SECTION 7.6 Variation 407
OBJECTIVES Functions in which y depends on a multiple of xor y depends on a number divided by
xare common in business and the physical sciences.
OBJECTIVE 1 Write an equation expressing direct variation.
The circumference of a circle is given by the formula
where ris the radius of the circle. See FIGURE 6. The circumference is
always a constant multiple of the radius. (Cis always found by multi-
plying rby the constant )
As the radius increases,the circumference increases.
As the radius decreases,the circumference decreases.
Because of these relationships, the circumference is said to vary directlyas the radius.
2 p.
C= 2 pr,
Variation
7.6
1 Write an equation
expressing direct
variation.
2 Find the constant of
variation, and solve
direct variation
problems.
3 Solve inverse
variation problems.
4 Solve joint variation
problems.
5 Solve combined
variation problems.
rC = 2r
FIGURE 6Direct Variationyvaries directly asxif there exists a real number ksuch that
yis said to be proportional tox. The number kis called the constant of variation.
ykx.
In direct variation, for as the value of x increases, the value of y increases.
Similarly, as x decreases, y decreases.
k>0,
OBJECTIVE 2 Find the constant of variation, and solve direct variation
problems.The direct variation equation defines a linear function, where
the constant of variation k is the slope of the line.For example, we wrote the fol-
lowing equation to describe the cost yto buy xgallons of gasoline.
See Section 3.3, Example 7.The cost varies directly as, or is proportional to, the number of gallons of gasoline
purchased.
As the number of gallonsof gasoline increases,the cost increases.
As the number of gallonsof gasoline decreases,the cost decreases.
The constant of variation kis 3.20, the cost of 1 gal of gasoline.
y= 3.20 x
ykx
Finding the Constant of Variation and the Variation
EquationEva Lutchman is paid an hourly wage. One week she worked 43 hr and was paid
$795.50. How much does she earn per hour?
Let hrepresent the number of hours she works and Prepresent her corresponding
pay. Write the variation equation.
Pvaries directly as h.