Notes for Lesson 5-2: Direct Variation
5-2.1 – Identifying Direct Variation from equations
Vocabulary:
Direct Variation – A relationship between two variables that can be written in the form y  kx
where k is a nonzero constant
Constant of variation – the non-zero constant k in a direct variation equation
A Direct Variation equation is vary similar to an arithmetic sequence except that instead of adding or
subtracting be a constant amount, you are multiplying the domain instead.
To tell whether an equation is a direct variation you must solve the equation for the y variable. That is, get the y
by itself. If it is equal to a constant times x then it is a direct variation. If you are adding or subtracting a
constant to the x variable then it is not.
Examples: Tell whether or not the equation represents a direct variation, If so, identify the constant of
variation.
 3x  5 y  0
5 y  3x
3
y x
5
3
Yes k 
5
y  4x
Yes k  4
2 x  y  10
y  2 x  10
No you are adding 10 to the end
5-2.2 – Writing and solving direct variation equations
If we know the direct variation equation, or know that an ordered pair is from a direct variation then we can use
the constant of variation to project another amount.
We can do this by writing the equation of direct variation or by setting up a proportion.
Example:
The value of y varies directly with x, and y = 6 when x = 12. Find y when x = 27
Write the equation
y  kx
6  k (12)
6
1
or  k
12 2
1
x
2
1
y  (27)
2
1
y  13
2
y
so
The value of y varies directly with the value of x, and y = 8 when x =  32 . Find y when x = 64
Method 1: Equation
1
y x
4
y  kx
1
8  k (32)
so y   (64)
4
8
1
y  16
or   k
 32
4
5-2.3 – Graphing direct variation
The three toed sloth is an extremely slow animal. On the ground, it travels at a speed of 6 feet per minute.
Write a direct variation equation for the distance (y) a sloth will travel in x minutes. Then graph
Distance = 6 feet per minute
y  6x
Find 3 ordered pairs and graph
X
1
Y
6
2
12
3
18
5-2.4 – Identifying direct variation from ordered pairs
y
. So if you find that
x
ratio from each ordered pair solution is the same every time then the equation that the ordered pair came from
represents a direct variation.
If you solve the constant of variation (k) variable for the equation y  kx you get that k 
Example: Tell whether each relationship is a direct variation.
X
Y
1
6
3
18
5
30
X
Y
2
-2
4
0
8
4
y 2 0 4

, ,
x
2 4 8
y 6 18 30
 , ,
x 1 3 5
All have the same value
So, it is a direct variation
These do not have same value
So, it is not a direct variation
2,64,12(6,18)
y 6 12 18
 , ,
x 2 4 6
All have same value so the equation is a direct variation