Inverse Variation - Loudoun County Public Schools

[Pages:2]Algebra 1 Notes SOL A.8 (12.1) Inverse Variation

Mrs. Grieser

Name: _______________________________ Date: _____________ Block: _______

Inverse Variation

Direct Variation

Two variables x and y show direct variation if y = kx and k 0.

Inverse Variation

Two variables x and y show inverse variation if y = k and k 0. x

Constant of Variation The non-zero constant "k" is called the constant of variation.

In direct variation, the variables vary in the same way ? if one increases, the other increases; if one decreases, the other decreases.

In inverse variation, the variables vary in the opposite way ? if one increases, the other decreases.

Example: Identify direct or inverse variation (or neither!).

To identify, see if you can put the equations in y = kx or y = k form. x

a) xy = 4

b)

y 2

= x

c) y = 2x + 3

Graphing Inverse Variations

In direct variation, the graph is a line that always passes through the origin (y = kx). In inverse variation, the graph is a hyperbola (y = k ).

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.

Example: Graph y = 4 x

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Algebra 1 Notes SOL A.8 (12.1) Inverse Variation

Example: Graph y = - 4 x

Using the tables we used to graph y = 4 , we notice that the yx

values will be the negatives of those found in the above tables. We get the graph to the right.

Mrs. Grieser

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.

y= k x

6= k 3

Inverse variation definition; we need to find the value of a. Substitute the values of y and x that we are given. Find a.

k = -18 We found a (constant of variation) !

y

=

- 18 x

Put our value of a into the equation.

b) Find the value of y when x = 4.

y = - 18 = - 18 = - 9 x 42

Example: Identifying an Inverse Variation From a Table

Given the table on the right, does it describe an inverse variation?

y =

k x

is our definition of inverse variation.

That means that xy = k is also true (why??)

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.

Review: to find if it is a direct variation, then divide y by x to see if there is a constant

value

Example: Word problem

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.

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