Download Graph the direct variation equation

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 this
question by recognizing situations
that represent direct variation and
identifying the constant of
variation.
Warm-Up Exercises
Rewrite the equation so y is a function of x.
1. 4x – 2y = –8
ANSWER
2.
y = 2x + 4
–9x + 3y = 21
ANSWER
y = 3x + 7
Warm-Up Exercises
3. You are traveling by bus. After 4.5 hours, the bus has
traveled 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-Up1Exercises
EXAMPLE
Identify direct variation equations
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
a. 2x – 3y = 0
b. –x + y = 4
Warm-Up1Exercises
EXAMPLE
Identify direct variation equations
SOLUTION
To tell whether an equation represents direct
variation, try to rewrite the equation in the form y = ax.
2x – 3y = 0
–3y = –2x
2x
y= 3
Write original equation.
Subtract 2x from each side.
Simplify.
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
Warm-Up1Exercises
EXAMPLE
Identify direct variation equations
b.
–x + y = 4
y = x+4
Write original equation.
Add x to each side.
ANSWER
Because the equation –x + y = 4 cannot be rewritten
in the form y = ax, it does not represent direct
variation.
Warm-Up
Exercises
GUIDED
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
Exercises
GUIDED
PRACTICE
for Example 1
Tell whether the equation represents direct variation. If
so, identify the constant of variation.
2.
2x + y = 0
ANSWER
direct variation; –2
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Tell whether the equation represents direct variation. If
so, identify the constant of variation.
3.
4x – 5y = 0
ANSWER
4
direct variation;
5
Warm-Up2Exercises
EXAMPLE
Graph direct variation equations
Can the y-intercept
Graph the direct variation equation.
for the graph
of a direct variation
2
a. y = x
b. y = –3x
equation be a
3
number other than
zero?
SOLUTION
a.
Plot a point at the origin.
The slope is equal to the
constant of variation, or 2 .
Find and plot a second 3
point, then draw a line
through the points.
What is the relationship of
slope to the constant of
variation?
Warm-Up2Exercises
EXAMPLE
Graph direct variation equations
b.
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.
Warm-Up3Exercises
EXAMPLE
Write and use a direct variation equation
The graph of a direct variation
equation is shown.
a. Write the direct variation equation.
b. Find the value of y when x = 30.
What can you tell about the
SOLUTION Constant of variation from looking at
the graph?
a. 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.
What do you need from a graph to
write a direct variation equation?
y = ax
Write direct variation equation.
Substitute.
2 = a (–1)
Solve for a.
–2 = a
Warm-Up3Exercises
EXAMPLE
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
Exercises
GUIDED
PRACTICE
4.
for Examples 2 and 3
Graph the direct variation equation.
y = 2x
ANSWER
Warm-Up
Exercises
GUIDED
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
3
y = x; 36
2
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
SALTWATER AQUARIUM
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.
•
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?
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
SOLUTION
STEP 1
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
100 = a(20)
5=a
Write direct variation equation.
Substitute.
Solve for a.
ANSWER
A direct variation equation that relates w and s is
s = 5w.
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
STEP 2
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 gallon
fish tank.
Warm-Up
Exercises
GUIDED
PRACTICE
6.
for Example 4
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-Up5Exercises
EXAMPLE
Use a direct variation model
ONLINE MUSIC
The table shows the cost C of
downloading s songs at an
Internet music site.
a.
Explain why C varies directly
with s.
b.
Write a direct variation
equation that relates s and C.
Warm-Up5Exercises
EXAMPLE
Use a direct variation model
SOLUTION
a.
To explain why C varies directly with s, compare the
ratios C for all data pairs (s, C ):
s
2.97 4.95 6.93
=
=
= 0.99.
3
5
7
Because the ratios all equal 0.99, C varies directly
with s.
b.
A direct variation equation is C = 0.99s.
Warm-Up
EXAMPLE
5Exercises
for of
Example
5
Find a rate
change
GUIDED
PRACTICE
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.
Daily
Homework
Quiz
Warm-Up
Exercises
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
1.
5x – 6y = 2
ANSWER
2.
no
x+y = 0
ANSWER
yes, –1
Daily
Homework
Quiz
Warm-Up
Exercises
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.
• A direct variation
equation is of
the form y 5 ax, with a not
equal
to 0.
• The graph of a direct
variation
equation is a line through
the
origin with slope a.
• Essential Question:
How do you write and
graph direct variation
equations?
Use facts in a real-world situation
or the coordinates of a point for
the values of x and y in the direct
variation equation y = ax. Then
solve for the constant of variation.
To graph a direct variation equation,
plot a point at the origin and
then use the constant of variation,
which is the slope, to plot a
second point.