Name:
Ms. D’Amato
Date:
Block:
Direct and Inverse Variation
A
is a function that can be represented in the form y = ax.

Say “
“

a is called the

The graphs will always pass through the

If x
If x
, then y
, so does the value of y.
. Both variables change in the same manner.
Equations of Direct Variation:
Determine whether each equation below represents a direct variation. If so, identify the
constant of variation.
1. 2x – 3y = 0
2. x + y = 5
3. 4x + 5y = 0
4. –x + y = 4
5. 2x + y = 0
6. 5x + y = 9
Graphs of Direction Variation:
Direct variation y = ax is a form of the equation y = mx + b (a is the slope and the y-intercept is
0). Graph the direct variation equations below:
1. y 
2
x
5
2. y  3x
y
3. y 
y
x
1
x
2
y
x
x
Determine whether the graph represents a direct variation. Explain your answer.

1.
y

2.








x









y
x





















Using a direct variation equation:
Given that y varies directly as x, use the specified values to write a direct variation equation:
1. x = 3 and y = 9
2. x = -18 and y = -4
3. x = -2 and y = -2
Given that y varies directly as x, write a direct variation equation to find the missing value:
1. If x = 24 and y = 64, find y when x = 15.
2. If x = 2 and y = 3, find x when y = 21.
Applications:
1. The number of pages (p) a student can read varies directly with the amount of time (t) in
minutes spent reading. The student can read 90 pages in 60 minutes.
a. Write an equation that relates p and t.
b. Predict the number of pages the student can read if 90 minutes is spent reading.
2. The amount of money earned at your job (m) varies directly with the number of hours (h) you
work. The first day of work you earned $57 after working 6 hours. You are trying to save
money to go to buy a new car to take to college next year. How many hours will you need to
work in order to save $4750?
3. 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 that relates w and s. How
many tablespoons of salt should be added to a 30 gallon saltwater fish tank?
A
is a function that can be represented in the form y =
𝑎
𝑥
.

Say “
“

a is called the

If x
, then y
. The values of the two variables change in
the opposite manner of direct variation. If x
, then
y
.
Tell whether the equation represents direct variation, inverse variation, or neither.
1. y = -2x
2. xy = 1
3. y = x + 5
4. xy = 5
5.
𝑦
𝑥
=4
6. 2x + y = 6
Graphs of Inverse Variation: Graph each inverse variation equation.
1. y =
x
-4
-2
-1
0
1
2
4
2
2. y =
y
𝑥
y
x
x
-4
-2
-1
0
1
2
4
−2
y
𝑥
y
x
Using an inverse variation equation:
Given that y varies inversely as x, use the specified values to write an inverse variation
equation:
1. x = 5 and y = 2
2. x = -5 and y = 4
3. x = -15 and y = -15
Given that y varies inversely as x, write an inverse variation equation and find the missing value:
1. If x = 3 and y = 7, find y when x = 2.
2. If x = 13 and y = -1, find y when x = 2.
Applications:
1. A theater company plans to hire people to build a stage set. The work time t (in hours per
person) varies inversely with the number p of people hired. The company estimates that 25
people working for 300 hours each can complete the job. Find the work time per person if the
company hires 30 people.
2. In science, one theory of life expectancy states that the lifespan of mammals varies
inversely to the number of heart beats per minute of the animal. If a gerbil's heart beats 360
times per minute and lives an average of 3.5 years, what would be the life expectancy of a
human with an average of 72 beats per minute? Does this theory appear to hold for humans?