Algebra – Unit 4 - Slope
Notes for HW 4 on Inverse Variation
Inverse Variation
Direct Variation
y = kx
Inverse Variation
y=
Constant of Variation
The non-zero constant “k” is called the constant of variation
k
x

direct variation: variables vary in the same way – either both variables increase or
both decrease

inverse variation: variables vary in the opposite way – when one increases, the other
decreases
Exercise: Solve y = kx and y 
y = kx
k
for k:
x
k = ______________
y
k
x
k = __________
Example: Identify direct or inverse variation (or neither!).

To identify, see if you can put the equations in y = kx or y =
a) xy = -4
b)
y
=x
2
k
form.
x
c) y = 2x + 3
Inverse Variations Graphs



direct variation: the graph is a line that always passes
through the origin (y = kx)
k

inverse variation: the graph is a hyperbola  y  
x

An inverse variation hyperbola never passes through the
origin

An inverse variation hyperbola never crosses the x or y axis;
the x and y axes are called the asymptotes of the hyperbola

If k is negative, the inverse variation branches are in opposite quadrants
Example: Identify which graphs show direct variation, inverse variation, or neither…
Example: Identifying an Inverse Variation From a Table
Given the table at right, does it describe an inverse
variation?

To find if a table of values describes an inverse variation, multiply the x and y values.

If we get the same constant value, then we have an inverse variation.

Inverse variation? ___________ What is k? __________ What is the equation? _________

Summary:
o direct variation: divide y by x to see if there is a constant value
o inverse variation: multiply x and y to see if there is a constant value
Example: Direct, Inverse, or Neither? If direct or inverse, what are the equations?
x
y
3
15
5
25
3.5
17.5
x -10
y 1.4
2
-7
-1
14
x
y
10
5
16
8
8
2
Example: Writing Inverse Variation Equations
The variables x and y vary inversely, and y = 6 when x = -3.
a) Write an inverse variation equation that relates x and y.
b) Find the value of y when x = 4.
Example: Word problems
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 can complete the job. Find the work time per person if the
company hires 30 people.
The length of a violin string varies inversely as the frequency of its vibrations. A violin string
14 inches long vibrates at a frequency of 450 cycles per second. Find the frequency of a 12inch violin string.