Algebra 2/Trig
Name: ______________________________ Date: __________ Block: ______
Section 9.1: Direct and Inverse Variation
Direct Variation:
 x and y show direct variation if

k is called the ___________________________________.

Another phrase for “Direct Variation” is ______________________________________.

The graph of y = kx goes through the _____________ .

An example formula is ________________
- Distance and time vary directly, which means as time increases, distance increases OR as time
decreases, distance decreases.
The variables x and y vary directly. Write an equation that relates the two variables.
Then find y when x = 6.
1.) x = 2, y = 7
2.) x = 24, y = 6
3.) x = .8, y = -1.6
4.) If y varies directly with x, and y = 3 when x = 2, then what is the value of y when x = 6?
A) 1
B) 2
C) 3
D) 6
E) 9
5.) Suppose y is directly proportional to x2. If y = 16 when x = 2, what is the value of y when x is 7?
6.) If y is directly proportional to x2 and y = 18 when x = 3, which of the following could be the value of x
when y = 8?
A) -4
B) -2
C) 1
D) 4/3
E) 6
7.) If y varies directly with x, and y = 24 when x = 8, then what is the value of x when y = 33?
8.) If y varies directly with x, and y = 52 when x = 13, then what is the value of x when y = 14?
9.) If y varies directly with the table to the right, then when x = 5.5, what is the value of y?
10.) Which of the following tables shows a direct variation of y in terms of x?
A)
B)
C)
D)
E)
x
1
2
3
4
y
3
6
9
12
Inverse Variation
 x and y show inverse variation if

Another phrase for “Inverse Variation” is ______________________________________.

An example formula is ________________
- Density and volume vary indirectly, which means as volume increases, density decreases.
The variables x and y vary inversely. Use the given values to write an equation relating x and y.
Then find y when x = 4.
11.) y = 6, x = 1.5
12.) y = -7.5, x = 2
13.) y = 2, x = 8
14.) If y is inversely proportional to x, and y = 3 when x = 2, then what is the value of y when x = 6?
A) 1
B) 2
C) 3
D) 6
E) 9
15.) Suppose y varies indirectly to the square of x. When x is 2, y is equal to 25. What is the value of y when x
is 10?
A) 1
B) 4
C) 5
D) 10
E) 25
16.) If y varies inversely with x and y = 4 when x = 2, then which of the following defines y in terms of x?
Choose all that apply.
2
8
A) y  2 x
B) xy  8
C) y 
D) y  8 x
E) y 
x
x
17.) If y varies indirectly with x 2 and y = 9 when x = 3, then what is the value of x when y = 18?
18.) Which of the following expresses a relationship where y is inversely proportional to x?
A)
B)
C)
D)
E)
Direct variation, Inverse variation or Neither?
19.) xy = 4.8
20.) x 
y
1.5
21.) y = 2x + 3
22.) x = 3y
Applications:
Use direct and inverse variation to solve the following problems.
23.)
The amount of money earned on a job is directly proportional to the number of hours worked. If $36 is
earned for 8 hours of work, how much is earned for 30 hours of work?
24.)
The height that a ball bounces varies directly with the height from which it is dropped. A certain ball
bounces 30 cm when dropped from a height of 50 cm. How high will the ball bounce if dropped from a
height of 120 cm?
25.)
The amount of fertilizer needed for a lawn varies directly with the area of the lawn. If 4 pounds of
fertilizer are needed for 500 square feet of lawn, how much fertilizer is needed for Bob’s lawn, which is
rectangular in shape and measures 25 feet by 50 feet?
26.)
For rectangles with the same area, the length varies inversely as the width. One rectangle has a length of
12 cm and a width of 5 cm. Find the length of another rectangle with the same area whose width is 4 cm.
27.)
The current in an electrical circuit varies inversely as the amount of resistance in the circuit. The current
is 10 amps when the resistance is 24 ohms. Find the current when the resistance is 30 ohms.
28.)
The cost per person to rent a mountain cabin is inversely proportional to the number of people who share
the rent. If the cost is $36 per person when 5 people share, what is the cost per person when 8 people
share?
9.1 Homework:
Name: ________________________________ Date: ______________
Direct and Inverse Variation
****** Show your work on a separate sheet of paper!
Tell whether x and y show direct variation, inverse variation, or neither.
2.) y 
1.) y  3x
2
x
3.) x  y  7
4.) xy  5
5.)
y
9
x
The variables x and y vary directly. Use the given values to write an equation that relates x and y. Then
find y when x = 3.
6.) x = 4, y = 32
7.) x = 36, y = 9
8.) x = -2, y = 8
9.) x = -8, y = 12
The variables x and y vary inversely. Use the given values to write an equation that relates x and y. Then
find y when x = 4.
10.) x = 2, y = 4
11.) x = -3, y = 3
12.) x = 16, y =
1
4
13.) x = 10, y =
1
2
Use direct and inverse variation to solve the following application problems. Identify x and y first.
14.) The amount that a spring stretches is directly proportional to the weight of the object attached to it. If a
spring is stretched 10 cm by a weight of 8 kg, how much will it be stretched by a weight of 3 kg?
15.)
The number of kilograms of water in a person’s body varies directly as the person’s mass. A person
with a mass of 90 kg contains 60 kg of water. How many kilograms of water are in a person with a mass
of 50 kg?
16.)
On a certain map, 25 km are represented by 2 cm. If two cities are 7 cm apart on the map, what is the
actual distance between them?
17.)
The number of calories in a container of milk is directly proportional to the amount of milk in the
container. If there are 160 calories in an 8-ounce glass of milk, find the number of calories in a 15ounce glass of milk.
18.)
The volume of a gas varies inversely as the pressure. A helium-filled balloon has a volume of 21 m3 at
sea level, where the pressure is 1 atmosphere. The balloon rises to an altitude where the pressure is 0.7
atmospheres. What is the volume?
19.)
The number of chairs in a ski lift is inversely proportional to the distance between them. A lift has 70
chairs when they are 24 m apart. If 80 evenly-spaced chairs are used on the lift, how much space will be
left between them?
20.)
For piano wires under the same tension, the number of vibrations per second (frequency) of each wire is
inversely proportional to the length of the wire. A wire 0.75 m long vibrates 480 times per second.
How long is a wire that vibrates 300 times per second?
21.)
The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If the
trip takes 6 hours at 900km/h, how long would it take at 800km/h?
Determine whether x and y show direct variation, inverse variation, or neither.
22.)
23.)
24.)
25.)