8.1 Model Inverse and Joint Variation
Goal  Use inverse variation and joint variation models.
RECALL: Back in Ch2 we learned about direct variation, where y = ax (y equals a number times x with
nothing added or subtracted, and a was the constant of variation)
VOCABULARY
a
Inverse variation: Two variables x and y show inverse variation if they are related as follows: y  , a  0.
x
(y equals a number divided by x with nothing added or subtracted)
Constant of variation : The nonzero constant a in a variation equation. Solving for a gives us some
insight about how x and y are related in an inverse variation:
Joint variation : When a quantity varies directly with the product of two or more other quantities
Example 1--Classify direct and inverse variation
Tell whether x and y show direct variation, inverse variation, or neither.
Given Equation
Rewritten Equation
Type of Variation
_______________
a.
y
x
9
___________
b.
xy = 3
____________
c.
5𝑦 = 𝑥
____________
_______________
Checkpoint Tell whether x and y show direct variation, inverse variation, or neither.
y
1. y = x + 2
2. yx = 5
x
3.
2.6
Example 2--Write an inverse variation equation
The variables x and y vary inversely, and y = 3 when x = 6.
A) Write an equation that relates x and y. B) Find y when x = 9.
Write general equation for inverse variation.
Substitute for y and for x.
Solve for a.
To finish part A), plug in your constant back into the general equation so it looks like this
To finish part B), use your new equation and plug in the new x value to find the new y value.
A) The inverse variation equation is y =________
B) When x = 9, y =
Checkpoint Complete the following exercise.
4.
The variables x and y vary inversely, and y = 4 when x = 5.
A) Write an equation that relates x and y.
B) Find y when x = 2.
Example 3--Check data for inverse variation
Determine whether m and n show inverse variation. If they do, write a model that gives n as a
function of m. Find n when m = 45.
m
5
10
15
20
25
n
45
22.5
15
11.25
9
Calculate the product m  n for each data pair in the table.
Are they the same each time?
What does this number represent?
Each product is equal to _______. So, the data __________ inverse variation. A model relating m and
n is
m  n = _______ or n =
______
The value of n when m = 45 is n =
Checkpoint Does the data below show inverse variation? If so, write a model that gives y as a function of x.
x
2
4
6
8
10
y
18 9
6
4.5 3.6
JOINT VARIATION
Joint variation occurs when a quantity varies directly with the product of
___________________ other quantities. In the equation below, a is a nonzero
constant.
z = ___________
z varies jointly with x and y.
Example 4--Write a joint variation equation
The variable z varies jointly with x and y. Also, z = 84 when x = 4 and y = 3.
A)Write an equation that relates x, y, and z.
B)Find z when x = 5 and y = 2.
Write the general joint variation equation. Use the given values of z, x, and y to find the constant of
variation a.
z = axy
Substitute for z, x, and y.
Simplify.
Solve for a.
Be sure to rewrite the general equation with only the constant substituted back in.
A) The joint variation equation is
B) Calculate z when x = 5 and y = 2 using substitution in your new equation.
z = ________ = _________ = ________
Example 5--Compare different types of variation
Write an equation for the relationship.
Relationship
a.
m varies jointly with n, p, and q.
b.
r varies inversely with s.
c.
x varies inversely with the cube of y.
d.
k varies jointly with x and y and inversely with m.
e.
t varies directly with u and inversely with w.
Equation
Checkpoint Complete the following exercises.
6.
The variable z varies jointly with x and y. Also, z = 44 when x = 4 and y = 1. A)Write an
equation that relates x, y, and z. B)Find z when x = 6 and y = 3.
7.
Write an equation for the relationship: x varies jointly with y and z and inversely with the
square of t.