GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
3 APPLY
1. Explain what it means for x and y to vary directly.
It means that there is a nonzero number k such that y = kx.
2. In a direct variation equation, how are the constant of variation and the
slope related? The slope is the constant of variation.
ASSIGNMENT GUIDE
BASIC
Graph the equation. State whether the two quantities have direct
variation. If they have direct variation, find the constant of variation
and the slope of the direct variation model. 3–8. Check graphs.
1
3. y = x
4. y = 4x
5. y = ᎏᎏx
2
1 1
direct variation. 1; 1
direct variation. 4; 4
direct variation. }}; }}
2 2
6. y = 2x
7. y = x º 4
8. y º 0.1x = 0
direct variation. 2; 2
not direct variation.
direct variation. 0.1; 0.1
SALARY In Exercises 9–11, you
Total pay, p
$18 $42 $48 $30
work a different number of hours
each day. The table shows your
Hours worked, h
3
7
8
5
total pay p and the number of
Ratio
?
?
?
?
hours h you worked.
9. Copy and complete the table by finding the ratio of your total pay each day to
the number of hours you worked that day. 6; 6; 6; 6
Day 1: pp. 237–239 Exs. 12–20
even, 21–31, 33, 38–41,
44–62 even
AVERAGE
Day 1: pp. 237–239 Exs. 12–20
even, 21–31, 33, 38–41,
44–62 even
ADVANCED
Day 1: pp. 237–239 Exs. 12–20
even, 21–31, 33, 36–43,
44–62 even
BLOCK SCHEDULE WITH 4.6
pp. 237–239 Exs. 12–20 even,
21–31, 33, 38–41, 44–62 even
10. Write a model that relates the variables p and h. p = 6h
EXERCISE LEVELS
Level A: Easier
12–14
Level B: More Difficult
15–31, 33, 36–41, 44–63
Level C: Most Difficult
32, 34, 35, 42, 43
11. If you work 6 hours on the fifth day, what will your total pay be? $36
PRACTICE AND APPLICATIONS
STUDENT HELP
DIRECT VARIATION MODEL Find the constant of variation and the slope.
Extra Practice
to help you master
skills is on p. 800.
12. y = 3x 3; 3
2
2
2
13. y = ºᎏᎏx º}5}; º}5}
5
y
y
3
1
1
⫺1
⫺1
⫺1
14. y = 0.75x 0.75; 0.75
1
3 x
y
3
x
3
1
⫺3
1
3
5 x
CONSTANT OF VARIATION Graph the equation. Find the constant of
variation and the slope of the direct variation model. 15–20. Check graphs.
15. y = º3x º3; º3
18. y = ºx º1; º1
STUDENT HELP
HOMEWORK HELP
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 12–20
Exs. 23–31
Exs. 21–22, 32
Exs. 36–37
16. y = º5x º5; º5
5
5 5
19. y = ᎏᎏx }}; }}
4 4
4
17. y = 0.4x 0.4; 0.4
1
1
1
20. y = ºᎏᎏx º}}; º}}
5
5
5
RECOGNIZING DIRECT VARIATION In Exercises 21 and 22, state whether the
two quantities have direct variation.
21.
BICYCLING You ride your bike at an average speed of 14 miles per hour.
The number of miles m you ride during h hours is modeled by m = 14h. yes
22.
The circumference C of a circle and its diameter d
are related by the equation C = πd. yes
GEOMETRY
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 12, 16, 21, 24, 32. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 4
Resource Book, p. 78)
•
Transparency (p. 33)
MATHEMATICAL REASONING
Marlene’s salary varies directly
with the number of hours she
works. If Marlene’s salary is
graphed against the number of
hours she works, what does the
slope of the line represent?
her hourly wage
CONNECTION
4.5 Direct Variation
237
237
FOCUS ON
PEOPLE
APPLICATION NOTE
EXERCISE 33 Since
23. x = 4, y = 12 y = 3x
weight on Earth
weight on moon
k = ᎏᎏ is the ratio that
Homework Help Students can
find help for Exs. 34–35 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
APPLICATION NOTE
EXERCISES 36–37 A direct
variation function has many uses as
a mathematical model. In these
exercises, the direct variation function only approximates the total
body length of the violins. This is
evident because a constant of proportionality can only be approximated. By comparison, the direct
variation function in Exercises 34–35
represents the exact relationship
between the two variables.
RE
L
AL I
minute varies directly with your pulse rate p. Each time your heart beats, it
pumps approximately 0.06 liter of blood.
VALENTINA V.
TERESHKOVA
a. Find an equation that relates V and p. V = 0.06p
In 1963 the Soviet astronaut
Valentina Tereshkova was
the first woman to travel
in space.
b. COLLECTING DATA Take your pulse and find out how much blood your
heart pumps per minute. Answers may vary.
33.
ASTRONAUTS Weight varies directly with gravity. With his equipment
Buzz Aldrin weighed 360 pounds on Earth but only 60 pounds on the moon.
If Valentina V. Tereshkova had landed on the moon with her equipment and
weighed 54 pounds, how much would she have weighed on Earth with
equipment? 324 lb
SWIMMING POOLS In Exercises 34 and 35, use the following
information about several sizes of pools. The amount of chlorine
needed in a pool varies with the amount of water in the pool.
STUDENT HELP
NE
ER T
Amount of water (gallons)
6250
5000
12,500
15,000
Chlorine (pounds)
0.75
0.60
1.50
1.80
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with scatter plots
in Exs. 34–35.
34. Use a graphing calculator or a computer to make a scatter plot of the data.
Sketch the graph. See margin.
35. Does the scatter plot you made in Exercise 34 show direct variation? yes
VIOLIN FAMILY In Exercises 36 and 37, use the following information.
The violin family includes the bass, the cello, the
viola, and the violin. The size of each instrument
determines its tone. The shortest produces the
highest tone, while the longest produces the
Total length, t
deepest (lowest) tone.
34. See Additional Answers beginning on page AA1.
Violin family
ADDITIONAL PRACTICE
AND RETEACHING
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
238
Bass
Cello
Viola
Violin
Total length t (inches)
72
47
26
23
Body length b (inches)
44
30
16
14
Body length, b
36. Write a direct variation model that you can use to relate the body length of a
member of the violin family to its total length. b = 0.62t
For Lesson 4.5:
• Practice Levels A, B, and C
(Chapter 4 Resource Book, p. 68)
• Reteaching with Practice
(Chapter 4 Resource Book, p. 71)
•
See Lesson 4.5 of the
Personal Student Tutor
2
25. x = 18, y = 4 y = }9}x
1
INT
STUDENT HELP NOTES
24. x = 7, y = 35 y = 5x
26. x = 22, y = 11 y = }2}x 27. x = 5.5, y = 1.1
28. x = 16.5, y = 3.3
y = 0.2x
y = 0.2x
1
1
29. x = º1, y = º1 y = x 30. x = 7ᎏᎏ, y = º9
31. x = º9, y = 3 y = º}3}x
5
5
y = –}}x
4
32. SCIENCE CONNECTION The volume V of blood pumped from your heart each
FE
describes the weight relationship
for both astronauts, students can
360 x
also solve the equation ᎏᎏ = ᎏᎏ
60 54
to find Tereshkova’s total weight
on Earth.
FINDING EQUATIONS In Exercises 23–31, the variables x and y vary directly.
Use the given values to write an equation that relates x and y.
37. Another instrument in the violin family has a body length of 28 inches. Use
your model from Exercise 36 to estimate the total length of the instrument. Is
its tone higher or lower than a viola’s tone? 45 in.; lower
238
Chapter 4 Graphing Linear Equations and Functions
Test
Preparation
38. MULTIPLE CHOICE The variables x and y vary directly. When x = 4,
y = 24. Which equation correctly relates x and y? D
A
¡
x = 4y
B
¡
y = 4x
C
¡
D
¡
x = 6y
4 ASSESS
y = 6x
DAILY HOMEWORK QUIZ
39. MULTIPLE CHOICE Which equation models the ratio form of direct
variation? A
A
¡
y
º4
ᎏᎏ = ᎏᎏ
x
3
B
¡
º3y = 4x º 1
C
¡
4
ᎏᎏ = x + y
º3
D
¡
4
3
y = ºᎏᎏx + 6
40. MULTIPLE CHOICE Find the constant of variation of the direct variation
model 3x = y. A
1
A 3
B ᎏᎏ
C y
D x
3
¡
¡
¡
¡
41. MULTIPLE CHOICE You buy some oranges for $.80 per pound. Which direct
variation model relates the total cost x to the number of pounds of oranges y? B
★ Challenge
EXTRA CHALLENGE
A
¡
42.
y = 8x
B
¡
x = 0.8y
C
¡
D
¡
y = 0.8x
GEOMETRY CONNECTION Write an equation for the perimeter
of the regular hexagon at the right. Does your equation
model direct variation? P = 6x ; yes
43. LOGICAL REASONING If a varies directly with b, then
does b vary directly with a? Explain your reasoning.
www.mcdougallittell.com
Yes; if a varies directly with b, then a = kb for some nonzero k,
1
and b = }}a, so b varies directly with a.
x = 8y
x
x
x
x
x
x
SOLVING EQUATIONS Solve the equation. (Review 3.4)
44. 7z + 30 = º5
º5
47. 9x + 65 = º4x º5
45. 4b = 26 º 9b 2
52.
3
48. 55 º 5y = 9y + 27 2 49. 7c º 3 = 4(c º 3) º3
EXTRA CHALLENGE NOTE
x = 0.7y + 1.5
51. 3x + 12 = 5(x + y)
5
x = º }}y + 6
2
HOURLY WAGE You get paid $152.25 for working 21 hours. Find your
hourly rate of pay. (Review 3.8) $7.25/h
53. ORDERED PAIRS Plot and label the points R(2, 4), S(0, º1), T(3, 6), and
U(º1, º2) in a coordinate plane. (Review 4.1) See margin.
CHECKING SOLUTIONS Decide whether the given point lies on the line.
Justify your answer both algebraically and graphically. (Review 4.2)
54–57. See margin for graphs.
54. x º y = 10; (5, º5)
55. 3x º 6y = º2; (º4, º2)
Yes; 5 º (º5) = 5 + 5 = 10
No; 3(º4) º 6(º2) = 0 and 0 ≠ º2
56. 5x + 6y = º1; (1, º1)
57. º4x º 3y = º8; (º4, 2)
No; º4(º4) º 3(2) = 10 and 10 ≠ º8
Yes; 5(1) + 6(º1) = º1
FINDING INTERCEPTS Find the x-intercept and the y-intercept of the graph
of the equation. (Review 4.3 for 4.6)
58. 2x + y = 6 3; 6
61. x + 5y = 10 10; 2
a. x = 5, y = 18 y = }15}8 x
0.4 min, or 24 sec
46. 2(w º 2) = 2
FUNCTIONS In Exercises 50 and 51, rewrite the equation so that x is a
function of y. (Review 3.7)
50. 15 = 7(x º y) + 3x
1
a. y = ᎏᎏx }14}; }14}
4
b. y = –2.5x –2.5; –2.5
2. The perimeter of a square
with side length s is modeled
by p = 4s. Do the side length
and perimeter have direct
variation? yes
3. The variables x and y vary
directly. Use the given values
to write an equation that
relates x and y.
b. x = –4, y = 6.8 y = –1.7x
4. Sound travels about 12.4 miles
in one minute. How long does
it take sound to travel 5 miles?
k
MIXED REVIEW
Transparency Available
1. Find the constant of variation
and the slope of the direct
variation model.
1
2
59. x º 1.1y = 10 10; º9}1}1 60. x º 6y = 4 4; º}3}
7
1
7
62. y = x º 5 5; –5
63. y = ᎏᎏx º ᎏᎏ 7; º}3}
3
3
4.5 Direct Variation
239
Challenge problems for
Lesson 4.5 are available in
blackline format in the Chapter 4
Resource Book, p. 75 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Explain how the
direct variation function y = kx
is a special case of the linear
function y = mx + b. y = kx is a
linear function with slope equal
to k and y-intercept 0.
2. OPEN ENDED Describe a
real-life situation that can be
modeled by a direct variation
function. For a fixed interest rate
and period of time, the amount of
simple interest varies directly
with the principal amount.
53–57. See Additional Answers
beginning on page AA1.
239