Direct Variation
What is it and how do I know when I see it?
Definition:
Y varies directly as x means that y = kx
where k is the constant of variation.
(see any similarities to y = mx + b?)
y
Another way of writing this is k =
x
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 = 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.
X
4
8
12
18
Y
6
12
18
27
Yes!
k = 6/4 or 3/2
Equation?
y = 3/2 x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
X
10
6
4
2
Y
25
15
10
5
Yes!
k = 25/10 or 5/2
k = 10/4 or 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!
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 to find unknowns (y = kx)
Given that y varies directly with x, and y = 28 when x=7,
Find x when y = 52.
HOW???
2 step process
1. Find the constant variation
k = y/x or k = 28/7 = 4
k=4
X
Y
7
28
?
52
2. Use y = kx. Find the unknown (x).
52= 4x or 52/4 = x
x= 13
Therefore:
X =13 when Y=52
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 3 when x=9,
Find y when x = 40.5.
HOW???
2 step process
1. Find the constant variation.
k = y/x or k = 3/9 = 1/3
K = 1/3
2. Use y = kx. Find the unknown (x).
y= (1/3)40.5
y= 13.5
X
Y
9
3
40.5
?
Therefore:
X =40.5 when
Y=13.5
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 6 when x=-5,
Find y when x = -8.
HOW???
2 step process
1. Find the constant variation.
k = y/x or k = 6/-5 = -1.2
k = -1.2
X
Y
-5
6
-8
?
2. Use y = kx. Find the unknown (x).
y= -1.2(-8)
Therefore:
x= 9.6
X =-8 when Y=9.6
Using Direct Variation to solve word problems
Problem:
A car uses 8 gallons of
gasoline to travel 290
miles. How much
gasoline will the car use
to travel 400 miles?
Step Two: Find the constant
variation and equation:
k = y/x or k = 290/8 or 36.25
y = 36.25 x
Step One: Find points in table
X (gas) Y (miles)
8
290
?
400
Step Three: Use the equation
to find the unknown.
400 =36.25x
400 =36.25x
36.25 36.25
or x = 11.03
Using Direct Variation to solve word problems
Problem:
Julio wages vary
directly as the number
of hours that he works.
If his wages for 5 hours
are $29.75, how much
will they be for 30 hours
Step Two: Find the constant
variation.
k = y/x or k = 29.75/5 = 5.95
Step One: Find points in table.
X(hours) Y(wages)
5
29.75
30
?
Step Three: Use the equation
to find the unknown. y=kx
y=5.95(30) or Y=178.50
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
No
Tell if the following graph is a Direct Variation or not.
No
Yes
Yes
No
REVIEW:
A proportional relationship is a linear relationship where the yintercept is equal to zero. In this type of relationship, x and y vary
directly.
A proportional relationship can be represented by an equation in
the form y = kx, where k is the constant of variation.
y=kx
--- ----k
k
y
k = --x
EX. The table below shows the number of pages Rich has read.
Hours Read:
x
Pages Read:
y
1
40
3
120
4
160
The Correct
answer is: X
Example 2:
The equation y = 6x describes the distance Robert can ride his bike in an hour,
where x is the number of hours and y is the distance he has traveled.
Solution:
The number of miles he rode
his bike varies directly with the
time. Choose two values for x
to determine the
corresponding values of y.
Let x = 1 and x = 2.
when x = 1, y = 6 × 1 = 6
when x = 2, y = 6 × 2 = 12
The two points selected are (1,
6) and (2, 12). The only graph
that contains both of these
points is graph
Z.