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
1
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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