The general equation for DIRECT VARIATION is y =kx with k≠0.

k is called the constant of variation.

We will do an example together.

If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation

(b) Find y when x=2

(a) Find the constant of variation

y=kx Write the general equation

24=k⋅3 Substitute

k=8

(b) Find y when x=2

First we find the constant of variation, which was k=8

Now we substitute into y=kx.

y=kxy=8⋅2

y=16

Another method of solving direct variation problems is to use proportions.

If y1 =kx1, then k=y1

x1

and

If y2 =kx2, then k=y2

x2

Therefore...

y1

x1

=y2

x2

So lets look at a problem that can by solved by either of these two methods.

If y varies directly as x and y=6 when x=5, then find y when x=15.

Proportion Method:65

=y

15Let x1 =5, y1 =6, x2 =15, y2 =y

5y=90

y=18

Now lets solve using the equation.

y=kx

6=k⋅5

k=65

y=kx

y=65

⋅15

y=18

Either method gives the correct answer, choose the easiest for you.

Now you do one on your own.

y varies directly as x, and x=8 when y=9. Find y when x=12.

Answer: 13.5

What does the graph y=kx look like?A straight line with a y-intercept of 0.

5

-5

-10 10

f x( ) = 3⋅x

Looking at the graph, what is the slope of the line?

Answer: 3

Looking at the equation, what is the constant of variation?

Answer: 3

The constant of variation and the slope are the same!!!!

We will apply what we know and try this problem.

According to Hook’s Law, the force F required to stretch a spring x units beyond its natural length varies directly as x. A force of 30 pounds stretches a certain spring 5 inches. Find how far the spring is stretched by a 50 pound weight.

F1

x1

=F2

x2

Set up a proportion

305

=50x

Substitute

30x=250

x=813

inches

Now try this problem.

Use Hook’s Law to find how many pounds of force are needed to stretch a spring 15 inches if it takes 18 pounds to stretch it 13.5 inches.

Answer: 20 pounds

Inverse Variation

y varies inversely as x if k≠0

such that xy=k or y=kx

Just as with direct variation, a proportion can be set up solve problems of indirect variation.

x1

y2

=x2

y1

A general form of the proportion

Lets do an example that can be solved by using the equation and the proportion.

Find y when x=15, if y varies inversely as x and x=10 when y=12

Solve by equation:

xy=k10⋅12=k

120=k

xy=k15⋅y=120

y=8

Solve by proportion:

x1

y2

=x2

y1

1512

=10y

15y=120

y=8

Solve this problem using either method.

Find x when y=27, if y varies inversely as x and x=9 when y=45.

Answer: 15

Lets apply what we have learned.

The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.

Step #1: Set up a proportion.

x1

y2

=x2

y1

40200

=x

600

200x=24000

x=120 pounds/ in2

Now try this one on your own.

A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle’s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.

Answer: 80 pounds/ in2

What does the graph of xy=k look like? Let k=5 and graph.

6

4

2

-2

-4

-6

-10 -5 5 10

f x( ) = 5

x

This is a graph of a hyperbola.

Notice: That in the graph, as the x values increase the y values decrease.

also

As the x values decrease the y values increase.