2.5
Model Direct Variation
Direct
variation
The
equation y = ax represents direct variation between x and y.
Constant
of variation
The
nonzero constant a in the direct variation equation y= ax
DIRECT
VARIATION
Equation The equation y = __ax__
represents direct variation between x and y, and y is
said to __vary directly__ with x. The nonzero constant a is called the constant of _variation_.
Graph The graph of a direct
variation equation y = ax is a __line__ with slope a and y-intercept 0. The family of direct
variation graphs consists of lines through the _origin_.
Example
1
Write
and graph a direct variation equation
Write
and graph a direct variation equation that has (-3, 2) as a solution.
Solution
Use
the given values of x and y to find the constant of variation.
|
y = ax |
Write
direct variation equation. |
|
|
Substitute
__2__ for y and __-3__for x. |
|
|
Solve
for a. |
The direct variation equation is y = x.
Complete
the following exercise.
1. 
Write and graph a direct
variation equation that has the ordered pair (4, -2) as a solution.
y = x
Example
2
Write
and apply a model for direct variation
According to Hooke's
law, the force that is needed to stretch a spring varies directly with the
amount the spring is stretched.
a.
If 64 pounds of force F stretches a
spring a distance d of 8 inches, write an equation that relates F
and d.
b. Predict
the amount of force that is needed to stretch the spring to 14 inches.
Solution
a.
Find the constant of variation.
|
F = ad |
Write
direct variation equation. |
|
__64__ = a(__8__) |
Substitute
__64__ for F and __8__ for d. |
|
__8__ = a |
Solve
for a. |
An equation that relates F and d is
F = __8__ d.
b.
To stretch the spring d = 14 inches, the
amount of force needed is
F = __8__ (__14__)
= __112__ pounds.
Complete the following exercise.
2.
In Example 2, suppose that the force being
applied to the spring is 92 pounds. Predict how far the spring is being
stretched.
11.5 in.
Example
3
Use
ratios to identify direct variation
The
table below gives sample cell phone bills, showing the total monthly cost and
the number of minutes used that month. Tell whether total cost and the number
of minutes show direct variation. If so, write an equation that relates the
quantities.
|
Total
cost, c (in dollars) |
35 |
45 |
80 |
15 |
30 |
|
Minutes
used, m |
100 |
129 |
229 |
44 |
86 |
Solution
Find the ratio of the
total cost c to the minutes used m for each month.
|
= __0.35__ |
|
» 0.35 |
||||
|
15 44 |
|
|
Because
the ratios are approximately equal, the data show direct variation.
An equation relating
total cost to minutes used is
= __0.35__ or c = __0.35__
m.
Complete the following exercise.
3.
Tell whether the data in the table below shows
direct variation. If so, write an expression relating x and y.
|
x |
-1 |
1 |
3 |
5 |
7 |
|
y |
-2 |
2 |
6 |
10 |
14 |
y = 2x