Download Chapter 4.8 Answers - Aspen High School - Algebra I

GUIDED PRACTICE
✓
Concept Check ✓
Skill Check ✓
Vocabulary Check
1. Yes; no; every function
is a set of ordered pairs
and, so, is a relation. A
relation is a function
only if for each input
there is exactly one
output.
3 APPLY
1. Is every function a relation? Is every relation a function? Explain. See margin.
2. Describe a line that cannot be the graph of a linear function. a vertical line
BASIC
Evaluate the function ƒ(x) = 3x º 10 for the given value of x.
3. x = 0 –10
4. x = 20 50
Day 1: pp. 259–260 Exs. 11–19,
20–28 even, 29–31,
32–48 even
Day 2: pp. 260–262 Exs. 33–49
odd, 50–52, 56–58, 65, 70,
72–79, Quiz 3 Exs. 1–19
2
6. x = ᎏᎏ º8
3
5. x = º2 º16
In Exercises 7–9, decide whether the relation is a function. If it is a function,
give the domain and the range.
7. Input Output
10.
10
100
20
200
30
300
40
400
50
500
8. Input Output
yes; 10, 20,
30, 40, 50;
100, 200, 300,
400, 500
9.
5
3
y
AVERAGE
15
no
4
5
3
1
6
3
5 x
2
no
Extra Practice
to help you master
skills is on p. 800.
Day 1: pp. 259–260 Exs. 11–19,
20–28 even, 29–31,
32–48 even
Day 2: pp. 260–262 Exs. 33–49
odd, 53–55, 59–62, 65, 70,
72–79, Quiz 3 Exs. 1–19
BLOCK SCHEDULE
pp. 259–262 Exs. 11–19, 20–28
even, 29–31, 32–49, 50–53, 56–58,
65, 70, 72–79, Quiz 3 Exs. 1–19
GRAPHICAL REASONING Decide whether the graph represents y as a
function of x. Explain your reasoning. 11º13. See margin.
y
11.
Day 1: pp. 259–260 Exs. 11–19,
20–28 even, 29–31,
32–48 even
Day 2: pp. 260–262 Exs. 33–49
odd, 50–53, 56–58, 65, 70,
72–79, Quiz 3 Exs. 1–19
ADVANCED
CAR TRAVEL Your average speed during a trip is 40 miles per hour.
Write a linear function that models the distance you travel d(t) as a function
of t, the time spent traveling. d(t) = 40t
PRACTICE AND APPLICATIONS
STUDENT HELP
ASSIGNMENT GUIDE
y
12.
y
13.
x
12. No; there are vertical
lines that pass through
two points on the
graph.
13. No; there are vertical
lines that pass through
two or more points on
the graph.
STUDENT HELP
HOMEWORK HELP
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 11–19
Exs. 20–28
Exs. 29–40
Exs. 50–52
EXERCISE LEVELS
Level A: Easier
11–13, 29–31, 63–66
Level B: More Difficult
14–28, 32–40, 42–52, 56–58, 67–79
Level C: Most Difficult
41, 53–55, 59–62
x
x
11. Yes; no vertical line
passes through two
points on the graph.
RELATIONS AND FUNCTIONS In Exercises 14–19, decide whether the
relation is a function. If it is a function, give the domain and the range.
14. Input Output yes; 1, 2,
3, 4; 5, 4,
5
1
3, 2
17.
2
4
3
3
4
2
15. Input Output yes; 1, 2,
3, 4; 0
1
2
0
3
yes; 0, 1, 2,
3; 2, 4, 6, 8
Input
Output
0
1
2
3
2
4
6
8
16. Input Output no
7
7
9
9
8
10
4
10
no
18.
Input
Output
0
2
4
2
1
2
3
4
19.
yes; 1, 3, 5,
7; 1, 2, 3
Input
Output
1
3
5
7
1
2
3
1
4.8 Functions and Relations
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 12, 16, 18, 22, 32, 29, 42, 50.
See also the Daily Homework Quiz:
• Blackline Master (Chapter 5
Resource Book, p. 11)
•
Transparency (p. 37)
259
259
EVALUATING FUNCTIONS Evaluate the function when x = 2, x = 0, and
x = º3.
STUDENT HELP NOTES
Homework Help Students can
find help for Exs. 42–49 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
20. ƒ(x) = 10x + 1
21. g(x) = 8x º 2
22. h(x) = 3x + 6
21; 1; º29
14; º2; º26
12; 6; º3
23. g(x) = 1.25x
24. h(x) = 0.75x + 8
25. ƒ(x) = 0.33x º 2
2.5; 0; º3.75
9.5; 8; 5.75
º1.34; º2; º2.99
3
2
2
26. h(x) = x º 4
27. g(x) = x + 7
28. ƒ(x) = x + 4
4
5
74
1
º2.5; º4; º6.25
7.8; 7; 5.8
4 }}; 4; 3 }}
7
7
GRAPHICAL REASONING Match the function with its graph.
1
A. ƒ(x) = 3x º 2
B. ƒ(x) = 2x + 2
C. ƒ(x) = x º 2
2
APPLICATION NOTE
EXERCISES 50–52 Although the
relationships between years and
enrollments represent functions,
they cannot be modeled by simple
linear functions. Each function
could be modeled by the union of a
set of linear functions over
restricted domains.
5 x
3
41. The points (1, 2) and
(3, º1) are on the
graph, so the slope
–1 – 2
3–1
5
INT
1
1
1
1
34.
3
5 x
NE
ER T
2
2
2
6
2
6
1
1
1
1
36. h(x) = ºx + 2
3
1
37. ƒ(x) = ºx + 1
2
38. ƒ(x) = 4x + 1
39. h(x) = 5
40. g(x) = º3x º 2
41.
Writing Describe how to find the slope of the graph of linear function ƒ
given that ƒ(1) = 2 and ƒ(3) = º1. See margin.
FINDING SLOPE Find the slope of the graph of the linear function f.
42. ƒ(2) = º3, ƒ(º2) = 5 º2
43. ƒ(0) = 4, ƒ(4) = 0 º1
44. ƒ(º3) = º9, ƒ(3) = 9 3
45. ƒ(6) = º1, ƒ(3) = 8 º3
46. ƒ(9) = º1, ƒ(º1) = 2 º0.3
47. ƒ(º2) = º1, ƒ(2) = 6 1.75
48. ƒ(2) = 2, ƒ(3) = 3 1
49. ƒ(º1) = 2, ƒ(3) = 2 0
number of high school students
in year x. Estimate ƒ(2005).
about 16 million
52. Let g(x) represent the projected
y
5
number of college students in
year x. Estimate g(2005).
3 y 1x 2
3
about 17 million
1
1
1
3
x
260
37–40, 59. See Additional Answers
beginning on page AA1.
260
Student Enrollment
51. Let ƒ(x) represent the projected
y 2x 3
36.
B
35. g(x) = 2x º 3
a function of the year? Explain.
3 x
1
x
A
50. Is the projected high school
50. Yes; yes; in both cases,
each input is paired with
enrollment a function of the year?
exactly one output.
Is the projected college enrollment
y
1
1
HIGH SCHOOL AND COLLEGE STUDENTS In Exercises 50–52, use the
graph below. It shows the projected number of high school and college
students in the United States for different years. Let x be the number of
years since 1980. 51–52. Estimates may vary.
x
y 5x 6
35.
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 42–49.
y
3
3
2
STUDENT HELP
y x 4
3
x
GRAPHING FUNCTIONS In Exercises 32–40, graph the function.
32–40. See margin.
33. g(x) = ºx + 4
34. h(x) = 5x º 6
32. ƒ(x) = º2x + 5
is }} = º}}.
y
1
Enrollment (millions)
33.
1
1
1
x
C
y 2x 5
1
y
1
1
5
1
1
31.
y
1
y
3
30.
y
Chapter 4 Graphing Linear Equations and Functions
INT
32.
29.
NE
ER T
18
16
14
12
High school
College
10
0
0
5
10
15
20
Years since 1980
25
x
DATA UPDATE of National Center for Education
Statistics data at www.mcdougallittell.com
53.
ZYDECO MUSIC The graph
shows the number of people who
attended the Southwest Louisiana
Zydeco Music Festival for different
years, where k is the number of years
since 1980. Is the number of people
who attended the festival a function
of the year? Explain.
Music Festival
Attendance
FOCUS ON
APPLICATIONS
APPLICATION NOTE
EXERCISE 53 Students should
recognize that the data do not
represent a linear function,
although a linear function can be
used for modeling purposes.
Additional information about
Zydeco Music is available at
www.mcdougallittell.com.
9000
6000
3000
0
䉴 Source: Louisiana Zydeco Music Festival
0
5 10 15 k
Years since 1980
Yes; each input is paired with exactly
one output.
54.
MASTERS TOURNAMENT The table shows information about the top
RE
FE
L
AL I
7 winners of the 1997 Masters Tournament in Augusta, Georgia. Graph the
relation. Is the money earned a function of the score? Explain. 䉴 Source: Golfweb
ZYDECO MUSIC
blends elements
from a variety of cultures.
The accordion has German
origins. The rub board was
invented by Louisiana
natives whose roots go back
to France and Canada.
Test
Preparation
Check graphs; yes; each input is paired with exactly one output.
Score
Prize ($)
55.
282
283
284
285
285
486,000 291,600 183,600 129,600 102,600 102,600
286
78,570
SCIENCE CONNECTION It takes 4.25 years for starlight to travel 25 trillion
miles. Let t be the number of years and let ƒ(t) be trillions of miles traveled.
Write a linear function ƒ(t) that expresses the distance traveled as a function
100
}t, or f(t) ≈ 5.88t
of time. f(t) = }
17
QUANTITATIVE COMPARISON In Exercises 56–58, choose the statement
that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
1
60. If f(x) = }}, there is no
x
output value corresponding to the input value 0;
1
if f(x) = }}, there is
x+3
no output value
corresponding to the
input value –3.
56.
61. Let the domain of f(x) =
1
}} be all real numbers
57.
x
except 0; let the domain
1
58.
of f(x) = }} be all
x+3
real numbers except º3.
★ Challenge
270
The number in column A is greater.
The number in column B is greater.
The two numbers are equal.
The relationship cannot be determined from the given information.
Column A
Column B
ƒ(x) = º7x + 4 when x = º4
ƒ(x) = º7x + 4 when x = º5
B
1
ƒ(x) = ᎏᎏx º 3 when x = 7
2
1
ƒ(x) = ᎏᎏx º 5 when x = 11
2
C
1
7
2
7
g(x) = 6x º ᎏᎏ when x = ᎏᎏ
1
7
2
7
h(t) = 6t º ᎏᎏ when t = ᎏᎏ
C
RESTRICTING DOMAINS In Exercises 59–62, use the equations
1
1
y = }} and y = }}. 60–62. See margin.
x+3
x
62. For x < 0 but close to 0,
the graph falls sharply.
For x > 0 but close to 0,
the graph rises sharply.
EXTRA CHALLENGE
www.mcdougallittell.com
ADDITIONAL PRACTICE
AND RETEACHING
59. Use a graphing calculator or computer to graph both equations.
See margin for graph.
60. Explain why each equation is not a function for all real values of x.
For Lesson 4.8:
61. How can you restrict the domain of each equation so that it is a function?
1
62. Describe what happens to the graph of y = ᎏᎏ near the value(s) of x for which
x
the graph is not a function.
4.8 Functions and Relations
261
• Practice Levels A, B, and C
(Chapter 4 Resource Book, p. 112)
• Reteaching with Practice
(Chapter 4 Resource Book, p. 115)
•
See Lesson 4.8 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
261
4 ASSESS
DAILY HOMEWORK QUIZ
Transparency Available
1. Does the graph represent a
function? Explain your answer.
y
3
⫺1
⫺1
冋
冋
9
63.
12
64.
–11
7
冋
冋
1
3 x
⫺3
Yes; no vertical line passes
through two points of the graph.
3
66. º6
º5
MATRICES Find the sum or the difference of the matrices. (Review 2.4)
63–66.
4 º8
º5
3 See margin.
º6.5 º4.2
2.4 º5.1
63.
º
64.
+
册
册
º4.1
4.3
冋
冋
º9.3
0.7
9.8
65. º12.3
12.4
1
⫺3
MIXED REVIEW
º17.9
º0.1
º13.2
4
º3
5
2. Evaluate ƒ(x) = –3x + 2 when
x = –1, 0, and 3. 5, 2, –7
3. Find the slope of the graph
of the linear function ƒ if
ƒ(3) = –2 and ƒ(0) = 1. –1
册
册
1
º3
0
7
0
6.2
65. º2.5
3.4
册 冋
º5
º7
册冋
册
冋
º12
º3.6
5.9
º4.4 º
9.8 º4.3
º5.8
º9
7.4
0
册 冋
9
66. º4
º5
3.7
1
º7
0
册 冋
4.3
º3
册冋
6
º6
1 + º2
º1
0
册
3
4
5
º5
º4
1
册
SOLVING EQUATIONS Solve the equation if possible. (Review 3.4)
67. 4x + 8 = 24 4
68. 3n = 5n º 12 6
70. º5y + 6 = 4y + 3
1
}} 71. 3b + 8 = 9b º 7
3
69. 9 º 5z = º8z º3
1
2}}
2
72. º7q º 13 = 4 º 7q
no solution
GRAPHING LINES In Exercises 73–78, write the equation in slope-intercept
form. Then graph the equation. (Review 4.6 for 5.1) 73–78. See margin.
73. 2x º y + 3 = 0
74. x + 2y º 6 = 0
75. y º 2x = º7
76. 5x º y = 4
77. x º 2y + 4 = 2
78. 4y + 12 = 0
79.
1
CHARITY WALK You start 5 kilometers from the finish line and walk 1ᎏᎏ
2
kilometers per hour for 2 hours, where d is your distance from the finish line.
Graph the situation. (Review 4.6 for 5.1) See margin.
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 4.8 are available in
blackline format in the Chapter 4
Resource Book, p. 119 and at
www.mcdougallittell.com.
QUIZ 3
Self-Test for Lessons 4.7 and 4.8
Solve the equation graphically. Check your solution algebraically. (Lesson 4.7)
1
2
2. 6x º 12 = º9 }}
1. 4x + 3 = º5 º2
1
2
3
3
5. ᎏᎏx + 5 = ºᎏᎏx º 8 º13 6. ᎏᎏx + 2 = ºᎏᎏx º 6
3
3
4
4
1
º5}}
3
Evaluate the function when x = 3, x = 0, and x = º4. (Lesson 4.8)
1
2
4. º5x º 4 = 3x º}}
ADDITIONAL TEST
PREPARATION
1. WRITING Explain the relationship between a relation and a
function. A function is a relation
7. h(x) = 5x º 9
8. g(x) = º4x + 3
9. ƒ(x) = 1.75x º 2
3.25; º2; º9
6; º9; º29
º9; 3; 19
2
1
4
3
10. h(x) = º1.4x
11. ƒ(x) = ᎏᎏx + 9 9}4}; 9; 8 12. g(x) = ᎏᎏx + ᎏᎏ
4
7 2 7
º4.2; 0; 5.6
2; }}; º2
7
In Exercises 13–18, graph the function. (Lesson 4.8)
13–18. See margin.
13. ƒ(x) = º5x
14. h(x) = 4x º 7
15. ƒ(x) = 3x º 2
in which each value in the domain
is associated with exactly one
value in the range.
2. OPEN ENDED Give an example of a linear function that models a real-world problem. Explain
the significance of one point on
the graph of the function.
2
16. h(x) = ᎏᎏx º 1
5
See sample answer at right.
19.
73–79. See Additional Answers
beginning on page AA1.
262
1
1
17. ƒ(x) = ᎏᎏx + ᎏᎏ
4
2
18. g(x) = º6x + 5
LUNCH BOX BUSINESS You have a small business making and
delivering box lunches. You calculate your average weekly cost y of
producing x lunches using the function y = 2.1x + 75. Last week your cost
was $600. How many lunches did you make last week? Solve algebraically
and graphically. (Lesson 4.7) 250 lunches
Chapter 4 Graphing Linear Equations and Functions
Additional Test Preparation Sample answer:
2. I send my film away for processing. The company
charges me $4.10 for shipping and $3.75 for processing
each roll of film. The equation y = 3.75x + 4.10, where
262
3. 8x º 7 = x 1
x = the number of rolls of film that are processed and
y = the total cost, models the situation; the point (4, 19.1)
represents the cost ($19.10) of processing 4 rolls of film.