Download Study guide

Name ———————————————————————
Lesson
3.6
Date ————————————
Study guide
For use with the lesson “Model Direct Variation”
goal
Write and graph direct variation equations.
Vocabulary
Two variables x and y show direct variation, provided y 5 ax and
a Þ 0.
The nonzero number a is called the constant of variation, and y is said
to vary directly with x.
ExamplE 1
Identify direct variation equations
Tell whether the equation represents direct variation. If so, identify the
constant of variation.
a. 6x 2 3y 5 12
b. 25x 1 2y 5 0
Solution
To tell whether the equation represents direct variation, try to rewrite the equation in
the form y 5 ax.
a. 6x 2 3y 5 12
Write original equation.
Subtract 6x from each side.
y 5 2x 2 4
Divide each side by 23.
Because the equation 6x 2 3y 5 12 cannot be rewritten in the form y 5 ax, it does not
represent direct variation.
b. 25x 1 2y 5 0
Write original equation.
2y 5 5x
Add 5x to each side.
5
Simplify.
y 5 }2 x
Because the equation 25x 1 2y 5 0 can be rewritten in the form y 5 ax, it represents
5
direct variation. The constant of variation is }2.
Exercises for Example 1
Tell whether the equation represents direct variation. If so, identify the
constant of variation.
1. 3x 1 5y 5 0
2.
x 1 2y 5 1
3. 7x 2 9y 5 0
Algebra 1
Chapter Resource Book
CS10_CC_A1_MECR710723_C3L06SG.indd 83
Lesson 3.6
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
23y 5 26x 1 12
3-83
5/10/11 2:59:50 AM
Name ———————————————————————
Lesson
3.6
Date ————————————
Study guide
continued
For use with the lesson “Model Direct Variation”
ExamplE 2
Write and use a direct variation equation
The graph of a direct variation equation
is shown.
4
3
2
1
a. Write the direct variation equation.
b. Find the value of y when x 5 12.
1
�4 �3 �2 �1
Solution
a. Because y varies directly with x, the equation
23 5 a(21)
35a
2
3
4 x
(21, 23)
has the form y 5 ax. Use the fact that y 5 23
when x 5 21 to find a.
y 5 ax
y
�4
Write direct variation equation.
Substitute.
Solve for a.
A direct variation equation that relates x and y is y 5 3x.
b. When x 512, y 5 3(12) 5 36. The value of y when x 5 12 is 36.
Use a direct variation model
The table shows the cost C of purchasing tickets for a rock concert.
Lesson 3.6
a. Explain why C varies directly
with t.
b. Write a direct variation equation
that relates t and C.
Number of tickets, t
Cost, C
2
$36
3
$54
5
$90
Solution
C
a. To explain why C varies directly with t, compare the ratios }
t for
36
54
90
all data pairs (t, C): }
5}
5}
5 18. Because the ratios all
5
2
3
equal 18, C varies directly with t.
b. A direct variation equation is C 5 18t.
Exercises for Examples 2 and 3
4. The graph of a direct variation equation passes through the point (5, 22). Write
a direct variation equation and find the value of y when x 5 20.
5. What if ? In Example 3, suppose the ticket distributor charges $5.50 for each
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
ExamplE 3
transaction, no matter how many tickets are purchased, and $18 per ticket. Is it
reasonable to use a direct variation model for this situation? Explain.
3-84
Algebra 1
Chapter Resource Book
CS10_CC_A1_MECR710723_C3L06SG.indd 84
5/10/11 2:59:50 AM
Lesson 3.6 Model Direct
Variation, continued
y
Teaching Guide
23
21
1
3 x
23
5
25
3
10. y 5 } x; 6 11. y 5 2} x; 2}
8
4
5
40
4
7 35
12. y 5 } x; } 13. y 5 2} x; 2}
3
3
4
2
10 100
3
14. y 5 } x; } 15. y 5 } x; 3
3
3
10
5
25
2 20
16. y 5 } x; } 17. y 5 2} x; 2}
4
2
7
7
5
1
18. y 5 } x; } 19. yes; y 5 6x
12 6
7
1
20. yes; y 5 2} x 21. no 22. yes; y 5 } x
2
16
1
23. y 5 2x 24. y 5 2} x 25. y 5 25x
5
8
1
26. y 5 } x 27. y 5 23x 28. y 5 } x
3
3
1
9
2
29. y 5 2} x 30. y 5 } x 31. y 5 } x
9
5
2
3
1
1
32. y 5 2} x 33. y 5 } x 34. y 5 2} x
8
8
5
35. no 36. a. Because the ratios for each data
pair is 12, s varies directly with n. b. s 5 12n
c. 18 lights d. The 200-watt transformer because
2.
y
110
100
90
80
70
60
50
6 8 10 12 14 16 18 x
Hours worked
y
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
60
120 180
Distance (miles)
x
The graphs are linear functions because the points
plotted form a line.
Investigating Algebra Activity
5. No, the equation cannot be written in y 5 ax
form.
1. Sample answer: The functions have different
slopes, but the same value of b. The graphs of the
functions have the same y-intercept, but different
(positive) slopes. 2. Sample answer: The
functions have the same slope, but different values
of b. The graphs of the functions have the same
slope, but different y-intercepts; that is, the lines
are parallel. 3. Sample answer: The graphs have
the same slope (m 5 1), but different y-intercepts.
The graphs of this family are vertical translations
of the graph of f(x) 5 x.
4. Sample answer: The graphs have different
(positive or negative) slopes, but the same
y-intercept (0, 0). The graphs of this family are
vertical stretches or shrinks (with or without
refections in the x-axis) of the graph of f(x) 5 x.
Real-Life Application
Practice A
you need at least a 180-watt transformer.
37. a. s 5 0.07p b. p 5 1.6d
c. s 5 0.07(1.6d) ⇒ s 5 0.112d
Study Guide
2
3
7
1. yes; } 2. no 3. yes; } 4. y 5 2} x, 28
5
5
9
1. c 5 2.15g 2. $40.85 3. The constant of
c
variation 4. 2.19 5 }g or 2.19g 5 c 5. $39.42
Challenge Practice
1. Yes, A and W have direct variation. A 5 30W
2. Yes, V and D have direct variation. V 5 400D
3. No, A does not vary directly as both L and W
vary. 4. No, V does not vary directly as X varies.
5. $72 6. 250 acres 7. $1,190,000 8. 50 cm2
A34
1. The first table represents a function because
for each input there is exactly one output. The
second table does not represent a function because
there are two ouput values for the input values 13,
14, and 15. The last table represents a function
because for each input there is exactly one output.
1. 227; 3; 23 2. 226; 25; 9 3. 7; 4; 2
4. 6; 9; 11 5. 10; 1; 25 6. 215; 23; 5
7. 17; 11; 7 8. 7; 28; 218 9. 23.3; 0; 2.2
1
3
2
10. 9.6; 0; 26.4 11. 21; 0; } 12. }; 0; 2}
2
3
4
3
13. 23 14. 3 15. 3 16. } 17. 7 18. 2
2
19. 2 20. 22 21. 21 22. 4 23. 22 24. 2
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
1
Time (hours)
3
Amount earned (dollars)
answers
9.
Lesson 3.7 Graph Linear
Functions
Algebra 1
Chapter Resource Book
CS10_CC_A1_MECR710723_C3AK.indd 34
5/21/11 1:57:15 AM