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What is a Direct Variation?
The equation that represents a direct variation is ___________, where
k is the ______________________________ (this is the factor that
changes the quantities.)
The equation can be changed to
proportionality.
y
 k to find the constant of
x
Let’s say you know the coordinate (2,4). This is the same as saying that x
is ____ and y is ______. Substitute your coordinate into the direct
variation equation to solve for the constant of proportionality.
y
4
k
k
Relationship:
k=2
y = 2x
x
2
Solving for k
(constant of
proportionality)
This is the
constant of
proportionality
Writing is as
an equation
A direct variation equation can be represented by a proportion: A direct
y
y
proportion 1  2 can represent 2 different coordinates ( x 1 , y1 ) and
x1 x 2
( x 2 , y2 )
A direct variation graph __________ passes through the _______ (0,0).
Direct Variation In Context
A direct variation in context reveals how an increase in 1 variable will result
in an ___________ in another variable. It can also reveal how a decrease in
1 variable will result in a ____________ in another variable. A direct
variation can represent an increase in 1 variable and a decrease in another
variable.
The more time I drive at a constant rate, the more miles I go.
If I increase a recipe for more people, the more of an
ingredient I need.
The more hours I work, the more money I make.
The more CD’s I purchase, the more money it costs.
If you buy half as much cheese at the deli, you pay half as
much.
Increase in one variable results in decrease in another.
Ex.
-The more time you spend working out, the less you weigh.
-The less time you work out, the more you weigh.
-
Direct Variation Proportions
A direct variation proportion has ratios that are equivalent. A coordinate
y
y
y
(x,y) is actually the ratio of
. A direct proportion 1  2 can represent 2
x
x1 x 2
different coordinates ( x 1 , y1 ) and ( x 2 , y2 ) or 2 different coordinates can
represent a direct proportion. Write 1-2 examples in your foldable.
For example:
3 2
 is the same as the coordinates(12,3), (8,2)
12 8
9 15

is the same as the coordinates (3,9),(5,15)
3 5
Direct Variation Tables
Look at the following tables that represent direct variation.
X
1
2
3
Y
2
4
6
X
10
6
4
Y
5
3
2
Y divided by X = k (constant)
What is the constant for each of the tables?
Table 1:
Table 2:
NOT Direct Variation Tables
Look at the following tables that do not represent a direct variation. Find
what all of the tables have in common.
X
1
2
3
Y
3
4
5
X
10
20
30
Y
6
4
2
Direct Variation Equations
Explain why each of the tables are not
direct variation.
Table 1:
Table 2:
The following are examples of direct variation equations. The
number in front of the x variable is called the constant of
proportionality or variation. Y=kx
y =
1
2
Why are the following equations direct
variation?
x
y = x
y = 2x
y = -2.5x
NON-Direct Variation Equations
The following are non-examples of direct variation equations.
y =
1
2
y = 2
Why are the following NOT equations direct
variation?
x-2
y = x+3
y = 2.5x-4
Practice:
Find the constant of proportionality for each of the following.
1. Find the constant of proportionality if x and y vary directly, and y = 60 when x = 15.
2. Find the constant of proportionality if x and y vary directly, and y = 18 when x = -2.
3. Find the constant of proportionality if x and y vary directly, and y = 225 when x = 9.
4. Find the constant of proportionality for the following table.
X
Y
3
7.5
5
12.5
7
17.5
Directions: First solve for the constant and then write your answer as a direct variation
equation.
5. The variables x and y vary directly. When x = 14, y = 56. What equation relates x and
y?
6. Find an equation of variation where y varies directly as x, when y = 630 and x = 175.
7. Find an equation of variation where y varies directly as x, when y = 400 and x = -125.
8. Does the ordered pair (8, 48) satisfy the direct equation y = -6x? Explain why
or why not.
9. Using y = -6x, what is the value of x if y = 42?
10. If y varies directly as x, and y = 64 when x = 4, find x when y = 48.
(Hint: you could either use a proportion to solve or find the equation and plug in
your number.)
11. Write a proportion to show how you would solve the problem-do not solve:
A rose bush grows 3 inches every 4 months. How many inches will the bush
grow in 1 year?
12. Which of the given values satisfies the equation y = 8x
A. x = 7, y = 28
B. x =4 , y = 2
C. x =3, y = 24
D. x = 1.5, y = 5.5
13. The variables x and y vary directly. When x = 14, y = 56. What equation
relates x and y?
A. y = 16x
B. y = 2x
C. y = 14x
D. y = 4x
14. Johnny scores 27 points in three basketball games. The season has 10
games. How many points can Johnny expect to score by the end of the season?
A. 90
B. 81
C. 84
D. 810
15. If y varies directly with x, and y =- 7 when x = 4, find x when y = 21.
16. Which of the following equations is a direct variation?
A. 2x = y
B. x - 1 = y
C. y + 2 = x D. xy = 4
17. Six packs of gum cost $2.00. At that rate, about how much will 40 packs of
gum cost?
A. $3.33
B. $12.00
C. $13.33
D. $24.00
1
2
18. Complete the following table for y = - x.
X
Y
0
1
2
3
2
3
6
10
19. Complete the following table for y = 3x.
X
Y
0
1
20. Complete the following table for y = -3x.
X
Y
0
2
21. Complete the following table for y = 3.5x
x
0
1
2
3
4
y
0
3.5
7
?
14
22. Which of the following tables represent a direct variation?
23. Suzy can walk 1 mile in 10 minutes. Create a table of values that represent
time and distance.
Distance (x)
Time (y)
What direct variation equation would represent the miles and time that Suzy can
walk?_________
24. How do you find the constant of proportionality in a direct variation?
25. What are two properties of a direct variation graph?
26. What value of n makes the equations true?
21
7

n
10