Inverse Variation

1
Chapter 9

• Section 1
• Inverse Variation

2
Direct Variation (from Ch. 2)

• Direct Variation a linear function defined by
  an equation of the form y
  kx, where k ? 0
• Constant of Variation is represented by k
• How would you solve for k?
• Notice that as x increases so does y

3
Identifying a Direct Variation from a Table
Remember

• For each function, determine whether y varies
  directly with x. If so, find the constant of
  variation and write the equation.

b)
a)
x y
1 4
2 7
5 16
x y
2 8
3 12
5 20
Since the ratios are not equal this is not a
direct variation. y does not vary directly with x.
Since the fractions are equal, the constant of
variation (k) is 4. The equation is y 4x.
4
Inverse Variation

• An inverse Variation is a function of the form
  where k ? 0
• How would you solve for k?
• k yx
• Notice that if x increases, y decreases

5
Suppose that x and y vary inversely, and x
3 when y -5. Write the function that models
this situation.

1. Write the general form
2. Substitute the values of x and y
3. Solve for k
4. Put it in the general form

6
Suppose that x and y vary inversely, and x
0.3 when y 1.4. Write the function that models
the variation.
7
Suppose that x and y vary inversely, and x
7 when y 4. Write the function that models the
variation.
8
Each ordered pair is from an inverse variation.
Find the missing value.

• (2,4) and (6,y)

• (4,6) and (x,3)

9
Identifying Direct and Inverse Variations from
x-y charts

1. Divide each pair to see if the relation is Direct
2. Multiply each pair to see if the relation is
   Inverse
3. If Neither work answer neither

10
Is the relationship between the variables in each
table a direct variation, an inverse variation,
or neither? Write the function to model the
direct and inverse variations.
x 0.5 2 6
y 1.5 6 18
A)
11
x 0.2 0.6 1.2
y 12 4 2
12
x 1 2 3
y 2 1 .5
13
Is the relationship between the variables in each
table a direct variation, an inverse variation,
or neither? Write the function to model the
direct and inverse variations.
x 0.8 0.6 0.4
y 0.9 1.2 1.8
A)
x 2 4 6
y 3.2 1.6 1.1
B)
x 1.2 1.4 1.6
y 18 21 24
C)
14
Combined Variations

• Combined Variations include direct and inverse
  variations in more complicated relationships
• Notice that all the general forms have a k, but
  it is not mentioned in the words
• DO NOT WRITE THE BULLETS BELOW
• Copy the chart on the next slide on a notecard
  you will be able to use this on the test
• Note that the variables and constants used are
  examples these are guidelines for the questions
  you will need to answer, not an all inclusive list

15
Combined Variation Equation Form
y varies directly with the square of x
y varies inversely with the cube of x
z varies jointly with x and y
z varies jointly with x and y and inversely with w
z varies directly with x and inversely with the product of w and y
16
Write the function that models each relationship.
Find z when x 4 and y 9

• z varies directly with the square of x and
  inversely with y. When x 2, and y 4, z 3.

1. Write a variation, using the chart if necessary.
   Dont forget the k.
2. Substitute known values
3. Solve for k
4. Rewrite general variation substituting in the
   number for k.

17
Write the function that models each relationship.
Find z when x 4 and y 9

• z varies directly with x and inversely with y.
  When x 6 and y 2, z 15.

• z varies jointly with x and y. when x 2 and y
  3, z 60.

18
Homework

• Workbook
• Practice 9.1 1-32 all
• Show all work, dont take the shortcut
• Remember that sometimes x and y are presented in
  (x, y) format. Use your head and think of the
  information given.