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13.1: Inverse Variation

Difficulty Level: At Grade Created by: CK-12
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This activity is intended to supplement Algebra I, Chapter 12, Lesson 1.

Part 1 - Enter the Data

Enter the data from the table into lists.

Press STAT ENTER. Enter the \begin{align*}x\end{align*}x column in \begin{align*}L1\end{align*}L1 and the \begin{align*}y\end{align*}y column in \begin{align*}L2\end{align*}L2 as shown.

\begin{align*}x\end{align*}x \begin{align*}y\end{align*}y
1 \begin{align*} 24\end{align*}24
2 \begin{align*}12\end{align*}12
3 \begin{align*}8\end{align*}8
4 \begin{align*}6\end{align*}6
5 \begin{align*}4.8\end{align*}4.8
6 \begin{align*}4\end{align*}4

Press \begin{align*}Y=\end{align*}Y=, and select Plot1.

Press ENTER to turn the plot On. Select scatter as the type of plot, \begin{align*}L1\end{align*}L1 for the Xlist, and \begin{align*}L2\end{align*}L2 for the Ylist.

Press WINDOW. Set the window to the following:

\begin{align*}Xmin = 0,\ Xmax = 10,\ Xscl = 2\end{align*}Xmin=0, Xmax=10, Xscl=2

\begin{align*}Ymin = 0,\ Ymax = 25,\ Yscl = 5\end{align*}Ymin=0, Ymax=25, Yscl=5

Press GRAPH.

Part 2 - Questions

  • How would you describe the relationship between \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y by examining this data?

Press STAT ENTER to return to the lists.

  • What relationships can you see by examining the numbers in the lists?
  • What is the product of each pair of numbers?

Arrow to the top of \begin{align*}L3\end{align*}L3. Enter a formula to multiply the entries in \begin{align*}L1\end{align*}L1 by the entries in \begin{align*}L2\end{align*}L2. Press \begin{align*}2^{nd}\ [L1]\end{align*}2nd [L1] for \begin{align*}L1\end{align*}L1 and press \begin{align*}2^{nd}\ [L2]\end{align*}2nd [L2] for \begin{align*}L2\end{align*}L2. \begin{align*}L3 = L1*L2\end{align*}L3=L1L2

Press ENTER to execute the formula. The product in each case is \begin{align*}24\end{align*}24. So, \begin{align*}L1 \cdot L2 = 24\end{align*}L1L2=24 or \begin{align*}x \cdot y = 24\end{align*}xy=24. This relationship, when \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y have a constant product, is called “inverse variation.”

  • What type of situation might this be a formula for?

Solve the equation \begin{align*}x \cdot y = 24\end{align*}xy=24 for \begin{align*}y\end{align*}. Press \begin{align*}Y=\end{align*}. Enter the equation into \begin{align*}Y1\end{align*}.

  • What is your equation?

Press GRAPH.

  • What other information can you find from the graph of the equation that you could not gather from the plot?
  • Does this graph appear to be a function? Explain.

Press \begin{align*}2^{nd}\end{align*} [TABLE] to examine the function table.

  • What is happening when \begin{align*}x = 0\end{align*}? Why?

Arrow up to the negative \begin{align*}x-\end{align*}values in the table.

  • What do you notice about the \begin{align*}y-\end{align*}values?
  • Why does this occur?
  • What do you think the graph of your equation looks like to the left of the \begin{align*}y-\end{align*}axis?

Press WINDOW. Set the window as shown to examine the graph when \begin{align*}x\end{align*} is negative.

Press GRAPH.

  • What appears to be happening when \begin{align*}x = 0\end{align*}?
  • Why does the graph of the equation not appear in Quadrants II or IV?
  • Do you think an inverse variation can ever be found in Quadrants II or IV? Why?
  • Does this graph appear to be a function now? Explain.

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Date Created:
Feb 22, 2012
Last Modified:
Oct 31, 2014
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