Algebra II Chapter 8 Note Packet 1 – Evaluation 1
Inverse Variations and combined variations
Recall a direct variation function in chapter 2 was a function that could be expressed in
the form “y = k x” where “k” was called the constant of proportionality. To find “k” you
could substitute a known “x” and “y” value into the general direct variation equation and solve
for “k”. Once you know the “k” value you can write a specific direct variation equation you can
use to find other values on the same model.
There are other types of variation equations besides direct variation equations. One of
which is called an Inverse Variation. An inversed variation: “y” varies inversely with “x”, is
k
a variation equation that has the form y  where k is the constant of variation that can be
x
determined by plugging in a known “x” and “y” value into the general variation equation form of
an inverse variation. Once you know the “k” value you can write a specific inverse variation
equation that you can use to find other values on the same model.
Two other types of variation equations are Joint Variations which has the form y=kxz.
A joint variation function is when y varies directly with more than one variable. The final type
kx
of variation equation is a combined variation of the form y  . A combined variation is
z
when y varies directly with one or more variables and inversely with one or more variables.
Solving an Inverse, Joint, or combined variation equation
To solve any variation equations including direct variation equations you can use the
following four steps:
Step 1
Write the general variation equation described
Step 2
Plug in known values into equation from step 1 to find the “k”
constant of variation.
Step 3
Rewrite the variation equation from step 1 replacing the “k”
value found in step 2 into the model.
Step 4
Use the equation from step 3 to help you solve the problem by
substituting known values into your specific variation model
1
Example1 Inverse variation problem: Suppose y varies inversely with x and y = 12 when
x=4. Find y when x = 7.
Solution
k
x
k
k  48
Step 2: Find “k” 12 
4
48
Step 3: Rewrite it: y 
x
48
6
y
y6
Step 4: Use it:
7
7
Step 1: Write it: y 
Example 2: Joint variation problem: Suppose “y” varies jointly with “x” and “z” and y= 24
when x = 3 and z= 4. Find “y” when x= 5 and z= -1.
Step 1: Write it: y  kxz
24  k (3)(4)
Step 2: Find it: 24  12k
2k
Step 3: Rewrite it: y  2 xz
y  2(5)( 1)
y  10
Example 3: Combined variation problem: Suppose y varies directly with “x” and inversely
with the square of “z”. If y= 12 when x = 2 and z=2, find y when x = 3 and z=4.
Step 4: Use it:
kx
z2
k (2)
12  2
2
2k
Step 2: Find it: 12 
4
48  2k
Step 1: Write it: y 
24  k
Step 3: Rewrite it:
Step 4: Use it:
24x
z2
24(3)
y
(4) 2
72
y
16
9
1
y  or 4
2
2
y
2
Graphing Inverse variations
k
is a unique graph in that there are
x
two branches to it. The two branches are in quadrants I and III when “k” is positive or quadrants
II and IV when “k” is negative. The graph also has boundary lines called asymptotes or holes.
There’s a vertical asymptote at “x=0” (y-axis) and a horizontal asymptote at “y=0” (x-axis) )
k
for any rational function of the form " y  "
x
The graph of an inverse variation of the form y 
Steps to Graphing y 
k
. (Easiest Variety)
x
Step 1: Select 3 positive “x” values and plug them into the function and solve for their
corresponding “y” values and graph.
Step 2: Find 3 points on the second branch of the function by graphing the opposite “x”
and “y” from step 1.
Step 3: Connect the 3 points of each branch making sure to have the branches converge
towards their vertical and horizontal asymptotes
Graph the following:
Example 1.
y
4
x
Example 2
y
6
x
3
Graphing Transformations of Inverse Variations
To graph a transformation of an inverse variation function of the form y 
a
 k you
xh
k
table and then apply the transformation rule to each
x
“x” value in the parent function using the “h” value in the new function. If the “h” value is
positive you will add “h” to each parent function “x” and if the “h” value is negative you will
subtract “h” from each parent function “x”. If the “k” value is positive you will add “c” to every
“y” value from the parent function and if the “k” value is negative you will subtract “c” from
every “y” of the parent function. If there is an “h” value in the inverse variation function the
vertical asymptote will shift “h” units. If there is a “k” value in the function the horizontal
asymptote will shift “k” units.
will start with the “parent function” y 
Examples of Graphing Inverse Variation transformations
Example 1:
y
3
2
x 3
Example 2:
y
4
3
x 1
4
Practice Problems
Solve the following Variation Equations
1. Suppose y varies inversely as x. If y = 12 when x = -1 find y when x = 3.
2. Suppose y varies directly as the square of x. If y = 4 when x =2, find y when x is –2.
3. Suppose y varies jointly with x and z. If y = 30 when x = 3 and z = 5, find y when x = 2 and z = 7.
4. Suppose y varies jointly with x cubed and z squared. If y = 8 when x = 2 and z = -1
find y when x = 1 and z=2.
5. Suppose y varies directly with x and inversely with z squared. If y = 12 when x = 4 and
z=2 find y when x = 3 and z = -3.
5
Graph the following inverse variations on the coordinate plane provided. Make sure to
show asymptotes and or holes in the graph.
6.
y
1
x
7.
y
5
3
x
8.
y
4
2
x 1
6