12513 2. 64

12 3. 32

15

4. 712 ? 7

12 5. (25)

13 ? (25)

13 ? (25)

13 6. 3

12 ? 75

12

7. 1113 ? 11

13 ? 11

13 8. 7

12 ? 28

12 9. 8

14 ? 32

14

10. 1212 ? 27

12 11. 12

13 ? 45

13 ? 50

13 12. 18

12 ? 98

12


13. x43 14. (2y)

13 15. a1.5

16. b15 17. z

23 18. (ab)

14

19. m2.4 20. t227 21. a21.6


22. "x3 23. !3 m 24. !5y

25. "3 2y2 26. Q!4 bR3 27. !26

28. "(6a)4 29. "5 n4 30. "4 (5ab)3

31. Th e rate of infl ation i that raises the cost of an item from the present value P to

the future value F over t years is found using the formula i 5 QFPR1

t 2 1. Round your answers to the nearest tenth of a percent.

a. What is the rate of infl ation for which a television set costing $1000 today will become one costing $1500 in 3 years?

b. What is the rate of infl ation that will result in the price P doubling (that is, F 5 2P) in 10 years?

5 8 2

7 25

1411

18

4

15

4230

14.5%

7.2%

3"x4

5!b

5"m12 17"t2

15"a8

3"z2 4!ab

3"2y "a3

x

32 m

13 (5y)

12

213y

23

36a2 n45 (5ab)

34

b34 (26)

12

Name Class Date


34


Rational Exponents

Write each expression in simplest form. Assume that all variables are positive.

32. Q8114R 4 33. Q32

15R5 34. A2564B14

35. 70 36. 823 37. (227)

23

38. x12 ? x

13 39. 2y

12 ? y 40. A82B13

41. 3.60 42. Q 116R1

4 43. Q278 R2

3

44. "8 0 45. Q3x12R Q4x

23R 46.

12y13

4y12

47. Q3a12 b

13R2 48. Qy2

3R29 49. Qa23b2

12R26

50. y25 ? y

38 51. ax

47

x23

b 52. Q2a14R3

53. 81212 54. Q2x

25RQ6x

14R 55. Q9x4y22R1

2

56. a27x6

64y4b13 57.

x12 y

23

x13 y

12

58. y58 4 y

12

59. x14 ? x

16 ? x

13 60. a x2

13 y

x23 y2

12

b2 61. a 12x8

75y10b12

62. In a test kitchen, researchers have measured the radius of a ball of dough made with a new quick-acting yeast. Based on their data, the radius r of the dough ball, in centimeters, is given by r 5 5(1.05)

t3 after t minutes. Round the

answers to the following questions to the nearest tenth of a cm. a. What is the radius after 5 minutes? b. What is the radius after 20 minutes? c. What is the radius after 43 minutes?

32

4

1

x 2

21

3

y 16

12x

1320

1y6

12x

76

12

2y

32

81

1

y

3140

19

9ab23

0

1

x

56

256

9

8a34

3x2

y

b3

a4

94

4

5.4 cm

3x2

4y

43

x

34

y3

x22x4

5y5

x

16 y

16 y

18

6.9 cm10.1 cm


35

Name Class Date

6-4 Practice Form K

Rational Exponents


1. 16

14 2. (23)

13 ? (23)

13 ? (23)

13 3. 5

12 ? 45

12

!4 16


4. x

14 5. x

45 6. x

29


7. !3 2 8. "3 2x2 9. "3 (2x)2

10. Bone loss for astronauts may be prevented with an apparatus that rotates

to simulate gravity. In the formula N 5a0.5

2pr 0.5, N is the rate of rotation in

revolutions per second, a is the simulated acceleration in m/s2, and r is the radius of the apparatus in meters. How fast would an apparatus with the following radii have to rotate to simulate the acceleration of 9.8 m/s2 that is due to Earth’s gravity?

a. r 5 1.7 m b. r 5 3.6 m c. r 5 5.2 m d. Reasoning Would an apparatus with radius 0.8 m need to spin faster or

slower than the one in part (a)?

2

4!x

2

13

23

0.382 rev/s0.263 rev/s0.218 rev/s

faster

5"x 4

A2x 2B

13

15

9"x

2

(2x)

23


36

Name Class Date


Rational Exponents

Simplify each number.

11. (2216)

13 12. 2431.2 13. 3220.4

!3 2216

Find each product or quotient. To start, rewrite the expression using exponents.

14. A!4 6B A!3 6B 15. "5 x2

10%x2 16. !20 ? !3 135

5 Q6

14RQ6

13R


17. (125)

23 18. (216)

23(216)

23 19. (2243)

25


20. Q16x28R234 21. Q8x15R2

13 22. a x2

x210b13

23. Error Analysis Explain why the following simplifi cation is incorrect. What is the correct simplifi cation?

5Q4 2 5

12R

5 5(4) 2 5Q5

12R 5 20 2 25

12 5 15

26

12"6 7

25

x 6

8

You cannot multiply 5 and 5

12 together by multiplying bases. You

have to rewrite 5 as 51 and combine the exponents; 20 2 5!5.

729

5!x

1296

12x

5

14

6 6"5

5

9

x 4


37

Name Class Date

6-4 Standardized Test Prep Rational Exponents

Multiple Choice


1. What is 1213 ? 45

13 ? 50

13 in simplest form?

!27,000 30 10713 27,000

2. What is x13 ? y


x3"y3 "xy3 "3 (xy)2 "3 xy2

3. What is x13 ? x

12 ? x


x 1312 x

124 x

19 x

524

4. What is £x23y

13

x12y

34

≥6

in simplest form?

xy 52 x 7y

52

1

xy 52

x

y 52

5. What is (232x10 y35)215 in simplest form?

2x2y7 2

2

x2y7 2

1

2x2 y7 2

x2 y7

Short Response

6. Th e surface area S, in square units, of a sphere with volume V, in cubic units, is given by the formula S 5 p

13(6V )

23. What is the surface area of a sphere with

volume 43 mi3? Show your work.

[2] S 5 π

13(6V)

23 5 π

13 c6Q43R d

23

5 π

13 (8)

23 5 4π

13 mi

2

[1] appropriate method but some computational errors[0] incorrect answer and no work shown OR no answer given

B

I

A

I

C


38

Name Class Date

6-4 Enrichment Rational Exponents

Power Games

Each problem below involves rational exponents. Some of the problems are tricky. Good luck!

1. Begin with any positive number. Call it x. Divide x by 2. Call the result r. Now follow these directions carefully. You may use a calculator.

a. Divide x by r. Call the result q. b. Add q and r. Call the result s. c. Divide s by 2. Call the result r. d. Go back to step a.

Repeat steps a –d until r no longer changes. What is the relationship between the original x and the fi nal result?

2. If we take the square root of a number 6 times, it would look like this:

'&$#"!x

Rewrite the expression above using rational exponents.

Simplify the expression above. Express the denominator of the exponent as a power of 2.

If you were to take the square root of a number 10 times, what would the denominator of the exponent be equal to if you use rational exponents? 12 times?

Choose any number and repeatedly take the square root. What number is the answer approaching?

Does the answer appear to approach the same number if you change the number you choose?

In Exercises 3–6, assume that the square roots and the operations inside them repeat forever.

3. How much is $2 3 #2 3 "2 3 !2 3 c? (Hint: Let

y 5 $2 3 #2 3 "2 3 !2 3 c . Th en use substitution and solve the

equation y 5 "2 3 y.)

4. How much is $2 1 #2 1 "2 1 !2 1 c?

5. How much is $2 2 #2 2 "2 2 !2 2 c?

6. How much is $2 4 #2 4 "2 4 !2 4 c?

fi nal r 5 !x

aaaaax12b1

2b12b1

2b12b1

2

x1

64; 26

210; 212

1

yes

2

2

1

3"2


39

Name Class Date

6-4 Reteaching Rational Exponents

You can simplify a number with a rational exponent by converting the expression to a radical expression:

x1n 5 n!x, for n . 0 9

12 5

2!9 5 3 813 5 !3 8 5 2

You can simplify the product of numbers with rational exponents m and n by raising the number to the sum of the exponents using the rule

am ? an 5 am1n

Problem

What is the simplifi ed form of each expression?

a. 3614 ? 36

14

3614 ? 36

14 5 36

14 1

14 Use am ? an 5 am1n.

5 3612 Add.

5 2!36 Use x1n 5

n!x .

5 6 Simplify.

b. Write Q6x23R Q2x

34R in simplifi ed form.

Q6x23R Q2x

34R 5 6 ? 2 ? x

23 ? x

34 Commutative and Associative

Properties of Multiplication

5 6 ? 2 ? x23 1

34 Use xm ? xn 5 xm1n.

5 12x1712 Simplify.

Exercises

Simplify each expression. Assume that all variables are positive.

1. 513 ? 5

23 2. Q2y

14R Q3y

13R 3. (211)

13 ? (211)

13 ? (211)

13

4. 2y23 y

15 5. 5

14 ? 5

14 6. Q23x

16R Q7x

26R

5 6y

712 211

221!x!52y

1315


40

Name Class Date


Rational Exponents

To write an expression with rational exponents in simplest form, simplify all exponents and write every exponent as a positive number using the following rules for a 2 0 and rational numbers m and n:

a2n 51

an 1

a2m 5 am (am)n 5 amn (ab)m 5 ambm

Problem

What is A8x9y23B223 in simplest form?

(8x9y23)223 5 A23 x9 y23B22

3 Factor any numerical coeffi cients.

5 A23B223 Ax9B22

3 Ay23B223 Use the property (ab)m 5 ambm.

5 222x26y2 Multiply exponents, using the property (am)n 5 amn.

5y2

22x6 Write every exponent as a positive number.

5y2

4x6 Simplify.

Exercises


7. A16x2 y8B212 8. Az23B19 9. Q2x

14R4

10. A25x26 y2B12 11. A8a23 b9B23 12. a16z4

25x8b212

13. a x2

y21b15 14. A27m9 n23B22

3 15. a32r2

2s4 b14

16. A9z10B32 17. (2243)215 18. ax

25

y12

b10

14xy4 16x

2r 12

s

5y

x34b6

a25x4

4z2

x4

y5213

n2

9m6

27z15

1

z 13

x25y

15


41

Name Class Date

6-5 Additional Vocabulary Support Solving Square Root and Other Radical Equations

Problem

Solve the equation 4"3 (y 1 2)2 1 3 5 19. Justify your steps. Th en check your solution.

4(y 1 2)23 1 3 5 19 Rewrite the radical using a rational exponent.

4(y 1 2)23 5 16 Subtract 3 from each side.

(y 1 2)23 5 4 Divide each side by 4.

c(y 1 2)23 d 3

2

5 432 Raise each side to the 32 power.

(y 1 2) 5 8 Simplify.

y 5 6 Solve for y.

Check 4 "3 (6 1 2)2 1 3 0 19 Substitute 6 for y.

4"3 82 1 3 0 19 Add.

4 ? 4 1 3 0 19 Simplify the radical.

19 5 19 Simplify.

Exercise

Solve the equation 9"(2x 2 4)4 1 2 5 38. Justify your steps. Th en check your solution.

9(2x 2 4)42 1 2 5 38

9(2x 2 4)2 5 36

(2x 2 4)2 5 4

c(2x 2 4)2 d 12

5 412

(2x 2 4) 5 2

x 5 3

Check 9"(2 ? 3 2 4)4 1 2 0 38

9"16 1 2 0 38

38 5 38

Rewrite the radical using a rational exponent.

Divide each side by 9.

Raise each side to the 12 power.

Simplify.

Solve for x.

Substitute 3 for x.

Simplify the expression under the radical sign.

Simplify.

Subtract 2 from each side and simplify the exponent.


42

Name Class Date

6-5 Think About a PlanSolving Square Root and Other Radical Equations

Traffi c Signs A stop sign is a regular octagon, formed by cutting triangles off the corners of a square. If a stop sign measures 36 in. from top to bottom, what is the length of each side?

Understanding the Problem

1. How can you use the diagram at the right to fi nd a relationship between s and x?

.

2. How can you use the diagram at the right to fi nd another relationship between s and x?

.

3. What is the problem asking you to determine?

Planning the Solution

4. What are two equations that relate s and x?

5. How can you use your equations to fi nd s?

.

Getting an Answer

6. Solve your equations for s.

7. Is your answer reasonable? Explain.

.

x

x

x

s

s

36 in.

Since the triangles are right triangles, use the

Pythagorean Theorem to relate s and x

The length of a side of the square, which is s 1 2x, is the same as the height of the

stop sign from top to bottom

Yes; the length of one side of the stop sign is a little more than a third of the total

height of the sign

the length s of each side of the stop sign

Solve the fi rst equation for x and substitute the result into the second equation

2x2 5 s2; 2x 1 s 5 36

about 14.9 in.

Name Class Date


43

6-5 Practice Form G

Solving Square Root and Other Radical Equations

Solve.

1. 5!x 1 2 5 12 2. 3!x 2 8 5 7 3. !4x 1 2 5 8

4. !2x 2 5 5 7 5. !3x 2 3 2 6 5 0 6. !5 2 2x 1 5 5 12

7. !3x 2 2 2 7 5 0 8. !4x 1 3 1 2 5 5 9. !33 2 3x 5 3

10. !3 2x 1 1 5 3 11. !3 13x 2 1 2 4 5 0 12. !3 2x 2 4 5 22

Solve.

13. (x 2 2) 13 5 5 14. (2x 1 1)

13 5 23 15. 2x

34 5 16

16. 2x 13 2 2 5 0 17. x

12 2 5 5 0 18. 4x

32 2 5 5 103

19. (7x 2 3)12 5 5 20. 4x

12 2 5 5 27 21. x

16 2 2 5 0

22. (2x 1 1)13 5 1 23. (x 2 2)

23 2 4 5 5 24. 3x

43 1 5 5 53

25. Th e formula P 5 4"A relates the perimeter P, in units, of a square to its area A, in square units. What is the area of the square window shown below?

26. Th e formula A 5 6V 23 relates the surface area A, in square units, of a cube to

the volume V, in cubic units. What is the volume of a cube with surface area 486 in.2?

27. A mound of sand at a rock-crushing plant is growing over time. Th e equation t 5 !3 5V 2 1 gives the time t, in hours, at which the mound has volume V, in cubic meters. When is the volume equal to 549 m3?

Perimeter: 24 ft

4

27

17

13

127

1

4

0

36 ft2

729 in.3

14 h

29, 225 8, 28

64 64

25 9

214 16

5 22

32 8

13 222

25 9

Name Class Date


44

28. City offi cials conclude they should budget s million dollars for a new library building if the population increases by p thousand people in a ten-year census. Th e formula s 5 2 1 13(p 1 1)

25 expresses the relationship between population and

library budget for the city. How much can the population increase without the city going over budget if they have $5 million for a new library building?

Solve. Check for extraneous solutions.

29. !x 1 1 5 x 2 1 30. !2x 1 1 5 23

31. (x 1 7)12 5 x 2 5 32. (2x 2 4)

12 5 x 2 2

33. !x 1 2 5 x 2 18 34. !x 1 6 5 x

35. (2x 1 1)12 5 25 36. (x 1 2)

12 5 10 2 x

37. !x 1 1 5 x 1 1 38. !9 2 3x 5 3 2 x

39. !3 2x 2 4 5 22 40. 2!5 5x 1 2 2 1 5 3

41. !4x 1 2 5 !3x 1 4 42. !7x 2 6 2 !5x 1 2 5 0

43. 2(x 2 1)12 5 (26 1 x)

12 44. (x 2 1)

12 2 (2x 1 1)

14 5 0

45. !2x 2 !x 1 1 5 1 46. !7x 2 1 5 !5x 1 5

47. (7 2 x)12 5 (2x 1 13)

12 48. (x 2 7)

12 5 (x 1 5)

14

49. !x 1 9 2 !x 5 1 50. !3 8x 2 !3 6x 2 2 5 0

51. A clothing manufacturer uses the model a 5 !f 1 4 2 !36 2 f to estimate the amount of fabric to order from a mill. In the formula, a is the number of apparel items (in hundreds) and f is the number of units of fabric needed. If 400 apparel items will be manufactured, how many units of fabric should be ordered?

52. What are the lengths of the sides of the trapezoid shown at the right if the perimeter of the trapezoid is 17 cm?



x � 1

x

2V x2V x

242,000

3

9

23

no solution

21, 0

22

2

10

8

22

16

32

x 5 4 cm, 2!x 5 4 cm, x 1 1 5 5 cm

21

11

3

4

4

6

0, 3

7

no solution

2, 4

9

Name Class Date


45

Solve. To start, rewrite the equation to isolate the radical.

1. !x 1 2 2 2 5 0 2. !2x 1 3 2 7 5 0 3. 2 1 !3x 2 2 5 6

!x 1 2 5 2

Solve.

4. 2(x 2 2)

23 5 50 5. 2(x 1 3)

32 5 54 6. (6x 2 5)

13 1 3 5 22

7. Th e formula d 5 2# Vph relates the diameter d, in units, of a cylinder to its

volume V, in cubic units, and its height h, in units. A cylindrical can has a diameter of 3 in. and a height of 4 in. What is the volume of the can to the nearest cubic inch?

8. Writing Explain the diff erence between a radical equation and a polynomial equation.

9. Reasoning If you are solving 4(x 1 3)

34 5 7, do you need to use the absolute

value to solve for x? Why or why not?

6-5 Practice Form K


223

127 and 2123 6 220

28 in.3

6

A radical equation has a variable in a radicand or a variable with arational exponent, while a polynomial equation has a variable with whole number exponents.

No; the numerator of the exponent 34 is not even.

Name Class Date


46



Solve. Check for extraneous solutions. First, isolate a radical, then square each side of the equation.

10. !4x 1 5 5 x 1 2 11. !23x 2 5 2 3 5 x 12. !x 1 7 1 5 5 x

A!4x 1 5B2 5 (x 1 2)2

13. !2x 2 7 5 !x 1 2 14. !3x 1 2 2 !2x 1 7 5 0 15. !2x 1 4 2 2 5 !xA!2x 2 7B2 5 A!x 1 2B2

16. Find the solutions of !x 1 2 5 x . a. Are there any extraneous solutions? b. Reasoning How do you know the answer to part (a)?

17. A fl oor is made up of hexagon-shaped tiles. Each hexagon tile has an area of 1497 cm2. What is the length of each side of the hexagon? (Hint: Six equilateral triangles make one hexagon.)

s

s!32

1 and 21

22 9

9

5 0 and 16

221

Substitute the solutionsinto the original equation. If a solution does not make the equation true, then the solution is extraneous.

about 24 cm


47

Name Class Date

Gridded Response

Solve each exercise and enter your answer in the grid provided.

1. What is the solution? !2x 2 4 2 3 5 1

2. What is the solution? 5x 12 2 8 5 7

3. What is the solution? !2x 2 6 5 3 2 x

4. What is the solution? !5x 2 3 5 !2x 1 3

5. Kepler’s Th ird Law of Orbital Motion states that the period P (in Earth years) it takes a planet to complete one orbit of the sun is a function of the distance d (in astronomical units, AU) from the planet to the sun. Th is relationship is P 5 d

32. If it takes Neptune 165 years to orbit the sun, what is the distance

(in AU) of Neptune from the sun? Round your answer to two decimal places.

6-5 Standardized Test Prep Solving Square Root and Other Radical Equations

1. 2. 3. 4. 5.

Answers

9

765

3210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

8

43

89

765

32

0

99876543210

987654

210

9876543210

9876543210

9876543210

–

3

8

43

8

11

9

765

3210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

8

43

89

765

3210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

8

43

89

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

8

1 0

10

9

9

3

3

22

2

3 0 . 0 8

3

0 0

8


48

Name Class Date

When solving radical equations you will often get an extraneous solution. You can use a graph to explain why an algebraic answer is not a solution.

1. Solve the equation !x 1 2 5 x 2 4. Is there an extraneous solution?

2. To analyze this equation with a graph, rewrite the equation as a system of two equations. What two equations can you write?

3. Graph the two equations.

4. Explain how you fi nd the solution to this system of equations on your graph. What is the solution?

5. How can you use the solution from the graph of the system of equations to help you solve the original equation !x 1 2 5 x 2 4?

6. How can you tell from your graph that one of your algebraic answers is an extraneous solution?

Solve each equation. Graph each equation as a system to determine if there are any extraneous solutions.

7. !4x 1 1 5 3 8. x 5 !6 2 x 9. !x 1 1 5 x 2 1

6-5 Enrichment Solving Square Root and Other Radical Equations

7; 2 is an extraneous solution.

y 5 !x 1 2 and y 5 x 2 4

Answers may vary. Sample: On a graph the solution to a system of equations is the point of intersection; the solution for this system is (7, 3).

Answers may vary. Sample: The x-coordinate of the solution to the system is the solution to the original equation.

Answers may vary. Sample: Because there is only one point of intersection, there can only be one solution to the equation.

2; no extraneous solutions



4 8

�8

�4

4

8

x

y

O�8 �4

4 8

�8

�4

4

8 y

xO�8 �4 �4 84

�8

�4

4

8

x

y

�4 4 8

�8

�4

4

8

x

y


49

Name Class Date

6-5 Reteaching Solving Square Root and Other Radical Equations

Equations containing radicals can be solved by isolating the radical on one side of the equation, and then raising both sides to the same power that would undo the radical.

Problem

What is the solution of the radical equation? 2!2x 1 2 2 2 5 10

2!2x 1 2 2 2 5 10

2!2x 1 2 5 12 Add 2 to each side.

!2x 1 2 5 6 Divide each side by 2.

(!2x 1 2)2 5 62 Square each side to undo the radical.

2x 1 2 5 36 Simplify.

2x 5 34 Subtract 2 from each side.

x 5 17 Divide each side by 2.

Check the solution in the original equation.

Check

2!2x 1 2 2 2 5 10 Write the original equation.

2!2(17) 1 2 2 2 0 10 Replace x by 17.

2!36 2 2 0 10 Simplify.

12 2 2 0 10

10 5 10 �

Th e solution is 17.

Exercises

Solve. Check your solutions.

1. x 12 5 13 2. 3!2x 5 12 3. !3x 1 5 5 11

4. (3x 1 4)12 2 1 5 4 5. (6 2 x)

12 1 2 5 5 6. !3x 1 13 5 4

7. (x 1 2) 12 2 5 5 0 8. !3 2 2x 2 2 5 3 9. !3 5x 1 2 2 3 5 0

169

7 23 1

211 523

8 12


50

Name Class Date

6-5 Reteaching (continued) Solving Square Root and Other Radical Equations

An extraneous solution may satisfy equations in your work, but it does not make the original equation true. Always check possible solutions in the original equation.

Problem

What is the solution? Check your results. !17 2 x 2 3 5 x

!17 2 x 2 3 5 x

!17 2 x 5 x 1 3 Add 3 to each side to get the radical alone on one side of the equal sign.

A!17 2 xB2 5 (x 1 3)2 Square each side.

17 2 x 5 x2 1 6x 1 9

0 5 x2 1 7x 2 8 Rewrite in standard form.

0 5 (x 2 1)(x 1 8) Factor.

x 2 1 5 0 or x 1 8 5 0 Set each factor equal to 0 using the Zero Product Property.

x 5 1 or x 5 28

Check

!17 2 x 2 3 5 x !17 2 x 2 3 5 x

!17 2 1 2 3 0 1 !17 2 (28) 2 3 0 28

!16 2 3 0 1 !25 2 3 0 28

1 5 1 2 2 28

Th e only solution is 1.

Exercises


10. !5x 1 1 5 !4x 1 3 11. !x2 1 3 5 x 1 1 12. !3x 5 !x 1 6

13. x 5 !x 1 7 1 5 14. x 2 3!x 2 4 5 0 15. !x 1 2 5 x 2 4

16. !2x 2 10 5 x 2 5 17. !3x 2 6 5 2 2 x 18. !x 2 1 1 7 5 x

19. !5x 1 1 5 !3x 1 15 20. !x 1 9 5 x 1 7 21. x 2 !x 1 2 5 40

�

5, 7 2

2

9 16 7

3solutionno

7 25

10

47


51

Name Class Date

6-6 Additional Vocabulary Support Function Operations

Darnell wrote the steps to compose the following functions on index cards, but the cards got mixed up.

Let f (x) 5 x 1 7 and g(x) 5 x3. What is (g + f )(24)?

Use the note cards to write the steps in order.

1. First,

.

2. Second,

.

3. Then,

.

4. Finally,

.

Subtract 4 from 7.

Raise 3 to the 3rd power.

Substitute 24 for x in f(x).

Substitute 3 into g(x).

substitute 24 for x in f(x)

subtract 4 from 7

substitute 3 into g(x)

raise 3 to the 3rd power


52

Name Class Date

6-6 Think About a Plan Function Operations

Sales A salesperson earns a 3% bonus on weekly sales over $5000. Consider the following functions.

g(x) 5 0.03x h(x) 5 x 2 5000

a. Explain what each function above represents.

b. Which composition, (h + g)(x) or (g + h)(x), represents the weekly bonus? Explain.

1. What does x represent in the function g(x)?

2. What does the function g(x) represent?

3. What does x represent in the function h(x)?

4. What does the function h(x) represent?

5. What is the meaning of (h + g)(x)?

.

6. Assume that x is $7000. What is (h + g)(x)?

7. What is the meaning of (g + h)(x)?

.

8. Assume that x is $7000. What is (g + h)(x)?

9. Which composition represents the weekly bonus? Explain

.

the sales amount used to calculate a 3% bonus

the bonus earned by the salesperson on sales

the total weekly sales made by the salesperson

the weekly sales over $5000 made by the salesperson

First multiply the value of x by 0.03, then subtract 5000 from the result

(g + h)(x) represents the weekly bonus because you must fi rst fi nd the sales amount

over 5000 by subtracting 5000 from the weekly sales, and then you multiply the result

by the bonus percent as a decimal, or 0.03

First subtract 5000 from the value of x, then multiply the result by 0.03

−4790

60

Name Class Date


53

6-6 Practice Form G

Function Operations

Let f (x) 5 4x 21 and g(x) 5 2x2 1 3. Perform each function operation and then fi nd the domain.

1. ( f 1 g)(x) 2. ( f 2 g)(x) 3. (g 2 f )(x)

4. ( f ? g) (x) 5. fg (x) 6.

gf (x)

Let f (x) 5 2x and g(x) 5 !x 21. Perform each function operation and then fi nd the domain of the result.

7. ( f 1 g)(x) 8. ( f 2 g)(x) 9. (g 2 f )(x)

10. ( f ? g)(x) 11. fg (x) 12.

gf (x)

Let ƒ(x) 5 23x 1 2, g(x) 5 x5 , h(x) 5 22x2 1 9, and j(x) 5 5 2 x. Find each value or expression.

13. ( f + j)(3) 14. ( j + h)(21) 15. (h + g)(25)

16. (g + f )(a) 17. ƒ(x) 1 j(x) 18. ƒ(x) 2 h(x)

19. (g + f )(25) 20. ( f + g)(22) 21. 3ƒ(x) 1 5g(x)

22. g( f (2)) 23. g( f (x)) 24. f (g(1))

25. A video game store adds a 25% markup on each of the games that it sells. In addition to the manufacturer’s cost, the store also pays a $1.50 shipping charge on each game.

a. Write a function to represent the price f (x) per video game after the store’s markup.

b. Write a function g(x) to represent the manufacturer’s cost plus the shipping charge.

c. Suppose the manufacturer’s cost for a video game is $13. Use a composite function to fi nd the cost at the store if the markup is applied after the shipping charge is added.

d. Suppose the manufacturer’s cost for a video game is $13. Use a composite function to fi nd the cost at the store if the markup is applied before the shipping charge is added.

2x2 1 4x 1 2; all real numbers

8x3 2 2x2 1 12x 23; all real numbers

2x 1 "x 2 1; x L 0

2x"x 2 2x; x L 0 2x!x 2 1; x L 0 and x u 1

2x 2 "x 1 1; x L 0 22x 1 "x 2 1; x L 0

"x 2 12x ; x S 0

22x2 + 4x 24; all real numbers

4x 2 12x2 1 3

; all real numbers 2x2 1 34x 2 1 ; all real numbers except 14


24

175

2 45

23x 1 25

165

23a 1 25

22

24x 1 7 2x2 2 3x 2 7

28x 1 6

75

7

f(x) 5 1.25x

g(x) 5 x 1 1.5

f(g(13)) N $18.13

g(f(13)) 5 $17.75

Name Class Date


54

26. Th e formula V 5 43π r 3 expresses the relationship between the volume V and radius r of a sphere. A weather balloon is being infl ated so that the radius is changing with respect to time according to the equation r 5 t 1 1, where t is the time, in minutes, and r is the radius, in feet.

a. Write a composite function f (t) to represent the volume of the weather balloon after t minutes. Do not expand the expression.

b. Find the volume of the balloon after 5 minutes. Round the answer to two decimal places. Use 3.14 for π.

27. A boutique prices merchandise by adding 80% to its cost. It later decreases by 25% the price of items that do not sell quickly.

a. Write a function f (x) to represent the price after the 80% markup. b. Write a function g(x) to represent the price after the 25% markdown. c. Use a composition function to fi nd the price of an item, after both price

adjustments, that originally costs the boutique $150. d. Does the order in which the adjustments are applied make a diff erence?

Explain.

28. A department store has marked down its merchandise by 25%. It later decreases by $5 the price of items that have not sold.

a. Write a function f (x) to represent the price after the 25% markdown. b. Write a function g(x) to represent the price after the $5 markdown. c. Use a composition function to fi nd the price of a $50 item after both price

adjustments. d. Does the order in which the adjustments are applied make a diff erence?

Explain.

Let g(x) 5 x2 2 5 and h(x) 5 3x 1 2. Perform each function operation.

29. (h + g)(x) 30. g(x) ? h(x) 31. 22g(x) 1 h(x)


Function Operations

f(t) 5 43 π(t 1 1)3

904.32 ft3

f(x) 5 1.8x

f(x) 5 0.75xg(x) 5 x 2 5

g(x) 5 0.75x

g(f (150)) 5 $202.50

No; it doesn’t matter whether you first multiply by 0.75 or by 1.8.

Yes; multiplying by 0.75 and then subtracting by 5 is different than subtracting by 5 and then multiplying by 0.75.

g(f (50)) 5 $32.50

3x2 2 13 3x3 1 2x2 2 15x 2 10 22x2 1 3x 1 12

Name Class Date


55

6-6 Practice Form K

Function Operations

Let f (x) 5 4x 1 8 and g(x) 5 2x 2 12. Perform each function operation and then fi nd the domain of the result.

1. ( f 1 g)(x) 2. ( f 2 g)(x) 3. ( f ? g)(x) 4. a fgb(x)

f (x) 1 g(x)

Let f (x) 5 x 1 2 and g(x) 5 !x 2 1. Perform each function operation and then fi nd the domain of the result.

5. ( f 1 g)(x) 6. ( f ? g)(x) 7. a fgb(x) 8. ag

f b(x)

Let f (x) 5 x 2 2 and g(x) 5 x2. Find each value. To start, use the defi nition of composing functions to fi nd a function rule.

9. (g ° f )(4) 10. ( f ° g)(21) 11. (g ° f )(23)

f (4) 5 4 2 2 5 2

Let f (x) 5 !x and g(x) 5 (x 1 2)2. Find each value.

12. ( f ° g)(25) 13. ( f ° g)(0) 14. (g ° f )(4)

(f 1 g) (x) 5 6x 2 4; all real numbers

(f 1 g)(x)5 x 1 !x 1 1;all x L 0

(f ? g)(x)5 x!x 2 x 1 2!x 2 2;all x L 0

Q fgR(x) 5 x 1 2!x 2 1

;

all x L 0, x u 1

Qgf R(x) 5!x 2 1x 1 2 ;

all x L 0

(f 2 g) (x)

5 2x 1 20; all real numbers

(f ? g) (x)

5 8x2 2 32x 2 96; all real numbers

Q fgR(x) 5 2x 1 4

x 2 6 ;

all real numbers, x u 6

421 25

3 2 16

Name Class Date


56


Function Operations

15. A car dealer off ers a 15% discount off the list price x of any car on the lot. At the same time, the manufacturer off ers a $1000 rebate for each purchase of a car.

a. Write a function f (x) to represent the price after discount. b. Write a function g (x) to represent the price after the $1000 rebate. c. Suppose the list price of a car is $18,000. Use a composite function to fi nd the price

of the car if the discount is applied before the rebate. d. Suppose the list price of a car is $18,000. Use a composite function to fi nd the price

of the car if the discount is applied after the rebate. e. Reasoning Between parts (c) and (d), will the dealer want to apply the

discount before or after the rebate? Why?

16. Error Analysis f (x) 5 2!x and g(x) 5 3x 2 6. Your friend gives a domain

for a fgb(x) as x $ 0. Is this correct? If not, what is the correct domain?

Let f (x) 5 2x2 2 3 and g(x) 5x 1 1

2 . Find each value.

17. f (g(2)) 18. g( f (23)) 19. ( f ° f )(21)

20. Reasoning A local bookstore has a sale on all their paperbacks giving a 10% discount. You also received a coupon in the mail for $4 off your purchase. If you buy 2 paperbacks at $8 each, is it less expensive for you to apply the discount before the coupon or after the coupon? How much will you save? before the coupon; $.40

f(x) 5 0.85xg(x) 5 x 2 1000

$14,300

$14,450

After; they will make more money selling the car for a higher price.

No; the correct domain is x L 0, x u 2.

32

8 21


57

Name Class Date

Multiple Choice

For Exercises 125, choose the correct letter.

1. Let f (x) 5 22x 1 5 and g(x) 5 x3. What is (g 2 f )(x)?

x3 2 2x 1 5 2x3 2 2x 1 5

x3 1 2x 2 5 2x3 1 2x 2 5

2. Let f (x) 5 3x and g(x) 5 x2 1 1. What is ( f · g)(x)?

9x2 1 3x 9x2 1 1 3x3 1 3x 3x3 1 1

3. Let f (x) 5 x2 2 2x 2 15 and g(x) 5 x 1 3. What is the domain of fg (x)?

all real numbers x 2 23

x 2 5, 23 x . 0

4. Let f (x) 5 !x 1 1 and g(x) 5 2x 1 1. What is (g + f )(x)?

2!x 1 3 !2x 1 1 1 1

2x!x 1 2x 1 !x 1 1 2x 1 !x 1 2

5. Let f (x) 5 1x and g(x) 5 x2 2 2. What is ( f + g)(23)?

179 1

7 2 179 2

73

Short Response

6. Suppose the function f (x) 5 0.035x represents the number of U.S. dollars equivalent to x Russian rubles and the function g(x) 5 90x represents the number of Japanese yen equivalent to x U.S. dollars. Write a composite function that represents the number of Japanese yen equivalent to x Russian rubles. Show your work.

6-6 Standardized Test PrepFunction Operations

[2] (g ° f )(x) 5 g(f(x)) 5 90(0.035x) 5 3.15x

B

H

C

F

B

[1] appropriate method but with one computational error

[0] incorrect answer and no work shown OR no answer given


58

Name Class Date

Composition and Linear FunctionsTwo functions f (x) and g(x) are equal if they have the same domains and the same value for each point in their domain. Suppose that f (x) 5 Ax 1 B and g(x) 5 Cx 1 D are two linear functions both of whose domains are the set of real numbers.

1. If f (x) 5 g(x), what can you conclude by examining the values of f and g at x 5 0?

2. Use your conclusion to eliminate D from the defi nition of g(x).

3. What equation results from examining the values of f and g at x 5 1?

4. What can you conclude about A and C?

5. When are two linear functions equal?

6. Compute (g + f )(x).

7. What type of function is the composite of two linear functions?

8. What is the coeffi cient of x in the expression for (g + f )(x)?

9. What is the constant term?

10. Compute ( f + g)(x) and express it in slope-intercept form.

11. What equation must be satisfi ed if f + g 5 g + f ?

12. What equations must be satisfi ed if f + g 5 f ?

13. What equations must be satisfi ed if f + g 5 g ?

14. What must occur if f + g 5 0?

15. Constant functions are a subset of linear functions in which the coeffi cient of x is zero. What type of function is the composite of two constant functions?

16. a. If h(x) and k(x) are two constant functions, under what circumstances does h + k 5 k + h?

b. Under what circumstances does h + k 5 k? c. Under what circumstances does h + k 5 h? d. Under what circumstances does h + k 5 0?

6-6 EnrichmentFunction Operations

g(x) 5 Cx 1 B

A 1 B 5 C 1 B

A 5 C

ACx 1 BC 1 D

AC

BC 1 D

ACx 1 AD 1 B

BC 1 D 5 AD 1 B

AC 5 A and AD 1 B 5 B

AC 5 C and AD 1 B 5 D

AC 5 0 and AD 1 B 5 0

constant

if h(x) 5 k(x)if h(x) 5 k(x)h ° k always equals hif h(x) 5 0

when the coeffi cients of x are equal and the constant terms are equal

B 5 D

linear


59

Name Class Date

When you combine functions using addition, subtraction, multiplication, or division, the domain of the resulting function has to include the domains of both of the original functions.

Problem

Let f (x) 5 x2 2 4 and g(x) 5 !x . What is the solution of each function operation? What is the domain of the result?

a. ( f 1 g)(x) 5 f (x) 1 g(x) 5 (x2 2 4) 1 (!x) 5 x2 1 !x 2 4

b. ( f 2 g)(x) 5 f (x) 2 g(x) 5 (x2 2 4) 2 (!x) 5 x2 2 !x 2 4

c. (g 2 f )(x) 5 g(x) 2 f (x) 5 (!x) 2 (x2 2 4) 5 2x2 1 !x 1 4

d. ( f ? g)(x) 5 f (x) ? g(x) 5 (x2 2 4)(!x) 5 x2!x 2 4!x

Th e domain of f is all real numbers. Th e domain of g is all x $ 0. For parts a2d, there are no additional restrictions on the values for x, so the domain for each of these is x $ 0.

e. fg

(x) 5f (x)g(x)

5x2 2 4!x

5(x2 2 4)!x

x

As before, the domain is x $ 0. But, because the denominator cannot be zero, eliminate any values of x for which g(x) 5 0. Th e only value for which !x 5 0 is x 5 0. Th erefore,

the domain of fg is x . 0.

f. gf (x) 5

g(x)f(x)

5 !xx2 2 4

Similarly, begin with x $ 0 and eliminate any values of x that make the denominator f(x) zero: x2 2 4 5 0 when x 5 22 and x 5 2. Th erefore, the domain of

gf is x $ 0

combined with x 2 22 and x 2 2. In other words, the domain is x $ 0 and x 2 2, or all

nonnegative numbers except 2.

Exercises

Let f (x) 5 4x 2 3 and g(x) 5 x2 1 2. Perform each function operation and then fi nd the domain of the result.

1. ( f 1 g)(x) 2. ( f 2 g)(x) 3. (g 2 f )(x)

4. ( f · g)(x) 5. fg (x) 6.

gf (x)

6-6 ReteachingFunction Operations

x2 1 4x 2 1; all real numbers

4x3 2 3x2 1 8x 2 6; all real numbers


4x 2 3x2 1 2

; all real numbers


x2 1 24x 2 3; x u 3

4


60

Name Class Date

• One way to combine two functions is by forming a composite.

• A composite is written (g + f ) or g( f (x)). Th e two diff erent functions are g and f.

• Evaluate the inner function f (x) fi rst.

• Use this value, the fi rst output, as the input for the second function, g(x).

Problem

What is the value of the expression g( f (2)) given the inner function, f (x) 5 3x 2 5

and the outer function, g(x) 5 x2 1 2?

Exercises

Evaluate the expression g( f (5)) using the same functions for g and f as in the Example. Fill in blanks 7–14 on the chart.

Use one color highlighter to highlight the fi rst input. Use a second color to highlight the fi rst output and the second input. Use a third color to highlight the second output, which is the answer.

Given f (x) 5 x2 1 4x and g(x) 5 2x 1 3, evaluate each expression.

15. f (g(2)) 16. g( f (2.5)) 17. g( f (25)) 18. f (g(25))


Function Operations

1st inputx � 2

f(x)

32nd output

11st output

1st output, 1,becomes 2nd input

2nd output

3

3x � 5 x2 � 2

g(x)

3(2) � 5

6 � 5

12 � 2

1 � 2

1st inputx � 5

3x � 5

3(—) � 5

— � 5

———

f(x)

x2 � 2

—2

� 2

———

g(x)1st output,

———,becomes 2nd input

2nd output

———

7.

8.

9.

11.

12.

13.

10. 14.

— � 2

77 35.5 13 21

5

15

10

10

10

100

102

102

Name Class Date


61

6-7 Additional Vocabulary Support Inverse Relations and Functions

Choose the word or phrase from the list that best matches each sentence.

inverse relation inverse function one-to-one function f21

1. In a , each y-value in the range corresponds to exactly one x-value in the domain.

2. A relation pairs element a of its domain to element b of its range. Th e pairs b with a.

3. Th e inverse of a function f is represented by .

4. If a relation and its inverse are functions, then they are .

5. Circle the inverse of y 5 2x 1 1. y 5 x 212 x 5

y2 2 2 y 5

x 2 12

6. Circle the inverse of y 5 (3 2 x)2. y 5 3 1 !x y 5 3 2 !x y 5 9 2 !x

7. Circle the inverse of y 5 5x2 1 4. y 5x 2 4

5 y 5 "x 245 y 5 "x 2 4

5

8. Explain each of the steps followed to fi nd f 21 of f (x) 5 !3x 2 2.

y 5 !3x 2 2

x 5 !3y 2 2

x2 5 3y 2 2

x2 1 2 5 3y

y 5x2 1 2

3

9. Find f21 for f(x) 5 4x 2 8, and explain the steps.

y 5 4x 2 8

x 5 4y 2 8

x 1 8 5 4y

y 5 x4 1 2

one-to-one function

inverse relation

inverse functions

Replace f (x) with y.

Switch x and y.

Square both sides.

Add 2 to both sides.

Replace f (x) with y.

Switch x and y.

Add 8 to both sides.

Divide both sides by 4 and solve for y.

Divide both sides by 3 and solve for y.

f21


62

Name Class Date

6-7 Think About a Plan Inverse Relations and Functions

Geometry Write a function that gives the length of the hypotenuse of an isosceles right triangle with side length s. Evaluate the inverse of the function to fi nd the side length of an isosceles right triangle with a hypotenuse of 6 in.

Know

1. An equation that relates the length of each side s and the length of the hypotenuse h of an isosceles right triangle is

z z.

Need

2. To solve the problem I need to:

.

Plan

3. A function that gives the length of the hypotenuse h in terms of the side length

s is z z.

4. An inverse function that gives the side length s in terms of the length of the

hypotenuse h is

.

5. What is the value of the inverse function for h 5 6 in.?

6. Is the side length reasonable? Explain.

.

s

sh

2s2 5 h2

write a function for the length of the hypotenuse in terms

of the side length and then fi nd the inverse function for

the side length in terms of the hypotenuse

Yes; the side length is less than the length of the hypotenuse

h 5 s"2

s 5 h"2

about 4.24 in.

Name Class Date


63

6-7 Practice Form G

Inverse Relations and Functions

Find the inverse of each relation. Graph the given relation and its inverse.

1.

y 23 22 21

x 22 21 0 1

0

2.

y 23 21 0

x 0 1 2 3

22

3.

y 21 0 1

x 23 21 1 2

3

4.

y 3 2 1

x 23 22 21 0

0

Find the inverse of each function. Is the inverse a function?

5. y 5 x2 1 2 6. y 5 x 1 2 7. y 5 3(x 1 1)

8. y 5 2x2 2 3 9. y 5 2x 2 1 10. y 5 1 2 3x2

11. y 5 5x2 12. y 5 (x 1 3)2 13. y 5 6x2 2 4

14. y 5 3x2 2 2 15. y 5 (x 1 4)2 2 4 16. y 5 2x2 1 4

Graph each relation and its inverse.

17. y 5x 1 3

3 18. y 512 x 1 5 19. y 5 2x 1 5

20. y 512 x2 21. y 5 (x 1 2)2 22. y 5 (2x 2 1)2 2 2

�2

2

�2 2

y

x

�2

�4

2

4

�2 2 4

y

x

y

21 0 1x

23 21 1 2

3

y

3 2 1x

23 22 21 0

0

y 5 w!x 2 2; no

y 5 w!2x 2 3; no

y 5 w"x5; no

y 5 w"x 1 23 ; no

y 5 x 2 2; yes

y 5 12x 1 1

2; yes

y 5 w!x 2 3; no

y 5 w!x 1 4 2 4; no

y 5 13 x 2 1; yes

y 5 w"1 2 x3 ; no

y 5 w"x 1 46 ; no

y 5 w!4 2 x; no

2 4 6�4

246

O

y

x2 6�4�6 �2

2

�2�4�6

46

O

y

x4�4�6 �2

2

�2�4�6

xO

y

4

4�4

�4

y

O x �6 �2 246

2

y

x

�2�3 21 3�2�3

123

O

y

x

y

23 22 21x

22 21 0 1

0�4

2

4

�4 2 4

y

x

�2

2

�2 2

y

x

y

23 21 0x

0 1 2 3

22

Name Class Date


64

For each function, fi nd the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.

23. f (x) 516 x 24. f (x) 5 2

15 x 1 2 25. f (x) 5 x2 2 2

26. f (x) 5 x2 1 4 27. f (x) 5 !x 2 1 28. f (x) 5 !3x

29. f (x) 5 3 2 x 30. f (x) 5 (x 1 1)2 31. f (x) 51!x

32. Th e equation f (x) 5 198,900x 1 635,600 can be used to model the number of utility trucks under 6000 pounds that are sold each year in the U.S. with x 5 0 representing the year 1992. Find the inverse of the function. Use the inverse to estimate in which year the number of utility trucks under 6000 pounds sold in the U.S. will be 6,000,000. Source: www.infoplease.com

33. Th e formula s 5 0.04n 1 2500 gives an employee’s monthly salary s, in dollars, after selling n dollars in merchandise at an appliance store.

a. Find the inverse of the function. Is the inverse a function? b. Use the inverse to fi nd the amount of merchandise sold if the employee’s

salary was $2820 last month.

34. Th e formula for the surface area A of a sphere of radius r is A 5 4pr2 for r $ 0. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to fi nd the radius of a sphere with surface area 10,000 m3.

Let f (x) 5 2x 1 5. Find each value.

35. ( f 21 + f )(21) 36. ( f + f

21)(3) 37. ( f + f 21)Q21

2R



$8000

n 5 25s 2 62,500; yes

f 21(x) 5 6x ;

The domain and range of f and f

21 is the set of all real numbers; f

21 is a function.

f 21(x) 5 3 2 x ; the domain

and range of f and f 21 is

the set of all real numbers; f

21 is a function.

f 21(x) 5

21 6 !x ; domain f 5 all real numbers 5 range f

21; range f 5 all nonnegative real numbers 5 domain f

21; f

21 is not a function.

f 21(x) 5

w!x 2 4; Domain of f 5 all real numbers 5 range of f

21; Range of f 5 all real numbers greater than or equal to 4 5 domain of f

21; f

21 is not a function.

f 21(x) 5

x2 1 1; Domain of f 5 all real numbers greater than or equal to 1 5 range of f

21; Range of f 5 all real numbers greater than or equal to 0 5 domain of f

21; f 21 is a function.

f 21(x) 5 25x 1 10; The

domain and range of f and f

21 is the set of all real numbers; f

21 is a function.

f 21(x) 5 w!x 1 2;

Domain of f 5 all real numbers 5 range of f

21; Range of f 5 the set of real numbers greater than or equal to 22 5 domain of f

21; f 21 is not a function.

f 21(x) 5 1

3 x2; The

domain and range of f and f 21

is the set of all real numbers greater than or equal to 0; f

21 is a function.

f 21(x) 5 1

x2; the domain and range of f and f

21 is the set of all positive real numbers; f

21 is a function.

f 21(x) 5

x 2 635,600198,900 in 2019

r 5 " A4p; yes

28.2 m

21221 3

Name Class Date


65

6-7 Practice Form K


Find the inverse of each relation. Graph the given relation and its inverse.

1. 2. 3.

Find the inverse of each function. Is the inverse a function? To start, switch x and y.

4. y 5x2 5. y 5 x2 1 4 6. y 5 (3x 2 4)2

x 5y2


7. y 5 3x 2 4 8. y 5 2x2 9. y 5 (3 2 2x)2

x y

0

1

2

3

21

1

3

5

x y

22

0

2

4

7

3

7

19

x y

23

22

21

0

2

2

2

2

y 5 2x ; yes y 5 w!x 2 4; noy 5 w!x 1 4

3 ; no

x

y4

2

�2�4�2

�4

42Ox

y

O

4

2

�2�2

�4

42

xO

y

4

6

2

4 62

x y

21135

0123

x

y

2

4

6

2 4 6O

x y

737

19

22024

x

y

5

10

15

5 10 15O

x y

2222

232221

0

x

y4

�2�4�2

�4

4O

Name Class Date


66




10. f (x) 5 (x 1 1)2 11. f (x) 52x3

5 12. f (x) 5 !3x 1 4

13. Multiple Choice What is the inverse of y 5 5x 2 1?

f 21(x) 5 5x 1 1 f 21(x) 5x 1 1

5 f 21(x) 5x5 1 1 f 21(x) 5

x5 2 1

For each function, fi nd its inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.

14. f (x) 5 !x 1 1 15. f (x) 5 10 2 3x 16. f (x) 5 4x2 1 25

17. Th e formula for the area of a circle is A 5 pr2. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to fi nd the radius of a circle that has an area of 82 in.2.

For Exercises 18−20, f (x) 5 5x 1 11. Find each value. To start, rewrite f (x) as y and switch x and y.

18. A f ° f 21B(5) 19. A f

21 ° f B(23) 20. A f 21 ° f B(0)

y 5 5x 1 11

y 5 w!x 2 1; no y 5 3Å52 x ; yes y 5

(x 2 4) 2

3 ; yes

B

5 23 0

f21(x) 5 x 2 2 1; domain

f(x): x L 21, range f(x): y L 0; domain f

21: x L 0, range f

21: y L 21;the inverse is a function.

f21(x) 5 213 x 1 10

3 ; domain f(x): all real numbers; range f(x): all real numbers; domain f

21: all real numbers; range f

21: all real numbers; the inverse is a function.

f21(x) 5 w!x 2 252 ; domain

f(x): all real numbers; range f(x): y L 25; domain f

21: x L 25; range f 21: all

real numbers; the inverse is not a function.

r 5 ÅAp ; yes

about 5.1 in.


67

Name Class Date

Multiple Choice


1. What is the inverse of the relation? y 3 1 21

x 22 21 0 2

22

y 2 0 21

x 22 21 1 3

22

y 22 21 1

x 22 21 0 2

3

y �3 �1 1

x �2 �1 0 2

2

y 2 1 21

x 22 21 1 3

22

2. What is the inverse of the function? y 5 5(x 2 3)

y 5x 1 3

5 y 515 x 1 3 y 5 5(x 1 3) y 5

15 x 2 3

3. What function with domain x $ 5 is the inverse of y 5 !x 1 5?

y 5 x2 1 5 y 5 x2 2 5 y 5 (x 2 5)2 y 5 (x 1 5)2

4. What is the domain and range of the inverse of the function? y 5 !x 2 5

domain is the set of all real numbers $ 0; range is the set of all real numbers $ 5

domain is the set of all real numbers $ 5; range is the set of all real numbers $ 0

domain and range is the set of all real numbers $ 5

domain and range is the set of all real numbers

Extended Response

5. A high school principal uses the formula y 5 150x 1 180 to predict a student’s score on a state achievement test using the student’s 11th-grade GPA number x.

a. What is the inverse of the formula? b. Is the inverse a function? c. Using the inverse, what GPA does a student need to get a passing score of

510 on the state exam?

6-7 Standardized Test Prep Inverse Relations and Functions

A

G

C

F

[4] a. y 5 1150

(x 2 180) or y 5 1150

x 2 65 b. yes c. 2.2 [3] most work is correct but

there are minor errors [2] student understands the problem and shows some correct work [1] student may understand the problem but doesn’t know how to proceed OR correct answers without work shown [0] no answers given


68

Name Class Date

Composition, Inverses, and Linear Functions

Solving an equation for one variable in terms of another is an important step in fi nding inverses. Th is step is also used in conversion formulas.

Consider the following linear functions. Let F denote the temperature in degrees Fahrenheit, C the temperature in degrees Celsius, and K the temperature in degrees Kelvin. Th e formula for converting degrees Fahrenheit to degrees Celsius is C 5

59

(F 2 32), and the formula for converting degrees Celsius to degrees Kelvin is K 5 C 1 273.

1. Use composition to determine the formula for converting degrees Fahrenheit to degrees Kelvin.

2. Solve this function for F.

3. Th is new equation converts degrees to degrees .

4. Derive a formula to convert degrees Celsius to degrees Fahrenheit.

5. Derive a formula to convert degrees Kelvin to degrees Celsius.

6. Compose these two functions to fi nd a formula for converting degrees Kelvin to degrees Fahrenheit.

Solve each of the following problems involving functions.

7. In 1940, the cost of a new house was $10,000. By 1980, this cost had risen to $90,000. Assuming that the increase is linear, fi nd a function expressing the cost c of a new house in terms of the year y. Solve this function for y. What does this new function enable you to do?

8. Between the ages of 5 and 15, a typical child grows at a fi xed annual rate. If Mary was 42 in. in height when she was 5 yr old and grew at a rate of 2 in. a year, fi nd a formula that expresses Mary’s height h in inches when her age is a years. Solve this function for a. What does this new function enable you to do?

9. Th e air temperature, in degrees Fahrenheit, surrounding an airplane on one particular day was modeled by T 5 2

1200a 1 110, where a is the altitude, in feet,

of the airplane. Solve this function for a. What does this new function enable you to do?

10. Th e formula L 5 0.25W 1 0.5 models the length of a certain spring, in inches, when a weight of W ounces is attached to it. Solve this function for W. What does this new function enable you to do?

6-7 EnrichmentInverse Relations and Functions

K 55(F 2 32)

9 1 273

F 5 95 (K 2 273) 1 32

F 5 95 (K 2 273) 1 32

Kelvin Fahrenheit

C 5 K 2 273

a 5 200(2T 1 110); fi nd the altitude of the airplane given the temperature

F 5 95C 1 32

c 5 10,000 1 2000( y 2 1940); y 5c 2 10,000

2000 1 1940; fi nd the year given the cost of a house

h 5 2(a 2 5) 1 42; a 5 h 2 422 1 5; compute Mary’s age given her height

W 5 L 2 0.50.25 ; fi nd the weight attached to the spring given the length of the stretched

spring


69

Name Class Date

• Inverse operations “undo” each other. Addition and subtraction are inverse operations. So are multiplication and division. Th e inverse of cubing a number is taking its cube root.

• If two functions are inverses, they consist of inverse operations performed in the opposite order.

Problem

What is the inverse of the relation described by f (x) 5 x 1 1?

f (x) 5 x 1 1

y 5 x 1 1 Rewrite the equation using y, if necessary.

x 5 y 1 1 Interchange x and y.

x 2 1 5 y Solve for y.

y 5 x 2 1 The resulting function is the inverse of the original function.

So, f 21

(x) 5 x 2 1.

Exercises

Find the inverse of each function.

1. y 5 4x 2 5 2. y 5 3x3 1 2 3. y 5 (x 1 1)3

4. y 5 0.5x 1 2 5. f (x) 5 x 1 3 6. f (x) 5 2(x 2 2)

7. f (x) 5x5 8. f (x) 5 4x 1 2 9. y 5 x

10. y 5 x 2 3 11. y 5x 2 1

2 12. y 5 x3 2 8

13. f (x) 5 !x 1 2 14. f (x) 523 x 2 1 15. f (x) 5

x 1 35

16. f (x) 5 2(x 2 5)2 17. y 5 !x 1 4 18. y 5 8x 1 1

6-7 ReteachingInverse Relations and Functions

f21 5 x 1 54

f21 5 2x 2 4

f 21(x) 5 5x

f21 5 x 1 3

f21 (x) 5 x2 2 2 for x L 22

f21 53"x 2 2

3

f 21(x) 5 x 2 3

f 21(x) 5 x 2 2

4

f21 5 2x 1 1

f 21(x) 5 3

2(x 1 1)

f21 53"x 2 1

f 21(x) 5 x 1 4

2

f21 5 x

f21 53"x 1 8

f 21(x) 5 5x 2 3

f 21(x) 5 5 6 "x

2 f21 5 (x 2 4)2 for x L 0 f21 5 x 2 18


70

Name Class Date

Examine the graphs of f (x) 5 !x 2 2 and its inverse, f

21(x) 5 x2 1 2, at the right.

Notice that the range of f and the domain of f 21 are the same:

the set of all real numbers x $ 0.

Similarly, the domain of f and the range of f 21 are the same:

the set of all real numbers x $ 2.

Th is inverse relationship is true for all relations whenever both f and f

21 are defi ned.

Problem

What are the domain and range of the inverse of the function f (x) 5 !3 2 x?

f is defi ned for 3 2 x $ 0 or x # 3.

Th erefore, the domain of f and the range of f 21 is the set of all x # 3.

Th e range of f is the set of all x $ 0. So, the domain of f 21 is the set

of all x $ 0.

Exercises

Name the domain and range of the inverse of the function.

19. y 5 2x 2 1 20. y 5 2 21x 21. y 5 !x 1 5

22. y 5 !2x 1 8 23. y 5 3!x 1 2 24. y 5 (x 2 6)2

25. y 5 x2 2 6 26. y 51

x 1 4 27. y 51

(x 1 4)2

6-7 Reteaching (continued) Inverse Relations and Functions

f �1

f

6

6

5

5

4

4

3

3

2

2

1

O 1

y

x

f

6

3

5

2

4

1

3

2

21

1

2223

y

xO

The domain and the range is the set of all real numbers.

domain: x L 8; range: y K 0

domain: x L 26; range: all real numbers

domain: x u 2; range: y u 0

domain: x L 2; range: y L 0

domain: x u 0; range: y u 24


domain: x L 0; range: all real numbers

domain: x S 0; range: y u 24


71

Name Class Date

6-8 Additional Vocabulary Support Graphing Radical Functions

1. Circle the radical functions in the group below.

y 5 2(x 1 5) 2 3 y 5 22!x 2 4 y 5 x 2 4 y 5 !3 x 1 8

2. Circle the square root functions in the group below.

y 5 !3 x 2 2 y 5 !x 1 4 y 5 2!x 2 3 y 5 3(x 1 6)

For Exercises 3−8, draw a line from each word or phrase in Column A to its matching item in Column B.

Column A Column B

3. parent function A. y 5 !x 2 k

4. translate k units downward B. y 5 2!x

5. stretch vertically by the factor k (k . 1) C. y 5 !x 2 k

6. translate k units upward D. y 5 !x

7. refl ection in x-axis E. y 5 k!x

8. translate k units to the right F. y 5 !x 1 k

Identify the meaning of the following terms in the function y 5 2!x 2 4 1 5.

9. 2: .

10. 4: .

11. 5: .

Stretch vertically by a factor of 2

Translate 4 units to the right

Translate 5 units upward


72

Name Class Date

6-8 Think About a Plan Graphing Radical Functions

Electronics Th e size of a computer monitor is given as the length of the screen’s diagonal d in inches. Th e equation d 5

56!3A models the length of a diagonal

of a monitor screen with area A in square inches.

a. Graph the equation on your calculator.

b. Suppose you want to buy a new monitor that has twice the area of your old monitor. Your old monitor has a diagonal of 15 inches. What will be the diagonal of your new monitor?

1. How can you use a graph to approximate the area of the old monitor?

.

2. Graph the equation on your calculator. Make a sketch of the graph.

3. What is the area of the old monitor?

4. How can you check your answer algebraically?

.

5. Show that your answer checks.

6. How can you fi nd the diagonal of a new monitor with twice the area of the old monitor?

.

7. Use your method to fi nd the diagonal of your new monitor.

8. What will be the diagonal of your new monitor?

0 100 300200 4000

10

20

30

40

Graph the equation and graph y 5 15. The x-coordinate of their intersection will be

the area of the old monitor

Substitute 15 for d and solve the equation for A

Substitute 2 times the area of the old monitor for A in the equation

15 5 56"3A, 15Q65R 5 "3A, 18 5 "3A, 182 5 3A, 324 5 3A, A 5 108

d 5 56"3A 5 5

6"3 ? 2 ? 108 5 56"648 N 21.2 in.

108 in.2

about 21.2 in.

d

A

Name Class Date


73

6-8 Practice Form G

Graphing Radical Functions

Graph each function.

1. y 5 !x 1 3 2. y 5 !x 2 1 3. y 5 !x 1 5

4. y 5 !x 2 3 5. y 5 22!x 2 2 6. y 514!x 2 1 1 5

Solve each square root equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why.

7. !x 1 6 5 9 8. !4x 2 3 5 5 9. !3x 2 5 5 !1 2 x

10. If you know the area A of a circle, you can use the equation r 5 ÄAp

to fi nd the radius r.

a. Graph the equation. b. What is the radius of a circle with an area of 350 ft2?


11. y 5 2!3 x 1 2 12. y 5 2!3 x 2 3 13. y 5 !3 x 1 3 2 1

2

2

4

4

6

6

y

xO

8

8

O

2

�2

�2

2 x

y

2

2

4

4 6

y

Ox

6

8

2

�22

4

4 6

y

xO

6

8

O

2

�2

�2

2 x

y

�8

�4

�6

�22 4 6

y

O x

�4 �2 2

4

1

�2

y

xO

6

O

2

�2�4

�2

2 x

y

2

2

4

4

6

6

y

x

8

8O

75 7no solution; 32 is extraneous

about 10.6 ft

A

r

O 100 200

4

8

300

12

Name Class Date


74

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph.

14. y 5 !81x 1 162 15. y 5 2!4x 1 20 16. y 5 !3 125x 2 250

17. y 5 2!64x 1 192 18. y 5 2!3 8x 2 56 1 4 19. y 5 !25x 1 75 2 1

20. y 5 !0.25x 1 1 21. y 5 5 2 !4x 1 2 22. y 5 !3 27x 2 54

23. To fi nd the radius r of a sphere of volume V, use the equation r 5 Å3 3V4p .

a. Graph the equation. b. A balloon used for advertising special events has a volume of 225 ft3.

What is the radius of the balloon?

24. An exercise specialist has studied your exercise routine and says the formula t 5 1.85!c 1 10 expresses the amount of time t, in minutes, it takes you to burn c calories (cal) while exercising.

a. Graph the equation. b. According to this formula, how long should it take you to burn

100 cal? 200 cal? 300 cal?

25. You can use the equation t 5 14!d to fi nd the time t, in seconds, it takes an object to fall d feet after being dropped.

a. Graph the equation. b. How long does it take the object to fall 400 feet?



y 5 0.5!x 1 4; graph of y 5 0.5!x shifted left 4 units

y 5 9!x 1 2; graph of y 5 9!x shifted left 2 units


y 5 5 2 2"x 1 12; graph

of y 5 22!x shifted left 12 unit and up 5 units


y 5 22 3!x 2 7 1 4; graph of y 5 22

3!x shifted right 7 units and up 4 units

y 5 3 3!x 2 2; graph of y 5 3 3!x shifted right 2 units

y 5 5 3!x 2 2; graph of y 5 5 3!x shifted right 2 units

y 5 5!x 1 3 2 1; graph of y 5 5!x shifted left 3 units and down 1 unit

3.77 ft

19.4 min, 26.8 min, 32.6 min

5 s

2

100

3

1

54

500300

r

VO

20

100O

30

10

5040

400

t

c

4

200O

6

2

10 t8

800

d

Name Class Date


75

6-8 Practice Form K



1. y 5 !x 1 3 2. y 5 !x 2 4 3. y 5 !x 2 7


4. y 5 4!x 5. y 5 22!x 1 1 6. y 5 5!x 2 4

Solve each square root equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why.

7. !x 1 2 5 7 8. !4x 1 1 5 5 9. 3!3 2 x 5 10

10. A periscope on a submarine is at a height h, in feet, above the surface of the water. Th e greatest distance d, in miles, that can be seen from the periscope on a clear day is given

by d 5 Å3h2 .

a. If a ship is 3 miles from the submarine, at what height above the water would the submarine have to raise its periscope in order to see the ship?

b. If a ship is 1.5 miles from the submarine, to what height would it have to be raised?

xO

y

4

6

2

4 62

xO

y

4

6

2

4 62

xO

y

�2

�4

�6

4 62

xO

y

2

4

6

4 62

xO

y

�6

�4

�24 62

xO

y

2

�2

�4

4

4 62

xO

2468

20 30 4010

y

xO

y

2

4

6

4 62

y

3

6

9

47 6 28.11

6 ft1.5 ft

Name Class Date


76



Graph each function. To start, graph the parent function, y 5 !3 x.

11. y 5 !3 x 2 4 12. y 5 3 2 !3 x 1 1 13. y 512!3 x 2 1 1 3

14. A center-pivot irrigation system can water from 1 to 130 acres of crop land. Th e length l in feet of rotating pipe needed to irrigate A acres is given by the function l 5 117.75!A.

a. Graph the equation on your calculator. Make a sketch of the graph. b. What length of pipe is needed to irrigate 40, 80, and 130 acres?

Graph each function. Find the domain and range.

15. y 5 2!3 x 2 4 16. y 5 2!3 8x 1 5 17. y 5 23!x 2 4 2 3

18. Open Ended Write a cube root function in which the vertical translation of y 5 !3 x is twice the horizontal translation. function in the form y 5 a 3!x 2 h 1 k where k 5 2h.

Answers may vary. Any cube root

x

y

�2�2

�4

�6

2

x

y6

�2

4

2 x

y6

�2

4

2

744.7 ft, 1053.2 ft, 1342.6 ft

xO

y4

2

2 4 64

�8

812

2�2 xO

y

�4�2

�8�6

24

24 6 8

xO

y

domain: all real numbers; range: all real numbers

domain: all real numbers; range: all real numbers


xO

y300

�2�4

200

100

2 4


77

Name Class Date

Multiple Choice


1. What is the graph of y 5 !x 1 4?

2. What is the graph of y 5 !x 2 3 2 2?

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

4. What is the description of y 5 !9x 2 3 to make it easy to graph using transformations of its parent function?

the graph of y 5 3!x , shifted right 3 units

the graph of y 5 3!x , shifted right 13 unit

the graph of y 5 !x , shifted right 3 units and up 9 units

the graph of y 5 !x , shifted right 13 unit and up 9 units

Short Response

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

6-8 Standardized Test Prep Graphing Radical Functions

4

2�2�4

68

4

y

xO

42

1

�2�3

8

y

xO

�2�2�4�6

�4

4y

xO

4

642O

6

2

8

8

y

x

�2 2�4

2

�4

4

4

y

x

2�2�4

2

�2�4

4

4

y

x

4

642O

6

2

8

8

y

x

642

2

�4

46

8

y

xO

�2�4�6�8

2

�4

4y

xO

4

2�2�4

68

4

y

xO

4

642O

6

2

8

8

y

x

�2�2 2�4�6

2

�4

4y

x

C

F

D

G

[2] The graph is correct.[1] One of the transformations (horizontal, vertical, or stretch) is incorrect.[0] no answer given

642O

4

2

6 y

x


78

Name Class Date

6-8 EnrichmentGraphing Radical Functions

Transformations of Other FunctionsYou can obtain the graph of any function of the form y 5 a ? f(x 2 h) 1 k by using the shifting rules similar to those used to obtain the graph of y 5 a!x 2 h 1 k. Note that the second function is a special case of the fi rst when f(x) 5 !x . To obtain the graph of y 5 a ? f(x 2 h) 1 k, given the graph of y 5 f(x), use the following general rules:

• If a , 0, refl ect the graph of y 5 f(x) across the x-axis.

• If ua u . 1, the graph of y 5 f(x) is stretched by a factor of a.

• If 0 , ua u , 1, the graph of y 5 f(x) is compressed by a factor of a.

• Th e graph of y 5 f(x) is shifted right h units if h . 0 and left h units if h , 0.

• Th e graph of y 5 f(x) is shifted up k units if k . 0 and down k units if k , 0.

1. Use the general rules to describe how the graph of y 5 23(x 2 5)2 1 7 can be obtained from the graph of f(x) 5 x2.

2. Write the equation for the graph that looks like y 5 !3 x but that is shifted right four units, refl ected across the x-axis, and shifted down six units.

3. Use the graph of y 5 f(x) given below to sketch the graph of y 5 f(x 1 2) 2 1.

4. Th e graph of y 5 f(x) and y 5 g(x) is given below. Th e graph of g is a transformation of the graph of f. Write the equation for the graph of g in terms of f.

O

4

�4

�4

4

x

y

y � f (x)

2

�2

�2

2

x

y

f

g

O

O

42

6

�4�6 �2

�4�6

42 6

x

y

refl ect across the x-axis, stretch by a factor of 3, shift right 5 units and shift up 7 units

y 5 23!x 2 4 2 6

g(x) 5 f(x 1 1) 23

O

4

�4

�4

4

x

y


79

Name Class Date

6-8 ReteachingGraphing Radical Functions

Th e graph of y 5 a!x 2 h 1 k is a translation h units horizontally and k units vertically of y 5 a!x . Th e value of a determines a vertical stretch or compression of y 5 !x .

Problem

What is the graph of y 5 2!x 2 5 1 3?

y 5 2!x 2 5 1 3

a 5 2 h 5 5 k 5 3

Translate the graph of y 5 2!x right fi ve units and up three units. Th e graph of y 5 2!x looks like the graph of y 5 !x with a vertical stretch by a factor of 2.

Exercises


1. y 5 !x 2 4 1 1 2. y 5 !x 2 4

3. y 5 !x 1 1 4. y 5 2!x 1 2 2 3

5. y 5 2!x 2 1 6. y 5 22!x 1 3 1 4

7. y 5 2!x 1 1 8. y 5 !x 1 3 2 4

9. y 5 3!x 1 2 10. y 5 2!x 2 2

O

2

4

6

62 4x

y

O

2

4

62 4x

y

O

2

�2�2

2x

y

O

2

�2�2

2x

y

O�2�2

�4

2xy

O�2

�2�4xy

O

2

4

6

�2�4x

y

O

2

�2

�4

4x

y

O

2

4

6

2 4x

y

O�2

�2�4

�4

xy


80

Name Class Date



Graphs can be used to fi nd solutions of equations containing radical expressions.

Problem

What is the minimum braking distance of a bicycle with a speed of 22 mph?

You can fi nd the minimum braking distance d, in feet, of a bicycle travelling s miles per hour using the equation s 5 5.5!d 1 0.002.

We want to fi nd the value of d when s 5 22. In other words, solve the equation 5.5!d 1 0.002 5 22. Graph Y1=5.5√(X+0.002) and Y2=22. Try diff erent values until you fi nd an appropriate window. Th en use the intersect feature to fi nd the coordinates of the point of intersection.

Th e minimum braking distance will be about 16 ft.

Exercises

Solve the equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why.

11. !3x 1 1 5 5 12. !4x 1 1 5 9

13. !2 2 5x 5 4 14. !3x 1 5 5 7

15. !7x 1 2 5 11 16. !2x 2 1 5 !1 2 2x

17. !x 2 2 5 !2 2 3x 18. 7!x 2 3 5 2!2x 1 1

19. !2x 2 5 5 !4 2 x 20. !2x 1 7 5 3!5x 1 2

Y � 22Intersectionx � 15.998

8 20

14.67

0.5

3.68

20.26

17

3

no solution; x 5 1 is extraneous

22.8

Name Class Date


81

Chapter 6 Quiz 1 Form G

Lessons 6-1 through 6-4

Do you know HOW?

Find all the real roots.

1. !36 2. !0.25 3. !3 264 4. Å3 28

125


5. "25y2 6. "49x4 7. "3 28x9 8. "3 20.125y6


9. 9x2 2 4 5 0 10. x4 5 0.0016

Multiply or divide and simplify. Assume that all variables are positive.

11. 2!2x ? "18xy2 12. "3 4xy7

"3 32x4y4

Simplify. Rationalize all denominators.

13. 3!180 1 !45 2 8!20 14. 5 1 !3

2 2 !3


15. (2125)23 16. 81

34 17. 320.6 18. 491.5

Do you UNDERSTAND?

19. Geometry What is the perimeter of the triangle at the right?

20. Reasoning Solve. !75 1 !3x 5 12!35

3 � V2

2

3 � V2

2

3 � V2

26, 6

5|y|

223,

23

12xy

5!5

25

49

27 2 !27

27 8 343

13 1 7!3

y2x

7x2 22x3 –0.5y2

20.5, 0.5

20.2, 0.2

24 225

Name Class Date


82

Chapter 6 Quiz 2 Form G

Lessons 6-5 through 6-8

Do you know HOW?


1. !x 2 2 5 x 2 8 2. !x 1 4 5 !3x 2 2

Let f (x) 5 x2 2 x 2 12 and g (x) 5 x 2 4. Perform each function operation and then fi nd the domain.

3. f (x) 2 2g(x) 4. f (x) ? g (x)

5. f (x)g (x)

6. g (x)f (x)

Let f (x) 5 2x 2 5 and g (x) 5 x2. Find each composition.

7. ( f + g)(x) 8. (g + f )(x)


9. g(x) 5 "x2 2 3 10. h(x) 5 5 2 x3

Graph. Find the domain and range of each function.

11. y 5 2 2 !x 12. y 5 !x 2 1 1 1

Do you UNDERSTAND?

13. Writing Explain why it is not true that !22 ? !22 5 !4, even though (22) ? (22) 5 4.

14. Reasoning Suppose the cost of an item is x dollars. It is marked up by n% and, later, that new cost is reduced by n%. Is the fi nal cost equal to x dollars? Use a composition of functions to justify your answer.

11


x 1 3; x u 4 1x 1 3; x u 4 and x u 23

x3 2 5x2 2 8x 1 48; all real numbers

3

2x2 2 5 4x2 2 20x 1 25

g21(x) 5 2x2 1 6, x L 6; yes

h21(x) 5 !3 5 2 x ; yes

domain: x L 0, range: y K 2

domain: x L 1, range: y L 1

Answers may vary. Sample: !22 ? !22 5 i!2 ? i!2 5 i2(!2)2 5 (21)(2) 5 22 u !4

No; answers will vary. Sample: Let n 5 20, then f(x) 5 1.2x and g(x) 5 0.8x and (g ° f)(x) 5 0.8(1.2x) 5 0.96x .

4

6

2

O

y

x

2 4 6

4

6

2

2 4 6O

y

x

Name Class Date


83

Chapter 6 Test Form G

Do you know HOW?


1. "400x2y6 2. "3 2125a9 3. "4 81x5y9

4. "3 64a6b2 5. "50s2t4 6. "256x16y28

Simplify each expression. Rationalize all denominators. Assume that all variables are positive.

7. "200x3y"2xy5

8. Q8 2 3!2RQ8 1 3!2R 9. 1!3 1 5

10. "8x3 ? "2x5 11. !63 1 2!28 2 5!7 12. !3 2 1 1!3 4

13. 21 1 !2

14. !4 5!4 4

15. !15Q1 2 !45RSimplify each expression. Assume that all variables are positive.

16. Q16x5y10

81xy2 R34 17. (264)22

3 18. a23 ? a

12

19. Q4x22y4R212 20. Q8ab2R21

2Q8ab2R 12 21. Qs2

5 t13RQs1

2 t12R

Solve each equation. Check for extraneous solutions.

22. !3 x 2 3 5 1 23. !x 1 7 5 x 1 1 24. !3x 2 8 5 2

25. (2x 1 1)13 5 3 26. "x2 2 5 5 4 27. 3(x 1 1)

43 5 48

Let f(x) 5 x2 1 5 and g(x) 5 x 2 7. Perform each function operation and then fi nd the domain.

28. f (x)g (x)

29. f (x) 2 2g (x) 30. f (x) ? g(x)

For each pair of functions, fi nd (f + g)(x) and (g + f )(x).

31. f(x) 5 3x 1 5, g(x) 5 x2 1 1 32. f(x) 5 x2 2 5x 1 2, g(x) 5 2x

33. f(x) 5 !2x 2 1, g(x) 5 5x 1 3 34. f(x) 5 22x2, g(x) 5 x 1 4

Let f(x) 5 5x 2 4 and g(x) 5 x2 2 1. Find each value.

35. (g + f )(21) 36. (f + g)(2) 37. (g + f )(0)

38. f(g(!6)) 39. f(g(0)) 40. g Qf Q45RR

20»xy3… 25a3 3»x…y2 4"xy

4a2 3"b2

10xy2

4x4

22 1 2!2

2!7

!15 2 15!34!20

2

8x3y6

27

x2y2

4

13

x2 1 5x 2 7 ; all real

numbers except 7

3x2 1 8; 9x2 1 30x 1 26

!10x 1 5; 5!2x 2 1 1 3

80

21 29 21

11 15

22x2 2 16x 2 32; 22x2 1 4

4x2 2 10x 1 2; 2x2 2 10x 1 4

x2 2 2x 1 19; all real numbers x3 2 7x2 1 5x 2 35;

all real numbers

w"21 29, 7

2 4

116 a

76

s9

10t56

3!4 13!2

2

46

1

5 2 !322

5»s…t2!2 16x8y14

Name Class Date


84


41. f(x) 5 (x 1 2)2 2 4 42. f(x) 5 4x3 2 1 43. f(x) 5 !x 1 4

44. f(x) 5 3x 1 2 45. f(x) 5 x2 2 5 46. f(x) 5 !3 x 1 2

Graph. Find the domain and range of each function.

47. y 5 !x 2 1 1 2 48. y 5 2!x 1 3 2 1 49. y 512!x 1 3 50. y 5 2!x 1 4 2 1

Rewrite each function to make it easy to graph using transformations. Describe the graph.

51. y 5 !9x 2 63 1 4 52. y 5 !3 8x 2 64 2 5

53. y 5 !3 227x 2 27 1 4 54. y 5 !16x 2 32

55. Th e children’s park has become very popular since your club built new play equipment. Use the equation f 5 4!A to calculate the amount of fence f you need to buy based on the area A of the playground.

a. Th e park currently has an area of 8100 ft2. How many feet of fencing currently encloses the park?

b. Suppose you want to increase the fenced play area to four times its current area. If you can reuse the fencing already at the park, how much new fencing do you need to buy?

Do you UNDERSTAND?

56. Writing Explain under what circumstances 2x1n 5 (2x)

1n and provide an

example to justify your answer.

57. Reasoning Graph y 5 !x and y 5 !3 x on the same coordinate grid. Notice that for 0 , x , 1, the graph of y 5 !x lies below the graph of y 5 !3 x but the opposite is true for x . 1. Explain why this is the case. Give an example.

Chapter 6 Test (continued) Form G

f21(x) 5 w!x 1 4 2 2; no

f21(x) 5 x 2 23 ; yes





y 5 3!x 2 7 1 4; graph of y 5 3!x shifted right 7 units and up 4 units

y 5 23 3!x 1 1 1 4; graph of y 5 23 3!x shifted left 1 units and up 4 units

y 5 2 3!x 2 8 2 5; graph of y 5 2 3!x shifted right 8 units and down 5 units

y 5 4!x 2 2; graph of y 5 !x shifted right 2 units

f21(x) 5 w!x 1 5; no f21(x) 5 x3 2 2; yes

f21(x) 5 3%x 1 1

4 ; yes f21(x) 5 x2 2 4, x L 24; yes

O

2

4

2 4

x

y

O�2

�2�4

�4

xy

O�2

�2�4

�4

xy

360 ft

360 ft

O

2

4

2 4

x

y

n is an odd integer; answers may vary. Sample: 2813 5 22 5 (28)

13

Answers may vary. Sample: x1n grows more rapidly (that is, as n increases) for

x R 1 and grows more slowly for x S 1; for example, Å 164 5 1

8 R14 5 3Å 1

64 but!64 5 8 S 4 53!64.


85

Name Class Date

Chapter 6 Quiz 1 Form K

Lessons 6–1 through 6–4

Do you know HOW?

Find each real root.

1. !49 2. !0.36 3. 3!2125


4. "600x6y3 5. "3 54xy5 6. "4 64x4y8

Divide and simplify.

7. "20x3

"5x 8.

"3 56x4y"3 7x3y 9.

"32a7b2

"2a3b

Simplify.

10. !7 1 3!7 11. !18 1 !32 12. 2!3 24 2 !3 81


13. x13 14. x

34 15. x1.5


16. "3 4x2 17. !5ab 18. "65x4y

Do you UNDERSTAND?

19. Writing Explain when absolute value symbols are needed when you are simplifying radical expressions.

20. An object is moving at a speed of A5 2 !3B mi/h. How long will it take the object to travel 35 mi?

7

10x3y!6y

2x

4!7

Q4x2R13

0.6

3y 3"2xy2

2 3!x

7!2

(5ab)

12

–5

2xy2 4!4

4a2!b

3!3

3!x 4"x3 x"x

x 2(65y)

12

use absolute value symbols. If the index is odd, you don’t need absolute value symbols for any terms.

For radicals in the form n"a

m, if the index is even,

175 1 35!322 h


86

Name Class Date

Do you know HOW?


1. 4!x 2 2 5 2 2. !x 1 5 5 4

3. !5x 2 1 1 3 5 x 4. !2 2 x 2 x 5 4

Let f (x) 5 x2 2 x and g (x) 5 2x 2 2 . Perform each function operation and then fi nd the domain of the result.

5. (f 1 g)(x) 6. a fgb (x)


7. y 5 x2 2 3 8. y 5 (x 2 2)3 1 1


9. y 523!x 10. y 5 2!3 8x 1 5

Do you UNDERSTAND?

11. Multiple Choice Th e graph of y 5 2!x is shifted 4 units up and 3 units right. Which equation represents the new graph?

y 5 2!x 2 4 1 3 y 5 2!x 1 3 1 4

y 5 2!x 2 3 1 4 y 5 2!x 1 4 1 3

12. Writing Explain the relationship between the domain of a function and the range of the function’s inverse.

13. A spherical water tank holds 6000 ft3 of water. What is the diameter of the tank to the nearest tenth of a foot? (Hint: V 5

p6

d3)

Chapter 6 Quiz 2 Form K

Lessons 6–5 through 6–8

x 5 1

x 5 10

y 5 6"x 1 3; no y 53!x 2 1 1 2; yes

B

They are equal

22.5 ft

(f 1 g)(x) 5 x2 2 x 2 2; all real numbers

afgb (x)5 x

2 ; all real numbers

except x 5 21

x 5 22

x 5 11

x

y

24

6

�2 2O

x

y

24

6

2 4 6O

domain: all real numbers range: all real numbers

domain: x L 0range: y L 0


87

Name Class Date

Do you know HOW?


1. "49x2y10 2. "3 264y9 3. "5 243x15

Multiply and simplify.

4. "3 15 3 "3 18 5. "7x3 ? "14x 6. 3"4 4x3 ? "4 8xy5

Rationalize each denominator. Simplify your answer.

7. 1!3 8.

!x!5 9.

!3 4!3 2x

Multiply.

10. (7 1 !5)(1 1 !5) 11. (6 1 !10)2 12. (5 1 !3)(2 2 !3)


13. 2723 14. 251.5 15. 2

34

Write each expression in simplest form.

16. ax32b22

17. Qx

34R

43 18. Qx

2 38y

14R

16

Solve.

19. !2x 1 1 5 5 20. (x 1 6)

34

5 8 21. Ax2 1 13B12 5 7

Chapter 6 Test Form K

7»xy5…

!33

12 1 8"5 46 1 2"10

!5x5

3!2x2x

3 3!10

9 125

24y3

7x2!2

3x3

6xy 4!2y

4!8

7 2 3!3

x

x 5 12 x 5 10

y 4

x 6

1

x 3

x 5 6 and x 5 26


88

Name Class Date

Let f (x) 5 "x 1 3 and g (x) 5 4 2 "x. Perform each function operation and then fi nd the domain.

22. ( f 2g) (x) 23. ( f ? g)(x)

Let f (x) 5 3x 1 1 and g (x) 5 x2 1 2. Find each value or expression.

24. ( f + g)(2) 25. (g + f )(23)


26. y 5 x 1 4 27. y 5 x2 2 2

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph.

28. y 5 "16x 2 32 29. y 53"8x 1 3

Do you UNDERSTAND?

30. Error Analysis Explain the error in this simplifi cation of radical expressions. What is the correct simplifi cation? !2 ? !3 8 5 !2(8) 5 !16 5 4

31. Reasoning Show that "6 x3 5 !x by rewriting "6 x3 in exponential form.

32. A store is having a sale with a 15% discount on all items. In addition, employees get a $20 discount on purchases of $100 or greater. Will an employee get a better deal if the $20 discount is applied fi rst or if the 15% discount is applied fi rst to their purchase of $100?

Chapter 6 Test (continued) Form K

The product property does not apply to different indexes; 2!2

6"x

3 5 x

36 5 x

12 5 !x

vertical stretch of y 5 "x by a factor of

4 and translation 2 units rightvertical stretch of y 5

3"x by a factor of

3 and translation 3 units up

y 5 4"x 2 2 ; y 5 23"x 1 3;

The employee will pay less if the 15% discount is applied fi rst.

(f 2 g)(x) 5 2"x 2 1; (f ? g)(x) 5 "x 2 x 1 12;all real numbers L 0

19 66

all real numbers L 0

�2�4 2

y � x � 4

y � x � 4

4

�4

�2

2

4

x

y

O �2�4 2

y � x2 � 2

4

�4

�2

2

4

x

y

y � ��x � 2

O


89

Name Class Date

Chapter 6 Performance Tasks

Give complete answers.

Task 1

a. Write a product of two square roots so that the answer, when simplifi edis 12x3y2. Show how your product simplifi es to give the correct answer.

b. Write a quotient of two cube roots so that the answer, when simplifi ed, is 3a2

4b3. Show how your quotient simplifi es to give the correct answer.

c. Write a product of the form Qa 1 !bRQa 2 !bR so that the answer, when simplifi ed, is 59. Show how your product simplifi es to give the correct answer.

Task 2

a. Find a radical equation of the form !ax 1 b 5 x 1 c so that one solution is extraneous. Show the steps in solving the equation.

b. Is there a value for h that makes it possible for the equation !x 1 h 1 5 5 0 to have any real number solutions? Explain.

c. Explain the relationship between the solutions to the equation !x 2 3 2 2 5 0 and the graph of the function y 5 !x 2 3 2 2.

[4] Check students’ work. All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found products and quotient correctly but followed through the process incorrectly using incorrect simplifi cation.

[2] Correct answer with no work shown OR student only able to complete part of Task.

[1] Student understood that squaring, cubing or multiplying was involved but was unable to correctly fi nd the products or quotient, or to simplify them.

[0] No attempt was made to solve this problem OR answer is incorrect with no work shown

[4] Check students’ work. All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found parts (a) and (b) correctly but could not explain (c) OR could not show steps in (a) OR could not explain (b).


[1] Student know to square each side of the equation, but could not complete the solution.

[0] No attempt was made to solve this problem OR answer is incorrect with no work shown.


90

Name Class Date

Chapter 6 Performance Tasks (continued)

Let f (x) 5 x2 1 x 2 12 and g(x) 5 x 2 2. Answer each of the following questions.

Task 3

a. Find g(x)f(x) and its domain. Explain how you determined the domain.

b. Find (g + f)(x) and (f + g)(x). Are they equal?

c. For what types of functions will (g + f)(x) and (f + g)(x) both equal x? Explain.

Give complete answers.

Task 4

a. Find the inverse of f(x) 5 !x 2 2 1 5. Show all steps in the process. What

is the domain of f21?

b. Choose a value for a and use the inverse to fi nd Qf + f21R(a) and

Qf21 + fR(a) for the value you chose. What can you conclude about

Qf + f21R(a) and Qf21 + fR(a)?

c. Graph f and f21 on the same axes. What relationships do you see between the two graphs?

[4] All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found all parts correctly but could not explain (a) OR could not explain (c) OR could not correctly fi nd the domain for (a).


[1] Student found part (a) but could not compose functions for parts (b) and (c).[0] No attempt was made to solve this problem OR answer is incorrect with no

work shown.

[4] All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found parts (a) and (b) correctly but could not describe the relationships between the graphs OR could not draw conclusions in part (b).


[1] Student found the inverse of f(x) but could not draw conclusions in part (b) and could not graph the functions.

[0] No attempt was made to solve this problem OR answer is incorrect with no work shown.

x 2 2x2 1 x 2 12

; all real numbers

except 3 and 24x2 1 x 2 14; x2 2 3x 2 10; no

when f and g are inverses

The graphs are a refl ection of each other across the line y 5 x.

f 21(x) 5 (x 2 5)2 + 2; domain: x L 5

(f ° f21)(a) 5 (f21 ° f)(a) 5 aO

2

4

6

62 4

x

y f

f�1


91

Name Class Date

Chapter 6 Cumulative Review

Multiple Choice


1. What is an equation for this graph?

y 5 (x 1 3)2 1 1 y 5 (x 2 1)2 2 3

y 5 (x 2 3)2 1 1 y 5 (x 2 1)2 1 3

2. Simplify 432.

24 32 6 8

3. Solve the system. e y 5 x2 2 3x 2 8

y 5 x 1 4

(22, 2), (2, 210) (22, 2), (6, 10)

(21, 24), (1, 5) (2, 6), (6, 10)

4. Which of the following polynomials has roots 0, 1, and 2?

p(x) 5 x2 1 x 1 2 p(x) 5 x3 2 2x 1 1

p(x) 5 x3 2 3x2 1 2x p(x) 5 x2 1 2x

5. Complete the square. 3x2 2 2x 1 u 1

9 13 1 3

6. Let f(t) 5 t2 1 5t 2 2 and h(t) 5 t2 2 2t 2 6. What is the function for h(t) 1 f(t)?

2t2 1 3t 2 8 4t2 1 t 2 8 22t2 1 3t 2 2 2t2 1 t 2 8

7. Simplify. (25 1 2i)(3 2 i)

22 1 i 217 1 i 213 1 11i 215 2 2i

8. What is the solution to x2 2 5x 1 1 5 0?

5 4 !21

2

25 4 !29

2

5 4 i!29

2

25 4 i!21

2

9. If h(x) 5 2x2 1 x 2 5 and g(x) 5 24x 2 1, what is the value of h(1) 2 g(3)?

-15 -11 11 15

x

y

O1 2 3 4

1234

5 6 7

567

B

I

C

G

F

C

F

B

C


92

Name Class Date

Chapter 6 Cumulative Review (continued)

10. Solve the system. e y 5 2x2 1 x 1 12

y 5 x2 1 3x

(24, 4), (0, 12) (23, 0), (0, 12)

(23, 0), (2, 10) (21, 22), (0, 12)

11. Which of the following relations defi nes y as a function of x?

x2 1 y2 5 6 y 5 4!4 2 x y 5 x2 2 3 x 5 y2

12. Simplify. 6 2 2i1 1 3i

22i 6 223i 6

5 1 85i 65 2 85i

Short Response

For Exercises 13 and 14, let f(x) 5 x2 1 2x 2 3 and g(x) 5 4x 2 1.

13. Simplify each expression. a. g(x 1 2) b. f(x 1 2)

14. Find each value. a. (g + f )(0) b. (f + g)(0)

15. Solve by graphing. e y , 2x2 2 2x 1 3

y . x2 2 x 2 12

16. To fi nd the time t, in seconds, it takes an object to fall h meters, use the

equation t 5 Å h4.9. A croissant dropped from the top of the Eiff el Tower takes

8.13 s to reach the ground. How tall is the Eiff el Tower?

Simplify each expression. Assume that all variables are positive.

17. a125a15b7

28a3b4 b43

18. Q4 2 5!3RQ4 1 5!3RExtended Response

19. Writing Explain how you can obtain the solutions to a system of equations by graphing.

20. Open-Ended Write the equation of a parabola with vertex at (22, 3).

G

C

F

4x 1 7

213 24

324 m

259625a16b4

16

x2 1 6x 1 5 2

6�6

�10

y

Ox

19. [4] Answers may vary. Sample: Graph both equations on the same grid. If the graphs do not intersect, then the system has no solution. If the graphs intersect in one or more points, then the intersection point(s) are the solution(s) of the system. [3] Explanation shows understanding of the concepts with some minor errors. [2] Explanation contains minor errors. [1] Explanation contains major errors and lacks detail. [0] Explanation is missing.

20. [4] Answers may vary. Sample: f(x) 5 (x 1 2)2 1 3. [3] Equation has minor errors. [2] Equation has major errors. [1] Equation is incomplete and the given answer has errors. [0] Equation is missing.


93

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

Chapter 6 Project Teacher Notes: Swing Time

About the ProjectTh e Chapter Project gives students an opportunity to conduct experiments involving a real-world application of physics and mathematics. Students solve a formula for a given variable to write the formula in a more useful form, then use the formula to determine the periods of pendulums.

Introducing the Project• Ask students if they have ever used a pendulum, or a swing-like motion.

Remind them that the motion of a playground swing is similar to a pendulum’s motion.

• Have students speculate as to whether a homemade pendulum could continue to swing forever, or if it would eventually stop swinging. Encourage them to discuss what might cause a pendulum to stop swinging.

Activity 1: ConstructingStudents use strings, coins, and binder clips to construct simple pendulums.

Activity 2: Investigating Students perform experiments to time the swings of their pendulums and record their observations in charts.

Activity 3: Analyzing Students solve a formula for a given variable, then use the formula to fi nd the theoretical periods of their pendulums. Th ey analyze their data and determine why their experimental results might diff er from their theoretical results.

Finishing the ProjectYou 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.

• Have students review their data and their calculations of the periods.

• Ask groups to share their insights that resulted from completing the project, such as any shortcuts they found for using the formula or calculating the periods.

SOURCES: Basic Physics: A Self-Teaching Guide, Second edition, p. 104, by Karl F. Kahn, 1996, John Wiley & Sons, Inc.; Physics, p. 242, by John D. Cutnell and Kenneth W. Johnson, 1989, John Wiley & Sons, Inc.


94

Name Class Date

Chapter 6 Project: Swing Time

Beginning the Chapter ProjectGalileo observed a swinging lantern and made an important discovery about the timing of a pendulum’s swing. A Dutch man named Christiaan Huygens discovered the relationship between the length of a pendulum and the time it takes to make a complete swing, which led to the use of pendulums in clock making.

You will construct pendulums using strings and weights and use your pendulums to investigate whether the length of the string or the amount of weight attached to a pendulum aff ects the time it takes the pendulum to make one full swing.

List of Materials• Calculator • Metric ruler or measuring tape

• Th read or thin string • Binder clips (2 medium)

• Coins (3 quarters, 3 nickels, or 3 pennies) • Stopwatch

Activities

Activity 1: ConstructingTo construct a simple pendulum, tie a medium binder clip to the end of a piece of string. Th e binder clip will be used to hold one or more coins for the experiments in Activity 2. Th e weight on the end of the string, which includes the binder clip and the coin(s) it holds, is called the pendulum bob. Th e period of a pendulum is the time it takes for the pendulum to complete one full swing (back and forth).

Activity 2: Investigating

Experiment 1

Tie the free end of the string of the pendulum to a stable object. Do this in such a way that neither the string nor the bob touch another object when the pendulum is swung. Insert one coin in the binder clip. Measure the length of the string (in centimeters) from the point where it is attached to the stable object to the center of the bob. Record this length. Th ree times, pull the pendulum back to an angle of about 20° and let it go. For each trial, use a stopwatch to record the number of seconds it takes for the pendulum to complete 10 full swings. Record each time in the fi rst column of the table provided on the next page. Next, fi nd and record the average of the three times you listed. Finally, divide the average time by 10 to determine the period of the pendulum. Repeat the procedure using two coins, then using three coins, recording the data in the second and third columns, respectively. Does it appear that the weight of the bob aff ected the period of the pendulum? What factors other than the weight might aff ect the period of the pendulum?


95

Name Class Date

Chapter 6 Project: Swing Time (continued)

Experiment 2

Cut a second string that is half the length of the original string. Repeat Experiment 1. Record data in a table. Does it appear that the pendulum string length aff ects its period? Explain.

Activity 3: Analyzing

Th e formula / 5980t2

4p2 represents the length O (in centimeters) of a simple pendulum with a period of t seconds. (In this formula, the acceleration due to gravity is given as 980 cm/s2.)

• Solve for t. According to the formula, how does changing the weight of the bob aff ect the period of a pendulum?

• Use the formula to fi nd the theoretical period for each pendulum. Record your experimental and theoretical periods for each.

Do your experimental results give the same period as the theretical models? What factors do you think would account for any diff erences? Explain your observations.

Finishing the ProjectPrepare a presentation for the class describing your results. Your presentation should include a chart showing your experimental and theoretical results.

Reflect and ReviseWhen you are sure your data are accurate, decide if your presentation is complete, clear, and convincing. If needed, make changes to improve your presentation.

Extending the ProjectResearch the use of clock pendulums. Find out how a pendulum keeps time. Research periods of the pendulums used in diff erent clock types.

Trial 1

1 coin 2 coins 3 coins

Trial 2

Trial 3

Average time to complete 10 full swings

Period of pendulum

Experiment 1

Length of Pendulum Experimental Period Theoretical Period

Experiment 2


96

Name Class Date

Chapter 6 Project Manager: Swing Time

Getting Started

Read the project. As you work on the project, you will need a calculator and materials on which you can record your results and make calculations. Keep all of your work for the project in a folder, along with this Project Manager.

Checklist Suggestions

☐ Activity 1: constructing a pendulum ☐ Use the lightest thread or string possible.

☐ Activity 2: determining the period ☐ Have one student swing the pendulum while another student keeps time.

☐ Activity 3: comparing experimental and theoretical periods

☐ Isolate t, then take the square root of each side of the equation. Substitute the string lengths into the new equation.

☐ pendulum experiment ☐ How would your results change if your pendulum were not able to swing freely, that is without contact with any other object? How would your results change if you pulled the pendulum back to an angle of 60°? What other changes would aff ect your results?

Scoring Rubric4 Your experimental results are reasonable. Calculations are correct.

Explanations are thorough and well thought out. Data, calculations, and conclusions are neatly presented.

3 Your experimental results are reasonable. Calculations are mostly correct with some minor errors. Explanations lack detail and accuracy. Data, calculations, and conclusions are not well organized.

2 Your experimental results are not reasonable. Calculations and explanations contain errors. Data, calculations, and conclusions are unorganized and lack detail.

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

Additional Vocabulary Support

Documents

Transcript of Additional Vocabulary Support