Chapter 2:

Lesson 1: Direct Variation

Mrs. Parziale

In New York, you can receive a refund for returning

aluminum cans. For each can returned, you receive a 5¢

refund. If r represents the refund and c represents the

number of cans returned, write an equation to show this

relationship.

___________________________________

• What happens to your refund if you double the number of cans returned?

________________________________• What happens to your refund if you triple the number of

cans returned?

________________________________• What happens to your refund if you only collect half of

the number of cans?

________________________________

The aluminum can problem is an example of direct variation.

refund r varies ________________ as c the number of cans

(Direct Variation means that two variables, such as r and c are so related that when one is multiplied by a constant value, so is the other.)

• When the number of cans ____________,

the refund ___________________.

• When the number of cans ____________,

the refund ________________________.

More Examples:• The cost of gas for a car varies directly as the

amount of gas purchased.• The price of breakfast cereal varies directly as

the number of boxes of cereal purchased.• The volume of a sphere varies directly as the

cube of its radius.• The deer population varies directly with the

amount of available vegetation.

Direct Variation Function –

• Independent variable is ____________.• Dependent variable is______________.• “k” is the ____________________________.

Example 2: The area of a circle varies directly with the square of the radius.

• As the size of the radius increases , the total area ______________.

• As the size of the radius decreases , the total area ______________.

• If the radius is 5, • If the radius is 10,

where k 0 and n 0ny kx

25A 100A

This means that two variables, such as radius and Area are so related that when one is multiplied by a constant value, so is the other.

Another way to express direct variation:

y varies directly as xn also means:y is directly ____________________ to xn

• In , area of a circle is directly proportional to the square of the radius.

• In the earlier equation for the refund of the cans our equation was -- r = 5c

where k 0 and n 0ny kx

2A r

y x k n

y x k n

Example 3:

Write an direct variation equation that describes the following (let k=constant of variation):

1. The cost c of gas varies directly as the amount of gas g purchased.________________________________

2. The price P of breakfast cereal varies directly as the number of boxes b of cereal purchased.

________________________________3. The volume V of a sphere is directly proportional to the

cube of its radius r.________________________________

Solving Direct Variations Problems

• Suppose that lightning strikes a known point 4 miles away, and that you hear the thunder 20 seconds later. Then, how far away has lightning struck if 30 seconds pass between the time you see the flash and hear its thunder? Here, the distance varies directly as the time it takes for us to hear the thunder.

How do we solve this?

Solving Direct Variation Problems:

Follow the following four steps:

1. Write an ___________________ that describes the function.

2. Solve for the __________________________.

3. __________________ the variation function using the constant of variation found in step 2.

4. __________________ the function for the desired value of the independent variable.

Solving Direct Variations Problems

• Suppose that lightning strikes a known point 4 miles away, and that you hear the thunder 20 seconds later. Then, how far away has lightning struck if 30 seconds pass between the time you see the flash and hear its thunder? Here, the distance varies directly as the time it takes for us to hear the thunder.

How do we solve this?

Example 4:

Suppose W varies directly as the third power of z and

W = 96 when z=2. Find W when z = 10.

Step 1:

Step 2

Step 3:

Step 4:

Example 5:• A certain car needs 25 feet to come to a stop after the brakes are

applied at 20 mph. Braking distance d (in feet) is directly proportional to the square of the speed s (in mph). Use the 4 steps to find the distance needed to stop the car after the brakes are applied at 60 mph. (Hint: find the constant of variation for the initial distance and braking speed and then use it with the 60 mph speed to find the distance.)

5-Minute Check

Translate into a direct variation equation. Let k be the constant of variation.

1. The distance d a star is from the earth is directly proportional to the length of time t it takes for its light to get here.

2. m varies directly as the fourth power of n.3. y varies directly as x. If y = 8 when x = -4, find y when x = 16True or False:4. The outdoor temperature varies directly as the time of

day.