Direct Variation
What is it and how do I know when I see it?
Direct Variation
When we talk about a direct
variation, we are talking about
a relationship where as
x increases, y increases
or x decreases, y decreases at
a CONSTANT RATE.
Definition:
A direct variation involving x and y is a
function in which the ratio y is a nonzero
x
constant.
y
Another way of writing this is k =
x
k is the constant of variation
also known as the slope of the
function.
Definition:
y varies directly as x means that
y = kx
where k is the constant of variation.
(see any similarities to y = mx + b?)
In other words:
* the constant of variation (k) in a direct
variation is the constant (unchanged) ratio of two
variable quantities.
Examples of Direct Variation:
X
6
7
8
Y
12
14
16
Note: X increases,
6,7,8
And Y increases.
12, 14, 16
What is the constant of variation of the table above?
y
Since y = kx we can say k 
Therefore:
x
12/6=k or k = 2
14/7=k or k = 2
16/8=k or k =2
Note k stays constant.
y = kx
y = 2x is the
equation!
Examples of Direct Variation:
Note: X decreases,
X
10
5
3
Y
30
15
9
10, 5, 3
And Y decreases.
30, 15, 9
What is the constant of variation of the table above?
y
Since y = kx we can say k 
Therefore:
x
30/10=k or k = 3
15/5=k or k = 3
9/3=k or k =3
Note k stays constant.
y = 3x is the
equation!
Examples of Direct Variation:
X
-4
-16
-40
Y
-1
-4
-10
Note: X decreases,
-4, -16, -40
And Y decreases.
-1,-4,-10
What is the constant of variation of the table above?
y
k

Since y = kx we can say
Therefore:
x
y = ¼ x is the
-1/-4=k or k = ¼
-4/-16=k or k = ¼
equation!
-10/-40=k or k = ¼ Note k stays constant.
What is the constant of variation for the
following direct variation?
Answer
Now
0%
0%
0%
0%
½
4.
-½
3.
-2
2.
2
-2
-½
½
Y
-8
-16
12
-6
2
1.
X
4
8
-6
3
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
Yes!
X
4
8
12
18
Y
6
12
18
27
k = 6/4 or 3/2
k = 12/8 or 3/2
k = 18/12 or 3/2
k = 27/18 or 3/2
Equation?
y = 3/2 x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
Yes!
X
10
6
4
2
Y
25
15
10
5
k = 25/10 or 5/2
K = 15/6 or 5/2
k = 10/4 or 5/2
k = 5/2
Equation?
y = 5/2 x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
X
15
3
1
2
Y
5
26
75
150
No!
k = 5/15 or 1/3
k = 75/1 or 75
The k values are
different!
Which of the following is a direct variation?
0%
0%
0%
0%
D
4.
Answer
Now
C
3.
B
2.
A
B
C
D
A
1.
Which is the equation that describes the
following table of values?
0%
0%
0%
0
20
=
xy
=
=
½
2x
x
0%
y
Answer
Now
y
4.
Y
5
1
6
10
-2
x
3.
X
10
2
12
20
=
2.
y = -2x
y = 2x
y= ½x
xy = 200
y
1.
Using Direct Variation
When x is 2 and y is 4, find an equation that
shows x and y vary directly.
2 Step Process
1st Find the constant variation
y
k=
x
k=2
or k = 4/2 = 2
2nd Use y = kx.
y = 2x
Using Direct Variation
When x is 3 and y is 12, find an equation that
shows x and y vary directly.
2 Step Process
1st Find the constant
variation
k = y/x or k = 12/3 = 4
k=4
2nd Use y = kx.
y = 4x
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = -30 when x=3, Find y when x = 8.
HOW???
2 step process
1. Find the constant variation
k = y/x or k = -30/-3 = 10
k=10
2. Use y = kx. Find the unknown (x).
y = 10x so y= 10(8)
y= 80
Therefore:
x = 8 when y = 80
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 20 when x=4,
Find y when x = 10.
HOW???
2 step process
1. Find the constant variation
k = y/x or k = 20/4 = 5
k=5
2. Use y = kx. Find the unknown (x).
y = 5x so y= 5(10)
y= 50
Therefore:
x = 10 when y = 50
Using Direct Variation to solve word problems
Problem:
To make mango salsa, you need 3
mangoes for 2 recipes (2 recipes = 8
serving). Write an equation relating
the number of servings y to the
number of mangoes x given so that y
varies directly with x. How many
servings of salsa can you make if
you have 5 mangoes?
Step Two: Find the constant
variation and equation:
k = y/x or k = 8/3
y = 8/3 x
Step One: Find points in table
Step Three: Use the equation
to find the unknown.
y = 8/3x
y = 8/3(5)
y = 40/3
y = 13.3
Using Direct Variation to solve word problems
Problem:
To make mango salsa, you need 3
mangoes for 2 recipes (2 recipes = 8
serving). Write an equation relating
the number of servings y to the
number of mangoes x given so that y
varies directly with x. How many
servings of salsa can you make if
you have 5 mangoes?
Step One: Find points in table
8(5) = 3y
40 = 3y
Y = 13.3
40=3y
Y = 13.3 servings
Use a proportion to solve.
x1 x 2

y1 y 2
3 5

8 y
Using Direct Variation to solve word problems
Problem:
The length that a spring will
stretch S varies directly with the
weight w attached to the spring.
If a spring stretches 1.4 inches
when a 20 pound weight is
attached, how far will it stretch
when a 10 pound weight is
attached?
Step Two: Find the constant
variation and equation:
k = y/x or k = 1.4/20
y = 0.07 x
Step One: Find points in table
Step Three: Use the equation
to find the unknown.
y = 0.07x
y = 0.07(10)
y = 0.7
y = 0. 7inches
Using Direct Variation to solve word problems
Problem:
The length that a spring will stretch
S varies directly with the weight w
attached to the spring. If a spring
stretches 1.4 inches when a 20
pound weight is attached, how far
will it stretch when a 10 pound
weight is attached?
Step One: Find points in table
Use a proportion to solve.
x x 20 10


y y 1.4 y
1.4(10) = 20y
14 = 20y
y = 0.7
y = 0.7 inches
Direct Variation and its graph
y = mx +b,
m = slope and b = y-intercept
With direction variation the equation
is y = kx
Note: m = k or the constant and b = 0 therefore the graph will
always go through…
the ORIGIN!!!!!
Tell if the following graph is a Direct Variation or not.
No
No
Yes- the line
passes
through the
origin.
No
Tell if the following graph is a Direct Variation or not.
No
Yes-the line
passes
through the
origin
Yes- the line
passes
through the
origin
No
Summary
An equation is a direct variation if:
• Its graph is a line that passes
through zero
OR
• The equation can be written in the
form y = kx