3.6 Model Direct Variation

• You will write and graph direct variation equations.

• How do you write and graph direct variation equations?

You will learn how to answer thisquestion by recognizing situationsthat represent direct variation andidentifying the constant of variation.

Predict

Warm-Up Exercises

1. 4x – 2y = –8

2. –9x + 3y = 21

ANSWER

ANSWER

y = 2x + 4

Rewrite the equation so y is a function of x.

y = 3x + 7

Warm-Up Exercises

3. You are traveling by bus. After 4.5 hours, the bus hastraveled 234 miles. Use the formula d = rt where d is distance, r is rate, and t is time to find the average rate of speed of the bus.

ANSWER 52 mi/h

Warm-Up ExercisesEXAMPLE 1 Identify direct variation equations

Tell whether the equation represents direct variation. If so, identify the constant of variation.

2x – 3y = 0a. –x + y = 4b.

Warm-Up ExercisesEXAMPLE 1 Identify direct variation equations

ANSWER

Because the equation 2x – 3y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is .2

3

SOLUTION

To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax.

Write original equation.

Subtract 2x from each side. –3y = –2x

y =23

x Simplify.

2x – 3y = 0

Warm-Up ExercisesEXAMPLE 1 Identify direct variation equations

–x + y = 4 Write original equation.

Add x to each side.y = x + 4

ANSWER

Because the equation –x + y = 4 cannot be rewritten in the form y = ax, it does not represent direct variation.

b.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. –x + y = 1

ANSWER

not direct variation

Warm-Up ExercisesGUIDED PRACTICE for Example 1

2. 2x + y = 0

ANSWER

direct variation; –2

Tell whether the equation represents direct variation. If so, identify the constant of variation.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

3. 4x – 5y = 0

Tell whether the equation represents direct variation. If so, identify the constant of variation.

ANSWER

direct variation;4

5

Warm-Up ExercisesEXAMPLE 2 Graph direct variation equations

Graph the direct variation equation.

a. y = x2

3y = –3xb.

SOLUTION

a. Plot a point at the origin. The slope is equal to the constant of variation, or . Find and plot a second point, then draw a line through the points.

2 3

Can the y-intercept for the graphof a direct variation equation be anumber other than zero?

What is the relationship of slope to the constant of variation?

Warm-Up ExercisesEXAMPLE 2 Graph direct variation equations

Plot a point at the origin. The slope is equal to the constant of variation, or –3. Find and plot a second point, then draw a line through the points.

b.

Warm-Up ExercisesEXAMPLE 3 Write and use a direct variation equation

The graph of a direct variation equation is shown.

y = ax Write direct variation equation.Substitute.2 = a (–1)

Write the direct variation equation.a.Find the value of y when x = 30.b.

SOLUTION

Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 2 when x = –1 to find a.

a.

Solve for a. –2 = a

What can you tell about the Constant of variation from looking atthe graph?

What do you need from a graph towrite a direct variation equation?

Warm-Up ExercisesEXAMPLE 3 Graph direct variation equations

ANSWER

A direct variation equation that relates x and y is y = –2x.

b. When x = 30, y = –2(30) = –60.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

4. Graph the direct variation equation.

y = 2x

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

5. The graph of a direct variation equation passes through the point (4, 6). Write the direct variation equation and find the value of y when x = 24.

ANSWER y = x; 3632

Warm-Up Exercises

SALTWATER AQUARIUM

EXAMPLE 4 Solve a multi-step problem

• Write a direct variation equation that relates w and s.

• How many tablespoons of salt should be added to a 30 gallon saltwater fish tank?

The number s of tablespoons of sea salt needed in a saltwater fish tank varies directly with the number w of gallons of water in the tank. A pet shop owner recommends adding 100 tablespoons of sea salt to a 20 gallon tank.

Warm-Up Exercises

SOLUTION

EXAMPLE 4 Solve a multi-step problem

Write a direct variation equation. Because s varies directly with w, you can use the equation s = aw. Also use the fact that s = 100 when w = 20.

s = aw Write direct variation equation.

100 = a(20) Substitute.

5 = a Solve for a.

ANSWER

A direct variation equation that relates w and s iss = 5w.

STEP 1

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

Find the number of tablespoons of salt that should be added to a 30 gallon saltwater fish tank. Use your direct variation equation from Step 1.

s = 5w Write direct variation equation.

s = 5(30) Substitute 30 for w.

s = 150 Simplify.

ANSWER

You should add 150 tablespoons of salt to a 30 gallonfish tank.

STEP 2

Warm-Up ExercisesGUIDED PRACTICE for Example 4

6. WHAT IF? In Example 4, suppose the fish tank is a 25 gallon tank. How many tablespoon of salt should be added to the tank?

ANSWER 125 tbsp

Warm-Up ExercisesEXAMPLE 5 Use a direct variation model

a. Explain why C varies directlywith s.

b. Write a direct variation equation that relates s and C.

ONLINE MUSIC

The table shows the cost C of downloading s songs at an Internet music site.

Warm-Up Exercises

SOLUTION

EXAMPLE 5 Use a direct variation model

Because the ratios all equal 0.99, C varies directly with s.

2.97 3 =

4.95 5

6.93 7= = 0.99.

Cs

To explain why C varies directly with s, compare the

ratios for all data pairs (s, C ):

a.

b. A direct variation equation is C = 0.99s.

Warm-Up ExercisesEXAMPLE 5 Find a rate of changeGUIDED PRACTICE for Example 5

7. WHAT IF? In Example 5, suppose the website charges a total of $1.99 for the first 5 songs you download and $.99 for each song after the first 5. Is it reasonable to use a direct variation model for this situation? Explain.

ANSWER

No; the equation that models this situation does not have the form y = ax.

Warm-Up ExercisesDaily Homework Quiz

Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. 5x – 6y = 2

ANSWER no

2. x + y = 0

ANSWER yes, –1

Warm-Up ExercisesDaily Homework Quiz

3. The number p of parts a machine produces varies directly with the time t (in minutes) the machine is in operation. The machine produces 84 parts in 14 minutes. Write a direct variation equation that relates t and p. how many parts does the machine produce in 25 minutes?

ANSWER p = 6t; 150 parts

• You will write and graph direct variation equations.

• Essential Question: How do you write and graph direct variation equations?

• A direct variation equation is ofthe form y 5 ax, with a not equalto 0.• The graph of a direct variationequation is a line through theorigin with slope a.

Use facts in a real-world situationor the coordinates of a point forthe values of x and y in the directvariation equation y = ax. Thensolve for the constant of variation.To graph a direct variation equation,plot a point at the origin andthen use the constant of variation,which is the slope, to plot asecond point.