Inverse Variation
Objectives: To solve inverse variation
Write an equation given a point.
Find the missing coordinate
To compare inverse variation and direct variation
Solve application problems involving inverse variation.
Suppose you are part of a volunteer crew constructing affordable housing. Building a house
requires a total of 160 workdays. For example, a crew of 20 people can complete a house in 8
days.
1. How long should it take a crew of 40 people?
2. Complete the table below.
Crew Size
(x)
2
5
8
20
40
Construction Days
(y)
80
Total Workdays
16
8
160
160
160
160
3. Graph the (x, y) data in the table above.
4. Describe what happens to construction time as the crew size increases.
Inverse Variation
k
, where k ≠ 0, is an inverse variation. The constant of
x
variation for direct variation k is the product of x and y for an ordered pair (x, y)
A function in the form xy = k or y 
Inverse Variations all have graphs with the same general shape.
How does k affect the shape of the graph?
Examples
Graph:
xy = 2
xy = 6
xy = 12
Describe the general shape of the graph of an
inverse variation.
How is it different than a graph with direct
variation? (Like y = 2x)
Examples – Suppose y varies inversely with x. Write an equation for the inverse variation.
a. y = 6 when x = 3
b. y = 7 when x = 8
d. y = 9 when x = 2
e. y = 7 when x = 5
c. y = 6 when x = 1/3
Suppose (x1,y1) and (x2,y2) are ordered pairs of an inverse variation. Each ordered pair
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Example – Finding the Missing Coordinate
Each pair of points is on the graph of an inverse variation. Find the missing value.
a. (3,8) and (2,y)
b. (3,y) and (5,9)
c. (75, 0.2) and (x, 3)
Example – Real world Problem Solving
a. The weight needed to balance a lever varies inversely with the distance from the fulcrum to
the weight. Where should Julio, who weighs 150 lb, sit to balance the lever?
b. A 100-lb weight is placed 4 ft from a fulcrum. How far from the fulcrum should a 75-lb weight
be placed to balance the lever?
c. An 80-lb weight is placed 9 ft from the fulcrum. What weight should you put 6 ft from the
fulcrum to balance the lever?
Comparing Direct and Inverse Variation
Direct Variation: y = kx
Inverse Variation: xy = k
Graph the following equations on the
graph to the right. Label and Identify
as Direct or Inverse Variation. (Make a
table of values to graph)
1. y = 3x
2. y 
4
x
3. y  
4. y 
x
2
8
x
Determine whether the data in each table represent a direct variation or an inverse variation.
How?? Look at the relationship between x and y values. Which equation does it fit:
y = kx or xy = k?
a.
b.
x
2
4
10
y
5
10
25
c.
x
3
6
9
x
5
10
25
y
20
10
4
x
3
5
8
y
12
20
32
d.
y
12
6
4
Examples – Application Problems
Explain whether each situation represents a direct or an inverse variation.
a. The cost of $20 worth of gasoline is split among several people.
b. You buy several markers for 70¢ each.
c. You are in a discount store. All sweaters are on sale for $15 each.
d. You walk 5 miles each day. Your speed and time vary from day to day.