Chapter Test
CHAPTER
6
ADDITIONAL RESOURCES
• Chapter 6 Resource Book
Chapter Test (3 levels) (p. 130)
SAT/ACT Chapter Test (p. 136)
Alternative Assessment (p. 137)
•
Test and Practice Generator
Simplify the expression. Tell which properties of exponents you used. 1–5. See margin.
1. x7 • ᎏ1ᎏ
x2
9
3. ᎏxᎏ
º2
x
2. (32x6)3
15x2y 6x3y2
5. ᎏᎏ • ᎏ
5xy
6 x 4 y5
4. (8x3y2)º3
1. x 5; quotient of powers property
Describe the end behavior of the graph of the polynomial function. Then
evaluate the function for x = º4, º3, º2, . . ., 4. Then graph the function. 6–8. See margin.
4
2
3
6. y = x º 2x º x º 1
7. y = º3x º 6x
2
2. 729x18; power of a product and
power of a power properties
8. y = (x º 3)(x + 1)(x + 2)
3. x 11; quotient of powers property
Perform the indicated operation.
2
2
9. (3x º 5x + 7) º (2x + 9x º 1)
x2 º 14x + 8
Factor the polynomial.
2
11. (x º 4)(x + 1)(x + 3) x3 º 13x º 12
10. (2x º 3)(5x º x + 6)
10x3 º 17x2 + 15x º 18
12. 64x3 + 343
(4x + 7)(16x2 º 28x + 49)
13. 400x 2 º 25
14. x4 + 8x 2 º 9
(x2 + 9)(x + 1)(x º 1)
25(4x + 1)(4x º 1)
15. 2x3 º 3x 2 + 4x º 6
(2x º 3)(x2 + 2)
Solve the equation.
2兹3苶
i, ±兹5苶 17. 81x4 = 16 ±}2}, ±}2}i
16. 3x4 º 11x 2 º 20 = 0 ±}
3
3
3
1
512x y
4. }
; power of a power,
9 6
power of a product, and negative exponent properties
3
y
5. ᎏ3 ; product of a power, quotient
1 1
18. 4x3 º 8x 2 º x + 2 = 0 º}}, }}, 2
2 2
15
Divide. Use synthetic division if possible. 19. 8x3 º 3x2 + 7x º 8 + }}
x+1
19. (8x4 + 5x3 + 4x 2 º x + 7) ÷ (x + 1)
20. (12x3 + 31x 2 º 17x º 6) ÷ (x + 3) 12x2 º 5x º 2
List all the possible rational zeros of ƒ using the rational zero theorem. Then
find all the zeros of the function.
±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36; º4, ±3i
22. ƒ(x) = x3 + 4x 2 + 9x + 36
23. ƒ(x) = x4 + x3 º 2x 2 + 4x º 24
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24; º3, 2, ±2i
0, ±1, ±2, ±7, ±14; º2, 0, 7
Write a polynomial function of least degree that has real coefficients, the given
zeros, and a leading coefficient of 1.
ƒ(x) = x3 º 5x2 + 4x º 20
ƒ(x) = x4 º 3x3 + 4x
25. 2, 2, º1, 0
26. 5, 2i, º2i
27. 3, º3, 2 º i
24. 1, º3, 4
ƒ(x) = x4 º 4x3 º 4x2 + 36x º 45
ƒ(x) = x3 º 2x2 º 11x + 12
3
2
28. Use technology to approximate the real zeros of ƒ(x) = 0.25x º 7x + 15.
about º1.428, about 1.505, about 27.923
29. Identify the x-intercepts, local maximum, and local minimum of the graph of
1
ƒ(x) = ᎏᎏ(x º 3)2(x + 3)2. Then describe the end behavior of the graph. See margin.
9
21. ƒ(x) = x3 º 5x 2 º 14x
of a power, zero exponent, and
negative exponent properties
6–8. See Additional Answers beginning on page AA1.
29. x-intercepts: –3, 3; local max at
(0, 9); local min at (–3, 0) and (3, 0);
ƒ(x) · +∞ as x · –∞, ƒ(x) · +∞
as x · +∞
30. ƒ(1) ƒ(2) ƒ(3) ƒ(4) ƒ(5) ƒ(6)
7
20
13
83 256 623 1292
63 173 367 669
50 110 194 302
60
84 108
24
24
30. Show that ƒ(x) = x4 º 2x + 8 has nonzero constant fourth-order differences. See margin.
31. The table gives the number of triangles that point upward that you can find in
ƒ(2) ⫽ 4
a large triangle that is n units on a side and divided into triangles that are each
one unit on a side. Find a polynomial model for ƒ(n).
32.
n
1
2
3
4
5
6
7
ƒ(n)
1
4
10
20
35
56
84
1
6
1
2
1
3
ƒ(n) = }}n3 + }}n2 + }}n
CELLS An adult human body contains about 75,000,000,000,000 cells.
Each is about 0.001 inch wide. If the cells were laid end to end to form a
chain, about how long would the chain be in miles? Give your answer in
scientific notation. about 1.1837 ª 10 6 mi
Chapter Test
391
391
CHAPTER
ADDITIONAL RESOURCES
• Chapter 6 Resource Book
Chapter Test (3 levels) (p. 130)
SAT/ACT Chapter Test (p. 136)
Alternative Assessment (p. 137)
•
Test and Practice Generator
6
Chapter Standardized Test
TEST-TAKING STRATEGY The mathematical portion of the SAT is based on concepts and skills taught in high
school mathematics courses. The best way to prepare for the SAT is to keep up with your day-to-day studies.
1. MULTIPLE CHOICE What is the value of º40? D
A
¡
D
¡
4
º1
B
¡
E
¡
C
¡
1
0
º4
2. MULTIPLE CHOICE What is the value of
ƒ(x) = 7x4 º 3x3 + 8x 2 + x º 9 when x = º1? A
A
¡
D
¡
8
º8
B
¡
E
¡
C
¡
4
2
º14
3. MULTIPLE CHOICE Which statement about the end
Qu. Standard
Qu. Standard
behavior of the graph of ƒ(x) = x4 + 1 is true? A
A
¡
B
¡
C
¡
D
¡
E
¡
ƒ(x) ˘ +‡ as x ˘ º‡.
ƒ(x) ˘ º‡ as x ˘ +‡.
number of grocery stores in the United States can be
modeled by G = 0.03t 2 º 1.5t + 171, where G is
the number of stores in thousands and t is the number
of years since 1990. The average sales per grocery
store can be modeled by S = 4.7t 2 + 49.1t + 2009,
where S is sales in thousands of dollars. What were
the approximate total sales in millions of dollars for
grocery stores in the United States in 1994? C
B
¡
D
¡
B
¡
D
¡
2x3 º 1
2x3 + 1
8x3 º 1
4x3 + 1
8x3 + 1
of the equation x 5 = 256x? A
392
D
¡
4x 2 º 27x + 99 º ᎏᎏ
E
¡
4x 2 º 27x + 99 º ᎏᎏ
39
xº4
4x 2 + 5x + 11 + ᎏᎏ
4x º 11x º 9x º 5
401
xº4
401
x+4
B
¡
D
¡
º2, º3, º7
2, º3, º7
2, 3, 7
0, 6, 7
º2, 3, 7
9. MULTIPLE CHOICE How many zeros does the
function ƒ(x) = º3x4 + x + 2 have? E
A
¡
D
¡
B
¡
E
¡
0
3
C
¡
1
0, ±4
0, ±4i
B
¡
E
¡
4, º4
C
¡
±4, ±4i
2
4
10. MULTIPLE CHOICE Which function is graphed? D
y
5
(0, 7)
(1, 0)
(⫺3, 0)
3
(4, 0)
x
3.8 ª 1011
6. MULTIPLE CHOICE What are all the real solutions
392
39
4x 2 + 5x + 11 + ᎏᎏᎏ
3
2
3.8 ª 108
factorization (2x + 1)(4x 2 º 2x + 1)? E
A
¡
D
¡
C
¡
3.8 ª 101
5. MULTIPLE CHOICE Which polynomial has the
A
¡
C
¡
E
¡
4x3 + 5x 2 + 11x + 39
A
¡
C
¡
E
¡
ƒ(x) ˘ º‡ as x ˘ 0.
3.8 ª 105
A
¡
B
¡
of ƒ(x) = x3 º 8x 2 + x + 42? E
ƒ(x) ˘ º‡ as x ˘ º‡.
3.8 ª 10º1
(4x3 º 11x 2 º 9x º 5) ÷ (x º 4)? B
8. MULTIPLE CHOICE What are all the rational zeros
ƒ(x) ˘ +‡ as x ˘ 0.
4. MULTIPLE CHOICE For 1992 through 1995, the
A
¡
C
¡
E
¡
7. MULTIPLE CHOICE What is the quotient of
A
¡
B
¡
C
¡
ƒ(x) = (x + 3)(x º 1)(x º 4)
ƒ(x) = 7(x + 3)(x º 1)(x º 4)
7
12
ƒ(x) = ᎏᎏ(x º 3)(x + 1)(x + 4)
D
¡
ƒ(x) = ᎏᎏ(x + 3)(x º 1)(x º 4)
E
¡
ƒ(x) = ºᎏᎏ(x + 3)(x º 1)(x º 4)
0, ±4, ±4i
Chapter 6 Polynomials and Polynomial Functions
7
12
7
12
QUANTITATIVE COMPARISON In Exercises 11 and 12, choose the statement
that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
13b.
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
14e.
ƒ(n)
(5, 336)
300
Column A
Column B
11.
xº2
x2
D
200
12.
Degree of ƒ(x) = x4 º 7x + 13
Degree of ƒ(x) = 4x3 + 2x2 º x + 1
A
100 (0, 6)
13. MULTI-STEP PROBLEM You are designing a monument for the city park.
0
(4, 210)
0
1
2
(3, 120)
(2, 60)
(1, 24)
3
4
5
6
7n
The monument is to be a rectangular prism with dimensions x + 1 feet,
x º 5 feet, and x º 6 feet.
a. Write a function ƒ(x) for the volume of the monument. ƒ(x) = x3 º 10x2 + 19x + 30
b. Use a graphing calculator to graph ƒ(x) for º10 ≤ x ≤ 20. See margin.
Writing Look back at your graph from part (b). Identify the local 13c. no; The local maximum occurs at
about (1.15, 40.15) and the local
maximums and local minimums. Do these values represent maximum
minimum occurs at about
and minimum possible volumes of the monument? Explain.
(5.52, º1.63), but x must be greater
d. If the volume of the monument is to be 220 cubic feet, what will the
than 6 for the side of length x º 6 to
dimensions be? 11 ft by 5 ft by 4 ft
have a positive measure.
14. MULTI-STEP PROBLEM The numbers in the table give the volumes of the
first six prisms in a sequence.
c.
ƒ(0) ⫽ 6
ƒ(2) ⫽ 60
ƒ(1) ⫽ 24
ƒ(3) ⫽ 120
Prism (n)
0
1
2
3
4
5r
Volume, ƒ(n)
6
24
60
120
210
336
a. Use finite differences to determine the degree of ƒ. 3
b. Use a system of equations to find a polynomial model for ƒ(n) in
standard form. ƒ(n) = n3 + 6n2 + 11n +6
Writing Factor the polynomial. Explain how the factors are related to the
dimensions of the prism. ƒ(n) = (n + 1)(n + 2)(n + 3);
for prism n, the dimensions are (n + 1) by (n + 2) by (n + 3).
d. Use your model to find the volume of the 50th prism in the sequence. 132,600
c.
e. Sketch a graph of your model and label the points that represent the first six
prisms. What is the domain of the function?
See margin for graph. The domain is all whole numbers.
Chapter Standardized Test
393
393
Chapter Test
CHAPTER
7
ADDITIONAL RESOURCES
• Chapter 7 Resource Book
Chapter Test (3 levels) (p. 106)
SAT/ACT Chapter Test (p. 112)
Alternative Assessment (p. 113)
•
Test and Practice Generator
Evaluate the expression without using a calculator.
3
1. 兹º
苶1苶0苶0苶0苶 º10
2. 45/2 32
1
4. 243º1/5 }3}
3. (º64)2/3 16
Simplify the expression. Assume all variables are positive.
3
6. (21/3 • 51/2)4 50兹2苶 7.
兹3 2苶7苶苶
x3苶
y6苶
z9 3xy 2z 3
冉 冊
3xyº1 x1/2
8. ᎏᎏ }
12x1/2y 4y 2
81x2 3/4 27x3/2
9. ᎏᎏ
}3/4
}
y
y
4
5. 兹1
苶6苶 2
10. 兹1
苶8苶 + 兹2苶0苶0苶 13兹2苶
21.
5 (10, 2)
Perform the indicated operation and state the domain.
4x º 8, x is all real numbers
13. ƒ • g; ƒ(x) = 5x + 7, g(x) = x º 9
5x 2 º 38x º 63, x is all real numbers
except 0
15. ƒ(g(x)); ƒ(x) = 4x2 º 5, g(x) = ºx
1
17. ƒ(x) = ᎏᎏx º 4
º1 3
(5, 4.71)
2x + 6x + 1, x is all real numbers
1
5
ƒº1(x) = º}}x + 1
ƒ (x) = 3x + 12
Graph the function. Then state the domain and range.
21–24. See margin.
y
2
18. ƒ(x) = º5x + 5
21. ƒ(x) = 兹x苶º
苶苶
6
3
22. ƒ(x) = 兹x苶 + 3
3
19. ƒ(x) = ᎏᎏx2, x ≥ 0
4
2
ƒº1(x) = }}兹3x
苶
3
20. ƒ(x) = x5 º 2
23. ƒ(x) = 3(x + 4)1/3 º 2
24. ƒ(x) = º2x1/2 + 4
º1
(⫺1, 2)
26. (x + 8)1/4 + 1 = 0
no solution
(1, 4)
⫺2
⫺2
23.
5
x
5
24.
, x ≥ 0, y ≤ 4
y
2/3
l = 24.1d
where d is the midshaft diameter of the bone (in millimeters). If the bone of
an antelope has a midshaft diameter of 20 millimeters, what is the length of
the bone? 䉴 Source: On Size and Life 177.57 mm
2
(0, 4)
(1, 2)
x
⫺2 (4, 0)
⫺2
ACADEMY AWARDS In Exercises 30–33, use the tables below which give
the ages of the Academy Award winners for best actress and for best actor
from 1980 to 1998.
30, 32, 35, 37, 37, 38, 39, 42, 43, 45,
45, 46, 51, 52, 52, 54, 60, 61, 76
(4, 4)
(⫺5, ⫺5)
CONNECTION Some biologists study the structure of animals.
By studying a series of antelopes, biologists have found that the length l
(in millimeters) of an antelope’s bone can be modeled by
21, 25, 26, 29, 31, 33, 33, 34, 34, 38,
39, 41, 42, 45, 49, 49, 61, 72, 80
, x and y are
all real
numbers
y
(⫺3, 1)
29. BIOLOGY
Best actor
x
2
ƒ (x) = (x + 2)
3
36
27. 兹7
苶x苶º
苶苶
9 + 11 = 14 }7} 28. 兹4苶x苶+
苶苶5
1苶 º 3兹x苶 = 0 3
Best actress
, x and y are
all real
numbers
1/5
Solve the equation. Check for extraneous solutions.
25. x 5/2 º 10 = 22 4
x
(6, 0)
22.
16. g(ƒ(x)); ƒ(x) = x2 + 3x, g(x) = 2x + 1
2
4x º 5, x is all real numbers
Find the inverse function.
⫺5
⫺5
12. ƒ º g; ƒ(x) = 2x1/4, g(x) = 5x1/4 º3x1/4, x ≥ 0
1
ƒ
}}, x is all real numbers
14. ᎏᎏ; ƒ(x) = xº1/5, g(x) = x3/5 x 4/5
g
11. ƒ + g; ƒ(x) = x º 8, g(x) = 3x
, x ≥ 6, y ≥ 0
y
30. Actresses: 41.16; 38; 33, 34, 49
(3 modes); 59; 15.2
Actors: 46.05; 45; 37, 45, 52
(3 modes); 46; 11.14
Actresses
31.
and
20
40
60
80
30. Find the mean, median, mode, range, and standard deviation of each data set. 30–33. See margin.
21 31 38
31. Draw a box-and-whisker plot of each data set.
49
80
Actors
32. Make a frequency distribution of each data set using six intervals beginning with
33.
21–30. Then draw a histogram of each data set.
30
Writing Compare the ages of the best actresses with the ages of the best
30 37
actors. Use statistics and statistical graphs to support your statements.
50
45
52
70
76
32–33. See Additional Answers
beginning on page AA1.
Chapter Test
459
459
Chapter Standardized Test
CHAPTER
ADDITIONAL RESOURCES
• Chapter 7 Resource Book
Chapter Test (3 levels) (p. 106)
SAT/ACT Chapter Test (p. 112)
Alternative Assessment (p. 113)
•
Test and Practice Generator
7
TEST-TAKING STRATEGY Some college entrance exams allow the optional use of calculators.
If you do use a calculator, make sure it is one you are familiar with and have used before.
1. MULTIPLE CHOICE If x4 = 625, what does x equal?
C
A
¡
D
¡
B
¡
E
¡
5
25
C
¡
º5
7. MULTIPLE CHOICE Which function is graphed? C
±5
y
±25
2
2. MULTIPLE CHOICE What is the simplified form of
the expression 兹1苶8苶 + 兹2苶0苶0苶 + 兹2苶 º 兹8苶? A
Qu. Standard
Qu. Standard
A
¡
C
¡
E
¡
B
¡
D
¡
12兹2苶
18兹2苶
14兹2苶 º 兹8苶
4兹2苶 º 4兹8苶
3. MULTIPLE CHOICE What is the simplified form of
the expression 兹5苶4苶苶
x3苶
y6苶
z10苶? (Assume all variables
are positive.) D
3
A
¡
C
¡
E
¡
xy 兹5苶4苶苶
z苶
2 3
10
3
3y3z7兹5苶
B
¡
D
¡
2 33
xy z 兹5苶4苶z苶
3
3xy2z3兹2苶z苶
18xy3z7
4. MULTIPLE CHOICE Which of the following is true
if ƒ(x) = 3xº1/2, g(x) = 6x3/4, and h(x) = 18x1/4? C
A
¡
C
¡
E
¡
h(x) = ƒ(x) + g(x)
h(x) = ƒ(x) • g(x)
B
¡
D
¡
h(x) = ƒ(x) º g(x)
ƒ(x)
g(x)
h(x) = ᎏ
h(x) = ƒ(g(x))
g(x) = x2 + 2, what is ƒ(g(x))? B
x4 + x2 + 17
x4 + x 2 º 9
A
¡
C
¡
E
¡
B
¡
D
¡
x4 + x2 + 5
x4 + x2 º 3
x4 + 7x2 + 5
ƒº1(x) = 2x º 10
E
¡
1
5
ƒº1(x) = ᎏᎏx + ᎏᎏ
2
2
460
460
3
y = 兹x苶º
苶苶
8 +3
3
y = 兹x苶+
苶苶
3 +8
3
y = 兹x苶+
苶苶
8 º3
3
y = 兹x苶+
苶苶
8 +3
8. MULTIPLE CHOICE What is the solution of the
equation (3x + 5)1/2 º 3 = 4? E
A
¡
ºᎏᎏ
B
¡
8
ᎏᎏ
3
D
¡
14
ᎏᎏ
3
E
¡
44
ᎏᎏ
3
4
3
C
¡
11
ᎏᎏ
3
9. MULTIPLE CHOICE What is the solution of the
3
equation 4兹x苶º
苶苶
5 = 20? B
A
¡
D
¡
B
¡
E
¡
120
2005
C
¡
130
220
4101
6, 4, 4, 10, 5, 12, 1? B
A
¡
D
¡
ļ1(x) = 2x + 5
ļ1(x) = 2x + 10
1
2
ƒº1(x) = ᎏᎏx + 5
Chapter 7 Powers, Roots, and Radicals
B
¡
E
¡
4
7
C
¡
5
6
10
11. MULTIPLE CHOICE Which data set matches the
box-and-whisker plot shown? D
6. MULTIPLE CHOICE Which function is the inverse
1
of ƒ(x) = ᎏᎏx º 5? C
2
A
¡
B
¡
C
¡
D
¡
B
¡
D
¡
3
y = 兹x苶º
苶苶
3 +8
10. MULTIPLE CHOICE What is the median of
5. MULTIPLE CHOICE If ƒ(x) = x2 º 3x + 7 and
A
¡
C
¡
E
¡
x
1
14兹2苶
0
¡
C
¡
E
¡
A
1
1
2
3
4
3
1, 1, 3, 5, 6, 7, 9
1, 2, 4, 5, 6, 8, 9
1, 3, 5, 5, 7, 8, 9
5
5
6
¡
D
¡
B
7
7
8
9
10
9
1, 2, 3, 5, 7, 8, 9
1, 3, 4, 5, 6, 7, 9
QUANTITATIVE COMPARISON In Exercises 12 and 13, choose the statement
that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
Column B
º2/3
ƒ(2) where ƒ(x) = xº2
ƒ(8) where ƒ(x) = x
12.
3
ƒ(ƒ(0)) where ƒ(x) = x + 1
ƒ(ƒ(0)) where ƒ(x) = 5x º 2
13.
14e. Specific metabolic rate
increases as body mass
decreases because the rate is
proportional to m –1/4. In other
words, as mass decreases, the
denominator gets smaller so the
rate increases.
C
B
14. MULTI-STEP PROBLEM The metabolic rate r (in kilocalories per day) of a
mammal can be modeled by r = km3/4 where k is a constant and m is the mass
(in kilograms) of the mammal. The specific metabolic rate s (the rate per unit mass)
km3/4
can be modeled by s = ᎏᎏ. 䉴 Source: Scaling: Why is Animal Size so Important?
m
a. A 922 kilogram cow has a metabolic rate of about 11,700 kilocalories per day.
What is the value of k in the model for metabolic rate? about 69.9
b. Using the k-value from part (a), simplify the model given for specific
69.9
metabolic rate. s = }
m1/4
c. What is the specific metabolic rate of a 922 kilogram cow? about 12.7 kilocalories per day per kilogram
d. What is the specific metabolic rate of a 16 gram mouse? about 197 kilocalories per day per kilogram
e.
Writing How does the specific metabolic rate change with decreasing
body mass? See margin.
15. MULTI-STEP PROBLEM Follow the steps below to find the relationship
between the number of pedal revolutions of a bicycle and the distance traveled.
a. The rear wheel of a bicycle has a diameter of 70 centimeters. Write the
function that describes the distance d traveled by the bicycle in terms of the
number w of rear-wheel revolutions. (Hint: When w = 1, the distance traveled
by the bicycle is equal to the circumference of the rear wheel.) d = ƒ(w) = 220w
b. The gear ratio of a bicycle is calculated by dividing the number of teeth in
the chainwheel by the number of teeth in the freewheel. The number w of rearwheel revolutions is equal to the product of the gear ratio and the number p of
pedal revolutions. A bicycle in first gear has 24 teeth in the chainwheel and 32
# of chainwheel teeth
}}
gear ratio (g) = }
teeth in the freewheel. Write the function that describes w in terms of p.
# of freewheel teeth
c. Use composition of functions to find the relationship between d and p.
so w(p) = gp = 0.75p
d = ƒ(w(p)) = 220gp
d. Shifting gears on a bicycle changes the gear ratio. Use the table below to find
how the distance traveled per pedal revolution changes as you shift gears. 1st gear: about 165 cm per pedal revolution
Gear
Number of teeth in chainwheel
Number of teeth in freewheel
5th
24
19
about 278 cm per pedal revolution
10th
40
22
about 400 cm per pedal revolution
15th
50
19
about 579 cm per pedal revolution
Distance traveled per pedal revolution increases as gear number increases; it becomes harder to pedal.
Chapter Standardized Test
461
461
Chapter Test
CHAPTER
8
ADDITIONAL RESOURCES
• Chapter 8 Resource Book
Chapter Test (3 levels) (p. 119)
SAT/ACT Chapter Test (p. 125)
Alternative Assessment (p. 126)
•
Test and Practice Generator
Graph the function. State the domain and range. 1–8. See margin.
冉冊
1 x
1. y = 2 6
2. y = 4x º 2 º 1
1
3. y = e x + 1
2
4. y = eº0.4x
5. y = log1/2 x
6. y = ln x º 4
7. y = log (x + 6)
2
8. y = 1 + 2eºx
1.
y
Simplify the expression.
º4e x º2
}}
10. 2e5x e4x
9. (2eº1)(3e2) 6e
11. e6 • e x • eº3x
12. log 10002 6
eº2x + 6
y2
13. 8log8 x x
1
3
Evaluate the expression without using a calculator.
14. log4 0.25 º1
15. log1/3 27 º3
17. ln eº2 º2
16. log 1 0
log 19 ºx5 ≈ º3.721
19. 12 = 10
+5
10
º7
eº2 ≈ 0.135
20. 5 º ln x = 7
21. log2 4x = log2 (x + 15) 5
x
4
22. = 3.3 0.617
1 + 2.5eº4x
x
1
domain: all real numbers;
range: y > 0
18. log3 2432
Solve the equation. Check for extraneous solutions.
1 x
6
()
2.
y
6
23. Tell whether the function ƒ(x) = 10(0.87) represents exponential growth or
y
exponential decay. exponential decay
4x 2 24. Find the inverse of the function y = log6 x. y = 6x
1
25. Use log2 5 ≈ 2.322 to approximate log2 50 and log2 0.4. 5.644; º1.322
1
26. Condense the expression 3 log4 14 º 3 log4 42. log4 }}
27
27. Expand the expression ln 2y2x. ln 2 + 2 ln y + ln x
x
2
2
domain: all real numbers;
range: y > –1
y
3.
28. Use the change-of-base formula to evaluate the expression log7 15. 1.392
29. Find an exponential function of the form y = ab x whose graph passes through
the points (4, 6) and (7, 10).
y = 3.036(1.186)x
1
30. Find a power function of the form y = ax b whose graph passes through the
1
points (2, 3) and (10, 21). y = 1.298x1.209
t
decreases by 10% per year. Write an exponential decay model for the value of
V = 24,900(0.90) ;
the car. After about how many years will the car be worth half its purchase price? after about 6.58 years
32.
33.
4.
EARNING INTEREST You deposit $4000 in an account that pays 7% annual
interest compounded continuously. Find the balance at the end of 5 years. $5676.27
y
y e0.4x
COD WEIGHT The table gives the mean weight w (in kilograms) and age x
(in years) of Atlantic cod from the Gulf of Maine.
x
1
2
3
4
5
6
7
8
w
0.751
1.079
1.702
2.198
3.438
4.347
7.071
11.518
x
1
domain: all real numbers;
range: y > 1
CAR DEPRECIATION The value of a new car purchased for $24,900
31.
y 2e x 1
1
1
1
1 x
domain: all real numbers;
range: y > 0
5.
y
a. Draw a scatter plot of ln w versus x. Is an exponential model a good fit for the
original data? yes; See margin for graph.
1
1
1
b. Find an exponential model for the original data. Estimate the weight of a cod
x
y log1/2 x
1
x
that is 9 years old. w = 0.509(1.460) ; 15.34 kg
Chapter Test
527
domain: x > 0;
range: all real numbers
6–8, 33a. See Additional Answers
beginning on page AA1.
527
Chapter Standardized Test
CHAPTER
ADDITIONAL RESOURCES
• Chapter 8 Resource Book
Chapter Test (3 levels) (p. 119)
SAT/ACT Chapter Test (p. 125)
Alternative Assessment (p. 126)
•
Test and Practice Generator
8
TEST-TAKING STRATEGY If you get stuck on a question, look at the answer choices for clues.
Or select an answer choice and check to see if it is a reasonable answer to the question.
1. MULTIPLE CHOICE Which function is graphed? E
2
equivalent to log2 7? D
y
(0, 1)
1
Qu. Standard
Qu. Standard
A
¡
B
¡
C
¡
D
¡
E
¡
x
ƒ(x) = 3(0.8)x
ƒ(x) = 2(0.8)x º 3
ƒ(x) = 2(0.8)x º 3
ƒ(x) = 4(0.8)x º 3
ƒ(x) = 4(0.8)x º 3
an investment account that pays 7% annual interest
compounded continuously. About how many years
will it take for your initial deposit to double? D
B
¡
E
¡
5
10
C
¡
7
9
14
of y = ln (x º 2)? A
B
¡
D
¡
x
y=e +2
x
y=e º2
y=e
x+2
y = ex º 2
y = eº2 x
y
(⫺3, 0)
1
¡
B
¡
C
¡
D
¡
E
¡
A
528
528
B
¡
27
C
¡
D
¡
log 7
ᎏᎏ
log 2
E
¡
lo g 2
ᎏᎏ
log 7
7 log 2
A
¡
B
¡
C
¡
D
¡
E
¡
log x + 2 log y + log z
log x + 2 log y º log z
log z º log x º 2 log y
2 log xy º log z
log z º 2 log xy
7. MULTIPLE CHOICE What is the solution of the
equation 2x + 14 = 16 2 x? A
A
¡
D
¡
B
¡
E
¡
2
16
C
¡
4
8
No solution
equation 0.5 log3 x = 2? C
A
¡
4
B
¡
64
D
¡
1
ᎏᎏ
64
E
¡
1
ᎏᎏ
81
C
¡
81
9. MULTIPLE CHOICE Which function does not have a
4. MULTIPLE CHOICE Which function is graphed? D
2
72
8. MULTIPLE CHOICE What is the solution of the
3. MULTIPLE CHOICE Which function is the inverse
A
¡
C
¡
E
¡
A
¡
6. MULTIPLE CHOICE Which of the following is
x y2
equivalent to log ᎏ
ᎏ? B
z
2. MULTIPLE CHOICE Suppose you deposit money in
A
¡
D
¡
5. MULTIPLE CHOICE Which of the following is
x
graph with asymptote y = 0? C
A
¡
C
¡
E
¡
ƒ(x) = 3x
ƒ(x) = log 6x
B
¡
D
¡
ƒ(x) = (0.25)x + 2
ƒ(x) = eº5x
1
1+e
ƒ(x) = ᎏ
º2x
10. MULTIPLE CHOICE What type of function is
ƒ(x) = log 4x
ƒ(x) = 4 + log x
ƒ(x) = º4 + log x
ƒ(x) = log (x + 4)
ƒ(x) = log (x º 4)
ƒ(x) = 4e0.5x? B
A
¡
B
¡
C
¡
D
¡
E
¡
Chapter 8 Exponential and Logarithmic Functions
Exponential decay function
Exponential growth function
Logarithmic function
Logistic growth function
Power function
QUANTITATIVE COMPARISON In Exercises 11 and 12, choose the statement
that is true about the given quantities.
Job Offers
The quantity in column A is greater.
Salary ($)
A
¡
B
¡
C
¡
D
¡
13c.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
Column B
11.
log 10,000
ln e4
C
12.
log2 4
log4 2
A
14a.
y
40,000
37,500
35,000
32,500
30,000
27,500
25,000
0
y 29,500(1.04)t
y 550t 32,000
(3.6, 33,984)
0
1
2
3 4
5 6 7
Years
8
9 10 t
ln w
13. MULTI-STEP PROBLEM You are considering two job offers. The first offer is a
salary of $32,000 with a $550 annual raise. The other offer is a salary of $29,500
with a 4% annual raise.
1
a. Write a linear model for the total salary with the first offer as a function of the
8
number t of years. y = 550t + 32,000
b.
t
ln w
b. Write an exponential model for the total salary with the other offer as a
function of the number t of years. y = 29,500(1.04)t
c. Graph the functions in the same coordinate plane with domain 0 ≤ t ≤ 8.
The intersection point (3.6, 33,984)
Find the point of intersection of the two graphs and tell what it represents. represents when the salaries are the
same.
d. Writing Explain the difference in the salaries over time.
Sample answer: After about 4 years, the salary from the second offer becomes greater than the first.
14.
MULTI-STEP PROBLEM The table gives the weight w (in pounds) of an average girl
for the first five years of life where t is her age in months. 䉴 Source: Your Baby & Child
t
2
4
6
8
10
12
24
36
48
60
w
10.5
13.5
16.0
18.5
20.0
21.5
27.5
31.5
35.5
39.0
1
15b.
ln t
y
y
1
14c. A power model is a better fit
a. Draw a scatter plot of ln w versus t. See margin.
for the original data. The
b. Draw a scatter plot of ln w versus ln t. See margin.
points in the graph of
(t, ln w) lie on a curve,
c. Analyze your scatter plots and decide whether an exponential model or a
while those in the graph of
power model is a better fit for the original data. Explain your choice.
(ln t, ln w) lie close to a
straight
line.
d. Using your answer from part (c), find a model for the data. Check your model
15c. asymptotes: x-axis and
by using the regression feature of a graphing calculator. y = 8.118x 0.383
1
1 1 1
1
y = 5; y-intercept: }};
e. Use your model to estimate a girl’s weight at 1, 2, 3, and 4 years old.
2
2 2 2
2
24.6 lb; 29.9 lb; 34.0 lb; 37.4 lb
pt. of max. growth: (2.197, 2.5)
5
15. MULTI-STEP PROBLEM Use the function ƒ(x) = .
e. Sample answer: At first the
1 + 9eºx
function grows very slowly,
a. Find ƒ(º1), ƒ(0), and ƒ(2). 0.196; 0.5; 2.254
then it grows faster and
faster until the growth rate
b. Sketch a graph of the function. See margin.
reaches its peak at the point
c. Identify the asymptotes, y-intercept, and point of maximum growth.
(ln 9, 2.5). After this, the
growth rate drops off until
d. Write and solve an equation to find the value of x when ƒ(x) equals 4. Label
the curve again becomes
this point on your graph. ln 36 ≈ 3.584
almost flat, approaching
e. Writing Describe how the growth represented by this function changes over time. y = 5 asymptotically.
Chapter Standardized Test
1
1
1
1
5
1 9ex
x
529
529
Chapter Test
CHAPTER
9
ADDITIONAL RESOURCES
• Chapter 9 Resource Book
Chapter Test (3 levels) (p. 93)
SAT/ACT Chapter Test (p. 99)
Alternative Assessment (p. 100)
•
Test and Practice Generator
The variables x and y vary inversely. Use the given values to write an equation
relating x and y. Then find y when x = 3.
1
2
5 5
8 8
2. x = ᎏᎏ, y = 5 y = }}; }} 3. x = 12, y = ᎏᎏ y = }}; }}
1. x = º4, y = 9 36
2x 6
x 3
2
3
y = º}}; º12
x
The variable z varies jointly with x and y. Use the given values to write an
equation relating x, y, and z. Then find z when x = º2 and y = 4.
4. x = 6, y = º1
6
y = º}}; º2
x
10.
y
2
x
2
5
1
3
7. x = ᎏᎏ, y = ᎏᎏ, z = ᎏᎏ
2
3
4
6. x = º3, y = 2, z = 18
5. x = 5, y = 4, z = 2 1
4
z = }}xy ; º}}
z = º3xy ; 24
10
5
Graph the function. 8–15. See margin.
º1
8. y = ᎏᎏ º 2
x+1
6
12. y = ᎏ
x2 + 4
z = 10xy ; º80
11.
4
9. y = ᎏᎏ
xº2
x
10. y = ᎏᎏ
2x + 5
4x º 3
11. y = ᎏᎏ
xº4
º3x 2
13. y = ᎏ
2x º 1
x2 º 2
14. y = ᎏ
x2 º 9
x 2 º 2x + 15
15. y = ᎏᎏ
x+1
⫺5
⫺5
Perform the indicated operation. Simplify the result.
x2 º 4 x2 + 4x + 3 (x + 2)(x + 1)
3x º 6
4x º 8
ᎏ ÷ ᎏᎏ
16. ᎏᎏ • ᎏᎏ }} 17. ᎏ
2
x+3
2x º 4
xº1
x2 º 3 x + 2
5
7
19. ᎏᎏ + ᎏᎏ
6x
18x
4
}
3(x º 2)
xº1
xº4
6x º 9
20. ᎏᎏ º ᎏᎏ }}
xº2
x + 1 (x º 2)(x + 1)
11
}}
9x
(x + 4)(x º 2)
x+4
ᎏ • (x 2 + 3x º 10) }}
18. ᎏ
xº5
x2 º 2 5
3x
5
ᎏ + ᎏᎏ
21. ᎏ
xº3
x2 º 10x + 21
12.
8x º 35
}}
(x º 3)(x º 7)
3
x
x(x + 3)
}
2x 2 º 5
4+x
ᎏᎏ
10
23. ᎏ
x2 º 16
ᎏᎏ
8
4
}
5(x º 4)
2
ᎏᎏ + 5
xº1
24. ᎏᎏ
x
ᎏᎏ
3
3(5x º 3)
}
x(x º 1)
36
25. ᎏ
7
1
ᎏᎏ + ᎏᎏ
2x
x
8x
10
13.
x
2
x
y
⫺2
⫺2
4x
37
ᎏ º 3 16
29. ᎏᎏ = ᎏ
x+3
x2 º 9
º}}, 4
7
30. SCIENCE
140
lb
14.
112
lb
y
2
6 ft
CONNECTION A sphere with radius r is inscribed in a cube as shown.
Find the ratio of the volume of the cube to the volume of the sphere. Write your
answer in simplified form. }6}
⫺4
?
31. GEOMETRY
π
32.
1
⫺1
⫺1
2
8
5
º15
x + 16
27. ᎏᎏ = ᎏᎏ º10, º6 28. ᎏᎏ = ᎏᎏ 13
x
4
x+3
xº3
CONNECTION A lever pivots on a support called a
fulcrum. For a balanced lever, the distance d (in feet) an
object is from the fulcrum varies inversely with the object’s
weight w (in pounds). An object weighing 140 pounds is
placed 6 feet from a fulcrum. How far from the fulcrum
must a 112 pound object be placed to balance the lever? 7.5 ft
x
y
Solve the equation using any method. Check each solution.
9
11
31
26. ᎏᎏ + ᎏᎏ = ᎏᎏ
x
5
x
10
1
Simplify the complex fraction.
1 + ᎏᎏ
x
22. ᎏ
5
2 º ᎏᎏ2
y
15.
r
4
⫺2
x
y
5
⫺5
⫺5
STARTING A BUSINESS You start a small bee-keeping business, spending
5
x
$500 for equipment and bees. You figure it will cost $1.25 per pound to collect,
clean, bottle, and label the honey. How many pounds of honey must you produce
before your average cost per pound is $1.79? about 920 lb
Chapter Test
579
579
CHAPTER
ADDITIONAL RESOURCES
• Chapter 9 Resource Book
Chapter Test (3 levels) (p. 93)
SAT/ACT Chapter Test (p. 99)
Alternative Assessment (p. 100)
•
Test and Practice Generator
9
Chapter Standardized Test
TEST-TAKING STRATEGY During a test, draw graphs and figures in your test booklet to help
you solve problems. Even though you must keep your answer sheet neat, you can make any
kind of mark you want in your test booklet.
1. MULTIPLE CHOICE The variable x varies inversely
6. MULTIPLE CHOICE Which function is graphed? A
with y. When x = 6, y = 6.5. Which equation relates
x and y? A
A
¡
D
¡
xy = 39
1
2
y = ᎏᎏx
B
¡
E
¡
xy = 11.5
C
¡
y
1
2
xy = ᎏᎏ
y = 39x
2
2. MULTIPLE CHOICE The variable z varies jointly
1
with x and y. When x = 6 and y = ᎏᎏ, z = 30.
3
Qu. Standard
Qu. Standard
Which equation relates x, y, and z? C
A
¡
D
¡
z = 30xy
1
z = ᎏᎏxy
30
B
¡
E
¡
30 = xyz
C
¡
z = 15xy
1
z = ᎏᎏxy
15
3. MULTIPLE CHOICE Which function is graphed?
B
y
2
y = ᎏᎏ º 3
10
x+5
B
¡
y = ᎏᎏ º 3
C
¡
y = ᎏᎏ + 3
10
x+5
D
¡
y = ᎏᎏ + 3
E
¡
y = ᎏᎏ
10
xº5
10
xº5
10
xº5
4. MULTIPLE CHOICE What is the quotient
x2 º 9x º 22
ᎏ? A
(x + 2) ÷ ᎏ
x2 º 121
A
¡
x + 11
B
¡
x + 11
ᎏᎏ
x+2
D
¡
x+2
ᎏᎏ
x º 11
E
¡
x+2
C
¡
x+2
ᎏᎏ
x + 11
5. MULTIPLE CHOICE What are all the solutions of
º10
x
the equation ᎏᎏ = ᎏᎏ ? E
xº9
2
¡
D
¡
580
º4, º5
5
¡
E
¡
B
º3x 2
x º 16
y=ᎏ
2
B
¡
y=ᎏ
2
C
¡
y=ᎏ
2
º3x 2
x º 25
º3x 2
y=ᎏ
x 2 + 25
D
¡
y=ᎏ
2
3x 2
x º 25
7. MULTIPLE CHOICE What is the difference
8x º 3
7
ºᎏ
? D
ᎏᎏ
x 2 + 2x º 35
x 2 º 25
A
¡
A
ºx 2
x º 25
A
¡
E
¡
x
2
580
x
2
4, º5
¡
C
4
4, 5
Chapter 9 Rational Equations and Functions
A
¡
2(4x 2 + 15x º 17)
ᎏᎏᎏ
(x 2 + 2x º 35)(x + 5)
B
¡
2(4x 2 + 15x + 32)
ᎏᎏᎏ
(x 2 + 2x º 35)(x + 5)
C
¡
2(4x 2 + 15x + 17)
ᎏᎏᎏ
(x 2 + 2x º 35)(x + 5)
D
¡
2(4x 2 + 15x º 32)
ᎏᎏᎏ
(x 2 + 2x º 35)(x + 5)
E
¡
2(4x 2 + 15x º 32)
ᎏᎏᎏ
2
(x + 2x º 35)(x 2 º 25)
8. MULTIPLE CHOICE What is the simplified form of
the following complex fraction? B
10
ᎏᎏ
x+1
ᎏᎏ
3
1
ᎏᎏ + ᎏᎏ
x+1
2
A
¡
20x
ᎏᎏ
x+7
B
¡
20
ᎏᎏ
x+7
D
¡
10(x + 7)
ᎏᎏ
x+1
E
¡
20
C
¡
10
ᎏᎏ
x+7
9. QUANTITATIVE COMPARISON Choose the statement that is true about the given
10a.
quantities. B
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
y
2
The quantity in column B is greater.
x
2
The two quantities are equal.
The relationship cannot be determined from the given information.
b.
Column A
y
Column B
xº4
x+1
7
2
The solution of ᎏᎏ = ᎏᎏ
5
x
8
3
1
12x
The solution of ᎏᎏ º ᎏᎏ = ᎏᎏ
⫺2
⫺2
10. MULTI-STEP PROBLEM For parts (a)–(d), graph the function and identify the
x
2
point at which the horizontal and vertical asymptotes intersect. 10a–d. See margin.
2
a. y = ᎏᎏ
x
(0, 0)
2
2
2
b. y = ᎏᎏ + 3 (1, 3) c. y = ᎏᎏ º 3 (1, º3) d. y = ᎏᎏ + 3
xº1
xº1
x+1
c.
(º1, 3)
y
2
e. Use your answers to parts (a)–(d) to predict the point of intersection of the
2
asymptotes of the graph of y = ᎏᎏ º 3. Check your prediction by graphing. (º1, º3)
x+1
⫺2
⫺2
x
f. CRITICAL THINKING Generalize your results for any function of the form
a
y = ᎏᎏ + k. The asymptotes will intersect at (h, k).
xºh
d.
11. MULTI-STEP PROBLEM Three tennis balls fit tightly in a can as shown. Recall
y
that the formula for the volume of a cylinder is V = πr 2h and the formula for the
4
3
volume of a sphere is V = ᎏᎏπr 3.
2
a. Write an expression for the height of the can, h, in terms of r. Rewrite the
⫺2
formula for the volume of a cylinder with r as the only variable. h = 6r ; V = 6πr 3
2
b. Find the ratio of the volume of the three tennis balls to the volume of the can. }3}
c.
Writing Do you think using a cylindrical can is an efficient way of
12d.
r
x
2
5⫹1
2
packaging tennis balls? Explain your reasoning. See margin.
12. MULTI-STEP PROBLEM The length l and width w of a golden rectangle satisfy
l
l
l+w
the equation ᎏᎏ = ᎏᎏ. The ratio ᎏᎏ is called the golden ratio. For centuries,
l
w
w
11c. Sample answer: A cylinder
is a fairly efficient way to
1
package tennis balls. Only }}
3
of the space inside the
golden rectangles have been known to be very pleasing to the human eye.
cylinder is wasted.
a. Rewrite the right side of the equation as a complex fraction by dividing each }l} + 1
term of the numerator and denominator by w.
5
1
1
2
w
ᎏ
l
l
b. Let g represent the golden ratio, so g = ᎏᎏ. Substitute g for each occurrence of
w
l
ᎏᎏ in the equation from part (a) and simplify the equation. g 2 º g º 1 = 0
w
}}
w
c. Solve the equation from part (b) for g. (Hint: Use the quadratic formula.)
1 + 兹5苶
Write an exact value and an approximate value for the golden ratio. }
≈ 1.618
2
d. GEOMETRY CONNECTION Use a ruler or graph paper to draw an accurate
golden rectangle of any size. Label the dimensions of your rectangle. See margin.
Chapter Standardized Test
581
581