The Power FunctionsThe Power Functions

Direct VariationWhat is it and how do I know when I see

it?

A power function - is any function in the form of y = kxn, where k is nonzero and n

is a positive number (1, 2, and 3).

The linear equation graph at the right shows

that as the x-value increases, so does the y-

value increase.

For instance, if x = 2, y = 4.If x = 6 (multiplied by 3), then y = 12 (also multiplied by 3).

Direct variation – y varies directly as x means that y = kx, where k is the

constant of variation.(see any similarities to y = mx + b ?)

• Another way of writing this is k = y/x

In other words:• As x increases in value, y increase or• As x decreases in value, y decrease.

This is a graph of direct variation. If the value of x is increased, then y increases as well. Both variables change in the

same manner.If x decreases, so does the value of y. It is said that y varies directly

as the value of x.A direct variation

between 2 variables, y and x, is a relationship that is expressed as:

y = kx

Direct variation with a power where n = 1.

Others are y =kx2 and y = kx3

The relationship between y and x that is expressed as

y = kx, k is called the constant of

proportionality.In most problems, the k value needs to be found using the first set of data

given.

Example:The power, P, of a gear varies directly with the

radius, r, of a gear. Find the constant of proportionality if

P = 300 when r = 50Start with the formula

P = krTypical problem, try it

In a factory, the profit, P, varies directly with the inventory, I. If P = 100 when I = 20, find P when I

= 50.

Step 1Set up the formula

P = IkStep 2

Find the missing constant, k, for the given data.

100 = (20)k, k = 5Step 3

Use the formula and constant to find the missing value.P = (50)(5) P = 250

It will be necessary to use the “first” set of data to find the value for the constant, k.

y = kx

What is the constant of variation of the table above?

Since y = kx, we can say k = y/x therefore,

12/6 = k or k = 2 14/7 = k or k = 216/8 = k or k = 2 Note: k stays constant

x y6 12

7 14

8 16

Note: x increases6, 7, 8

And y increases12, 14, 16

y = 2x is the equation!!!Another example

What is the constant of variation of the table above?

Since y = kx, we can say k = y/x therefore:

30/10 = k or k = 3 15/5 = k or k = 39/3 = k or k = 3 Note: k stays constant.

X Y30 10

15 5

9 3

Note: x decreases, 30, 15, 9

And y decreases.10, 5, 3

y = 3x is the equationAnother example

What is the constant of variation of the table above?

Since y = kx, we can say k = y/x therefore:-1/-4 = k or k = ¼ -4/-16 = k or k = ¼

-10/-40 = k or k = ¼ Note: k stays constant

x y-4 -1

-16 -4

-40 -10

Note: x decreases,-4, -16, -40

And y decreases.-1, -4, -10

y = ¼x is the equation

Use direct variation to solve word problems

• A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?

x y8 290

? 400

Step 1: find points in the table

Step 2: find the constant variation and equation.

k = y/x or k = 290/8 or 36.25

y = 36.25x

Step 3: use the equation to find the unknown.

400 = 36.25x 400 = 36.25x36.25 36.25or x = 11.03

Direct variation and its graph

y = mx + b,m = slope and b = y-intercept

With direct variation the equation is y = kx

Note: m = k or the constant and b = 0, therefore the

graph will always go through…

The ORIGIN!!!!!

Solve the following variation problems using the formula.

1) y varies directly as x. If y = 75 when x = 10, find y when x = 16.

2) Your distance, d, from lightning varies directly with the time, t, it takes you to hear thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. How far are you away from the lightning if you hear thunder in 3 seconds.

3) The distance, d, a cyclist travels varies directly with the time, t, it takes in hours. If a cyclist travels 40 km in 2 hours, how far will he have

traveled in 6 hours.

More problems

4) The amount of sales tax on a new car is directly proportional to the purchase price of the car. If a $25,000 car cost $1750 in sales tax, what is the purchase price of a new car which has a $3500 sales tax? Hint: sales tax = k(purchase price)

5) The cost of a house in Florida is directly proportional to the size of the house. If a 2850 ft2 house cost $182,400, then what is the cost of a 3640 ft2 house?

Other Power Functions

Other Power Functions

y = kx2 and y = kx3

y = kx2

The distance required for a moving car to stop varies directly as the square of the car’s speed. Therefore, the formula (equation) d = ks2 represents the stopping distance for the auto. A car traveling 50 miles per hour has a stopping distance of 135 feet. What would be the stopping distance of an auto travel 70 miles per hour.

d = ks2 d = ks2

135 = k(502) d = .054(702)

k = .054 d = 264.6 ft

y = kx3

The volume of a cube varies directly as the cube of the side lengths. If the volume (V) of a cube is 36cm with side lengths (s) of 2cm, what is the volume of a cube with side lengths (s) of 5.

V = ks3 V = 4.5(53)36 = k(23) y = 562.5 cmk = 4.5