Advanced Algebra 2: CH variation and other common functions
DIRECT and INVERSE VARIATION FUNCTION INTRODUCTION
Direct Variation
: ( k is the constant of proportionality) y varies directly with x . y
 kx y
Note that this is a linear function through the origin.
varies directly with the square of x . y
 kx
2
Note that this is a quadratic function through the origin. y varies directly with the cube of x . y
 kx
3 y
Note that this is a cubic function through the origin.
varies directly with x the nth power. y
 kx n
Note that this is a nth degree polynomial through the origin.
Example 1 : Suppose y varies directly with x and y = 25 when x = 10. Write the function of this relationship and determine the value of y when x

16 .
Solution : y
 kx and with substitution: 25
 k which implies k

25
10

5
2
So the function is y

5
2 x and y

5
2

20 when x

16 .
Example 2 : Suppose y varies directly with the cube of x and y = 100 when x = 4. Write the function of this relationship and determine the value of y when x

10 .
Solution : y
 kx 3 and with substitution: 100
 k
 
3 
100
 k which implies k

100
64

25
.
16
So the function is y

25 x
3
16 and y

25
3
16
 

25 
1000

16

3125 when x

10 .
2
Complete problems 1 and 5 from section 7.11

Advanced Algebra 2: CH variation and other common functions
DIRECT and INVERSE VARIATION FUNCTION INTRODUCTION
Inverse Variation:
( k is the constant of proportionality) y varies inversely with x . y
 k x
Note that this is a rational function with a vertical asymptote at x

0 . y varies inversely with the square of x . y
 x k
2
Note that this is a rational function with a vertical asymptote at x

0 . y varies inversely with the cube of x . y
 x k
3
Note that this is a rational function with a vertical asymptote at x

0 . y varies inversely with x the nth power. y
 x k n
Note that this is a rational function with a vertical asymptote at x

0 .
Example 3 : Suppose y varies inversely with the square of x and y = 4 when x = 5. Write the function of this relationship and determine the value of y when x

9 .
Solution : y
 k x
2 and with substitution: 4
 k
 
2
which implies k

100 .
So the function is y

100 x
2 and y

100
9
2

100
 
81 when x

9 .
Complete problems 7abcd, and 9 from section 7.11

Advanced Algebra 2: CH variation and other common functions
DIRECT and INVERSE VARIATION FUNCTION INTRODUCTION
Lets investigate the response to doubling, tripling or halving x in direct and inverse variation functions.
Illustration 1
: Consider y varies directly with x : y

1 kx
Question :
What happens to y when we double x , that is, what happens when we replace ( x ) with (2 x )?
Solution :
Let y be a result of doubling x , that is
2 twice that of y
1
. In short, doubling x y
2
 k
 
doubles y .

( 2 ) kx

2 y
1
. We see from this that y is
2
Illustration 2
: Consider y varies inversely with the cubed of x : y
1

3 x k
Question :
What happens to y when we double x , that is, what happens when we replace ( x ) with (2 x )?
Solution :
Let y be a result of doubling x , that is
2 y
2
 k
 
3

2
3 k
   k
2
3 x
3

1
2
3
 k x
3

1
2
3 y
1
. We see from this that
1 y is
2
2
3
(or one-eighth) that of y
1
.
Complete the chart on the following page.

Advanced Algebra 2: CH variation and other common functions
DIRECT and INVERSE VARIATION FUNCTION INTRODUCTION
For each type of variation, indicate the effect doubling, tripling, or halving x has on y . If done correctly a patterns should arise.
1. y varies directly with x . y
 kx
Doubling x : Tripling x :
2. y varies directly with the square of x . y
 kx
2
Halving x :
Doubling x : Tripling x :
3. y varies directly with the cube of x . y
 kx
3
Halving x :
Doubling x : Tripling x :
4. y varies directly with x the nth power. y
 kx n
Doubling x : Tripling x :
5. y varies inversely with x . y
 k x
Doubling x : Tripling x :
6. y varies inversely with the square of x . y
 x k
2
Doubling x : Tripling x :
7. y varies inversely with the cube of x . y
 x k
3
Doubling x : Tripling x :
8. y varies inversely with x the nth power. y
 x k n
Doubling x : Tripling x :
Halving
Halving
Halving
Halving
Halving
Halving x x x x x x
:
:
:
:
:
: