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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.