Variation Functions
Variation functions are equations in which y is
equal to a constant multiplied or divided by a
power of x.
If the constant is multiplied by the variable, then
y varies directly with the power of x.
If the constant is divided by the variable, then y
varies inversely with the power of x.
Variation Functions
If k and n are constants, then “y varies directly
th
with the n power of x” means:
y = kx
n
th
and “y varies inversely with the n power of x
means:
k
y= n
x
Variation Functions
The constant is called the proportionality
constant.
The letter k is often used as the constant.
• y varies directly with x:
y = kx
• y varies directly with the square of x:
• y varies directly with the cube of x:
• y varies indirectly with x:
y = kx
y = kx
k
y=
x
• y varies indirectly with the square of x:
2
3
k
y= 2
x
Variation Functions
Direct Variation: y gets bigger as x increases
Inverse Variation: y gets smaller as x increases
• x- and y-axes are the asymptotes
Variation Functions
To solve a variation problem:
1) Look at the ordered pairs given to you in the
problem. Figure out how x and y are changing in
relation to each other.
2) Write the general rule you will use.
3) Set your variables.
4) Write the particular equation.
5) Use the particular equation to make predictions.
Example 1 (Page 387)
• The pressure required to force water through a garden
hose depends on the number of gallons per minute
(gpm) you want to flow. By experiment, you find that
for a flow of 3 gpm, a pressure of 10 pounds per square
inch (psi) is required. For 6 gpm, a pressure of 40 psi is
required.
• Predict the following:
a) the pressure required for a flow rate of 12 gpm.
b) the pressure required for a flow rate of 4.2 gpm
c) the number of gpm you would get with a pressure of 5
psi.
Example 2 (page 389)
• The intensity of light reaching you from a light
bulb depends on how far from the bulb you are
standing. Suppose that at 3 meters the intensity
is 120 units, and at 6 meters it is 30 units.
a) How does the intensity vary with distance? Write
the general equation.
b) Find the particular equation.
c) Predict the intensity at 10 meters
d) Predict the intensity at 12 meters
e) Predict the distance at which the intensity with be 4
units.