SAT Mc Graw Hill 2011

422 McGRAW-HILL’S SAT

Lesson 4: Variation

Direct Variation

The statement “yvaries directly asx” means

that the variables are related by the equation

y= kx, where kis a non-zero constant. This

equation implies that xand ygo up and down

proportionally. For instance, whenever xis

multiplied by 3, yis also multipliedby 3.

The table and graph above show three examples

of direct variation functions. Notice that (1) every

graph passes through the origin, (2) as kincreases, so

does the slope of the graph, and (3) for any given k,

whenever xis doubled (or tripled or halved), so is the

corresponding value of y.

Example:

If xvaries directly as yand x= 20 when y= 60,

then what is the value of xwhen y= 150?

First find the value of the constant kby substituting

the values of xand yinto the equation y= kx.

y= kx

Substitute: 60 = k(20)

Divide by 20: 3 = k

Now we know that the equation isy= 3x.

Substitute y= 150: 150 = 3x

Divide by 3: 50 = x

Inverse Variation

The statement “yvaries inversely asx” means

that the variables are related by the equations

y= or xy= k, where kis a non-zero constant.

These equations imply that xand ygo up and

down inversely. For instance, whenever xis

multiplied by 3, yis dividedby 3.

The table and graph above show an example of an

inverse variation function. Notice that (1) the graph

never touches the x-or y-axis, (2) as xincreases, yde-

creases, and (3) for every point on the graph, the

product of xand yis always the constant k, in this

case k= 1.

Example:

If xvaries inversely as yand x= 40 when y= 10,

then what is the value of xwhen y= 25?

First find the value of the constant kby substituting

the values of xand yinto the equation xy= k.

xy= k

Substitute: (40)(10) = k

Simplify: 400 = k

Now we know that the equation isxy= 400.

Substitute y= 25: x(25) = 400

Divide by 25: x= 16

Joint and Power Variation

A variable can vary with more than one other

variable or with powers of a variable. For in-

stance, the statement “yvaries directly as x

and inversely as w” means that and “y

varies inversely as the square of x” means that

y.

k

x

= 2

y

kx

w

= ,

k

x