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Direct Variation
Section 5-2
Goals
Goal
• To write and graph an
equation of a direct
variation.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Direct Variation
• Constant of variation for a direct variation
Direct Variation
A recipe for paella calls for 1 cup of rice to make 5 servings. In
other words, a chef needs 1 cup of rice for every 5 servings.
The equation y = 5x describes this relationship. In this
relationship, the number of servings varies directly with the
number of cups of rice.
Definition
• Direct Variation – a special type of linear
relationship that can represented by a function in
the form y = kx, where k ≠ 0.
• Constant of Variation – is k, the coefficient of x, in
the function y = kx.
– By dividing each side of y = kx by x, you can see that
the ratio of the variables is constant: y/x = k.
Direct Variation
Direct Variation
Determining Direct Variation
• To determine whether an equation represents a direct
variation;
1)
Given an equation.


2)
Given a table of values.


3)
Solve for y.
If you can write the equation in the form y = kx, where k ≠ 0, then it
is a direct variation.
Find y/x for each ordered pair.
If the ratio y/x is constant for all ordered pairs, then it is a direct
variation.
Given a graph.


The graph is a line and passes through the origin, then it is a direct
variation.
The slope of the line is k.
Example: Determine Direct
Variation from an Equation
Tell whether the equation represents a direct variation. If so, identify
the constant of variation.
y = 3x
This equation represents a direct variation because it is in the
form of y = kx. The constant of variation is 3.
Example: Determine Direct
Variation from an Equation
Tell whether the equation represents a direct variation. If so, identify
the constant of variation.
3x + y = 8
–3x
–3x
y = –3x + 8
Solve the equation for y.
Since 3x is added to y, subtract 3x from
both sides.
This equation is not a direct variation because it cannot be
written in the form y = kx.
Example: Determine Direct
Variation from an Equation
Tell whether the equation represents a direct variation. If so, identify
the constant of variation.
–4x + 3y = 0
+4x
+4x
3y = 4x
Solve the equation for y.
Since –4x is added to 3y, add 4x to both
sides.
Since y is multiplied by 3, divide both
sides by 3.
This equation represents a direct variation because it is in the
form of y = kx. The constant of variation is .
Your Turn:
Tell whether the equation represents a direct variation. If so, identify
the constant of variation.
3y = 4x + 1
This equation is not a direct variation because it is not written in
the form y = kx.
Your Turn:
Tell whether the equation represents a direct variation. If so, identify
the constant of variation.
Solve the equation for y.
3x = –4y
–4y = 3x
Since y is multiplied by –4, divide
both sides by –4.
This equation represents a direct variation because it is in the
form of y = kx. The constant of variation is
.
Your Turn:
Tell whether the equation represents a direct variation. If so, identify
the constant of variation.
y + 3x = 0
– 3x –3x
y = –3x
Solve the equation for y.
Since 3x is added to y, subtract 3x from
both sides.
This equation represents a direct variation because it is in the
form of y = kx. The constant of variation is –3.
Determine Direct Variation
from a Table
What happens if you solve y = kx for k?
y = kx
Divide both sides by x (x ≠ 0).
So, in a direct variation, the ratio is equal to the constant of
variation.
Another way to identify a direct variation is to check whether
is the same for each ordered pair (except where x = 0).
Example: Determine Direct
Variation from a Table
Tell whether the relationship is a direct
variation. Explain.
Find
for each ordered pair.
This is a direct variation because
pair.
is the same for each ordered
Example: Determine Direct
Variation from a Table
Tell whether the relationship is a direct
variation. Explain.
Find
for each ordered pair.
…
This is not direct variation because
ordered pairs.
is the not the same for all
Your Turn:
Tell whether the relationship is a direct
variation. Explain.
Find for each ordered pair.
This is not direct variation because
ordered pairs.
is the not the same for all
Your Turn:
Tell whether the relationship is a direct
variation. Explain.
Find for each ordered pair.
10
 4
2.5
20
 4
5
30
 4
7.5
This is a direct variation because
pair.
is the same for each ordered
Your Turn:
Tell whether the relationship is a direct
variation. Explain.
Find for each ordered pair.
This is not direct variation because
ordered pairs.
is the not the same for all
Example: Writing a Direct
Variation Equation
The value of y varies directly with x, and y = 3, when x = 9. Find
y when x = 21.
Find the value of k and then write the equation.
y = kx
Write the equation for a direct variation.
3 = k(9)
Substitute 3 for y and 9 for x. Solve for k.
Since k is multiplied by 9, divide both sides by 9.
The equation is y = x. When x = 21, y = (21) = 7.
Your Turn:
The value of y varies directly with x, and y = 4.5 when x = 0.5. Find
y when x = 10.
Find the value of k and then write the equation.
y = kx
4.5 = k(0.5)
9=k
Write the equation for a direct variation.
Substitute 4.5 for y and 0.5 for x. Solve for k.
Since k is multiplied by 0.5, divide both sides by
0.5.
The equation is y = 9x. When x = 10, y = 9(10) = 90.
Example: Graphing Direct
Variation
A group of people are tubing down a river at an average speed of 2
mi/h. Write a direct variation equation that gives the number of
miles y that the people will float in x hours. Then graph.
Step 1 Write a direct variation equation.
distance
y
=
=
2 mi/h
2
times
hours

x
Example: Continued
Step 2 Choose values of x and generate ordered pairs.
x
y = 2x
(x, y)
0
y = 2(0) = 0
(0, 0)
1
y = 2(1) = 2
(1, 2)
2
y = 2(2) = 4
(2, 4)
Step 3 Graph the points and connect.
Your Turn:
The perimeter y of a square varies directly with its side
length x. Write a direct variation equation for this
relationship. Then graph.
Step 1 Write a direct variation equation.
perimeter
=
y
=
4 sides
4
times
•
length
x
Your Turn: Continued
Step 2 Choose values of x and generate ordered pairs.
x
y = 4x
(x, y)
0
y = 4(0) = 0
(0, 0)
1
y = 4(1) = 4
(1, 4)
2
y = 4(2) = 8
(2, 8)
Your Turn: Continued
Step 3 Graph the points and connect.
Joke Time
• How would you describe a frog with a broken leg?
• Unhoppy
• What did the horse say when he got to the bottom
of his feed bag?
• That’s the last straw!
• What kind of music do chiropractors listen to?
• Hip - Pop
Assignment
• 5-2 Exercises Pg. 325 - 327: #8 – 52 even