13.1: Inverse Variation
This activity is intended to supplement Algebra I, Chapter 12, Lesson 1.
ID: 8203
Time required: 30 minutes
Activity Overview
Students will examine a set of ordered pairs that vary inversely. They will plot the ordered pairs and explore the graph and table for relationships. Students will also graph the inverse variation and determine whether it is a function.
Topic: Rational Functions & Equations
- Given the values of two variables that vary inversely, determine the rational function that relates them.
Teacher Preparation
- This activity is appropriate for an Algebra 1 classroom or review for an Algebra 2 class.
Associated Materials
- Student Worksheet: Inverse Variation, http://www.ck12.org/flexr/chapter/9622
Students are to enter the data from the table on the worksheet into lists \begin{align*}L1\end{align*} and \begin{align*}L2.\end{align*}
Be sure that students have cleared all lists before beginning. To do this, press STAT, then select ClrAllLists.
Students will then create a scatter plot of the data using Plot1.
They are to describe the relationship of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. As the \begin{align*}x-\end{align*}values increase, the \begin{align*}y-\end{align*}values decrease. The rate is not constant; the amount of change in \begin{align*}y\end{align*} decreases as \begin{align*}x\end{align*} increases.
For list \begin{align*}L3\end{align*}, students will enter the formula \begin{align*}L1*L2,\end{align*} multiplying the \begin{align*}x-\end{align*}values by the \begin{align*}y-\end{align*}values.
The formula \begin{align*}x \cdot y =24\end{align*} might be used for the area of \begin{align*}24 \ units\end{align*}, where \begin{align*}x\end{align*} is the length and \begin{align*}y\end{align*} is the width.
When students solve the equation \begin{align*}x \cdot y=24\end{align*} for \begin{align*}y\end{align*}, they should get \begin{align*}y=\frac{24}{x}\end{align*} and enter this in \begin{align*}Y1\end{align*}.
As students view the function table, they should see that the function is undefined at \begin{align*}x = 0\end{align*}.
They should also see that the negative \begin{align*}x-\end{align*}values have the same \begin{align*}y-\end{align*}values, although negative, as the positive \begin{align*}x-\end{align*}values.
When students view the entire graph, they should confirm that \begin{align*}x\end{align*} is undefined at zero (vertical and horizontal asymptotes at zero).
The graph does not appear in Quadrants II or IV because the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}values have the same sign.
An inverse variation could be in Quadrants II or IV, if the coefficient is negative.