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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
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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
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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
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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
Roots and Radical Expressions
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
Roots and Radical Expressions
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…
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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?
Find the two real solutions of each equation.
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
Roots and Radical Expressions
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
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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
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8
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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
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
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
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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
Simplify each radical expression. Use absolute value symbols when needed.
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
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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
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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
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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
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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
6-2 Practice (continued) Form G
Multiplying and Dividing Radical Expressions
"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
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6-2 Practice Form K
Multiplying and Dividing Radical Expressions
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.
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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?
6-2 Practice (continued) Form K
Multiplying and Dividing Radical Expressions
2x
!783
c 5 ÅEm; c 5
!Emm
3a 2!2a4 seconds
3x 4!x
3!4x2
x 3"3y
2
x !35x5
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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
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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
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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
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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)
Multiplying and Dividing Radical Expressions
"5xx
3"3ba
4"9x3yx
"30xy6
k 3"k4
x2"15xy5y
4"10z2
z
3"19ac2
c2
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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
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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 �
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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
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6-3 Practice (continued) Form G
Binomial Radical Expressions
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
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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
Binomial Radical Expressions
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.
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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.
Simplify. Assume that all variables are positive.
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.
6-3 Practice (continued) Form K
Binomial Radical Expressions
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.
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Multiple Choice
For Exercises 1−5, choose the correct letter.
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
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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
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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
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• 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
6-3 Reteaching (continued)
Binomial Radical Expressions
3 "3 1 12
20 1 15!6
40 2 28!2 2!5 2 !15211 1 4"73
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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
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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
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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
Write each expression in radical form.
13. x43 14. (2y)
13 15. a1.5
16. b15 17. z
23 18. (ab)
14
19. m2.4 20. t227 21. a21.6
Write each expression in exponential form.
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
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6-4 Practice (continued) Form G
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
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Name Class Date
6-4 Practice Form K
Rational Exponents
Simplify each expression.
1. 16
14 2. (23)
13 ? (23)
13 ? (23)
13 3. 5
12 ? 45
12
!4 16
Write each expression in radical form.
4. x
14 5. x
45 6. x
29
Write each expression in exponential form.
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
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36
Name Class Date
6-4 Practice (continued) Form K
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
Simplify each number.
17. (125)
23 18. (216)
23(216)
23 19. (2243)
25
Write each expression in simplest form. Assume that all variables are positive.
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
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37
Name Class Date
6-4 Standardized Test Prep Rational Exponents
Multiple Choice
For Exercises 1−5, choose the correct letter.
1. What is 1213 ? 45
13 ? 50
13 in simplest form?
!27,000 30 10713 27,000
2. What is x13 ? y
23 in simplest form?
x3"y3 "xy3 "3 (xy)2 "3 xy2
3. What is x13 ? x
12 ? x
14 in simplest form?
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
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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
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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
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Name Class Date
6-4 Reteaching (continued)
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
Write each expression in simplest form. Assume that all variables are positive.
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
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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.
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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
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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
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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?
6-5 Practice (continued) Form G
Solving Square Root and Other Radical Equations
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
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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
Solving Square Root and Other Radical Equations
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
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6-5 Practice (continued) Form K
Solving Square Root and Other Radical Equations
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
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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
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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
2; 23 is an extraneous solution.
3; 0 is an extraneous solution.
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
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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
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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
Solve. Check for extraneous solutions.
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
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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
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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
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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
2x2 2 4x 1 4; all real numbers
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
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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)
6-6 Practice (continued) Form G
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
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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
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6-6 Practice (continued) Form K
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
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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
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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
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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
2x2 1 4x 2 5; all real numbers
4x 2 3x2 1 2
; all real numbers
x2 2 4x 1 5; all real numbers
x2 1 24x 2 3; x u 3
4
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• 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))
6-6 Reteaching (continued)
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
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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
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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
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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
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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
6-7 Practice (continued) Form G
Inverse Relations and Functions
$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
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6-7 Practice Form K
Inverse Relations and Functions
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
Graph each relation and its inverse.
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
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x y
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5
10
15
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Name Class Date
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6-7 Practice (continued) Form K
Inverse Relations and Functions
Find the inverse of each function. Is the inverse a function?
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.
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Multiple Choice
For Exercises 1−4, choose the correct letter.
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
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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
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• 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
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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: y L 25
domain: x L 0; range: all real numbers
domain: x S 0; range: y u 24
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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
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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
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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?
Graph each function.
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
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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?
6-8 Practice (continued) Form G
Graphing Radical Functions
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 28!x 1 3; graph of y 5 28!x shifted left 3 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!x 1 5; graph of y 5 22!x shifted left 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
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75
6-8 Practice Form K
Graphing Radical Functions
Graph each function.
1. y 5 !x 1 3 2. y 5 !x 2 4 3. y 5 !x 2 7
Graph each function.
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
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6-8 Practice (continued) Form K
Graphing Radical Functions
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
domain: x L 4; range: y K 23
xO
y300
�2�4
200
100
2 4
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Name Class Date
Multiple Choice
For Exercises 1−4, choose the correct letter.
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
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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
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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
Graph each function.
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
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O
2
4
6
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y
O
2
�2
�4
4x
y
O
2
4
6
2 4x
y
O�2
�2�4
�4
xy
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6-8 Reteaching (continued)
Graphing Radical Functions
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
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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
Simplify each radical expression. Use absolute value symbols when needed.
5. "25y2 6. "49x4 7. "3 28x9 8. "3 20.125y6
Find the two real solutions of each equation.
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
Simplify each expression.
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
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Chapter 6 Quiz 2 Form G
Lessons 6-5 through 6-8
Do you know HOW?
Solve. Check for extraneous solutions.
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)
Find the inverse of each function. Is the inverse a function?
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
x2 2 3x 2 4; all real numbers
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
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Chapter 6 Test Form G
Do you know HOW?
Simplify each radical expression. Use absolute value symbols when needed.
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
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Find the inverse of each function. Is the inverse a function?
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
domain: x L 1; range: y L 2
domain: x L 23; range: y K 21
domain: x L 24; range: y K 21
domain: x L 0; range: y L 3
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.
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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
Simplify. Assume that all variables are positive.
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
Write each expression in radical form.
13. x13 14. x
34 15. x1.5
Write each expression in exponential form.
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
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Name Class Date
Do you know HOW?
Solve. Check for extraneous solutions.
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)
Find the inverse of each function. Is the inverse a function?
7. y 5 x2 2 3 8. y 5 (x 2 2)3 1 1
Graph each function.
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
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Name Class Date
Do you know HOW?
Simplify each radical expression. Use absolute value symbols when needed.
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)
Simplify each number.
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
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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)
Graph each relation and its inverse.
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
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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).
[2] Correct answer with no work shown OR student only able to complete part of Task.
[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.
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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).
[2] Correct answer with no work shown OR student only able to complete part of Task.
[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).
[2] Correct answer with no work shown OR student only able to complete part of Task.
[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
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Chapter 6 Cumulative Review
Multiple Choice
For Exercises 1−12, choose the correct letter.
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
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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.
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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.
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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?
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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
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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