Algebra I
Name:
SOL A.8 Direct and Inverse Variation
Block:
Date:
Inverse Variation
Essential Quet
Essential Question(s): What are the characteristics of an inverse variation equation? How can you tell
if data in a table represents inverse variation? Compare/contrast direct and inverse variation.
Inverse Variation: ____________________________
k 0
We read this “y varies inversely as x” or “y varies inversely with x”
“k” is called the constant of variation or constant of proportionality
When “k” is positive:
OR
x
increases
As
As
y
decreases
x
decreases
y
increases
Examples:
1)
y=
3
𝑥
is inverse variation.
The constant of variation is ________.
2) xy = 5 is inverse variation
The constant of variation is ________.
Graphs of Direct Variation and Inverse Variation Equations
Direct Variation
y= kx, k > 0
y = kx, k < 0
Inverse Variation
k
x
y= ,k>0
k
x
y= ,k<0
Unlike the graph of direct variation, the graph of inverse variation is not linear.
The graph of an inverse variation is a hyperbola.
Note that the lines never cross the axes -- they get closer and closer to x = 0 and y = 0,
but x and y never equal zero.
Branches of the hyperbola: the two symmetrical parts of the graph
Asymptotes: the lines the hyperbola approaches but doesn’t intersect. The asymptotes
k
of the graph of y = x are the x-axis and the y-axis.
Identify Direct and Inverse Variation
Tell whether the equation represents direct variation, inverse variation, or neither.
If it represents direct or inverse variation, identify the constant of variation.
1)
xy = 8
2)
𝑦
4
=x
3) 3x + y = 5
Write and Use an Inverse Variation Equation:
Given that y varies inversely with x, use the specified values to write an inverse variation
equation that relates x and y.
1)
x = 3, y = 6
2) The variables x and y vary inversely, and y = 8 when x = 4.
a) Write an inverse variation equation that relates x and y.
b)
Find the value of y when x = 5.
3) In kick boxing, it is found that the force, f, needed to break a board, varies inversely
with the length, L, of the board. If it takes 5 lbs of pressure to break a board 2 feet long,
how many pounds of pressure will it take to break a board that is 5 feet long?
How to identify the type of variation from a table.
Direct Variation: ________________. Solve for k:
A table shows direct variation if
𝑦
𝑥
𝑦
𝑥
gives the same answer for each x,y pair.
= constant
x
2
4
6
8
y
4
8
12
16
Inverse Variation: ________________. Solve for k:
A table shows inverse variation if xy gives the same answer for each x,y pair.
xy = constant
x
-5
-3
4
8
y
2.4
4
-3
-1.5
Practice: Tell whether the table represents direct variation, inverse variation, or neither.
If it is direct or inverse variation, write the equation.
1)
2)
3)
x
y
-3
-2
-1
-6
0.5
12
6
1
x
y
-7
-21
1
3
3
9
6
18
x
y
-1
-2
1
2
2
6
4
7
4) Select three points that will create a relation that has inverse variation.
(1,20) (2, 10) (2, 40) (4, 5) (4, 80) (5, 100)
Summary:
Extra Practice:
Tell whether the equation represents direct variation, inverse variation, or neither.
x
3
1)
xy = 4
2)
2x – y = 1
3)
y
4)
y
3
x
5)
4y = 3x
6)
x = 4y
7) If y varies inversely with x and y = 4 when x = 3, find y when x = 2.
8) If y varies inversely with x and y = -2 when x = -2, find y when x = 12.
9) A theatre company plans to hire people to build a stage set. The work time t (in hours per person)
varies inversely with the number of p people hired to work. The company estimates that 25 people working
for 300 hours can complete the job.
a) Write an equation that relates “t” and “p.”
b) How long will it take in hours if the company hires 30 people?
10) The number of hours, h, it takes for a block of ice to melt varies inversely with the temperature, t.
If it takes 2 hours for a square inch of ice to melt at 65º, how long will it take a square inch of ice to melt at 52º.