# 13.1: Inverse Variation

**At Grade**Created by: CK-12

*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*} column in \begin{align*}L1\end{align*} and the \begin{align*}y\end{align*} column in \begin{align*}L2\end{align*} as shown.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

1 | \begin{align*} 24\end{align*} |

2 | \begin{align*}12\end{align*} |

3 | \begin{align*}8\end{align*} |

4 | \begin{align*}6\end{align*} |

5 | \begin{align*}4.8\end{align*} |

6 | \begin{align*}4\end{align*} |

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

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

Press **WINDOW**. Set the window to the following:

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

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

Press **GRAPH**.

## Part 2 - Questions

- How would you describe the relationship between \begin{align*}x\end{align*} and \begin{align*}y\end{align*} 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*}. Enter a formula to multiply the entries in \begin{align*}L1\end{align*} by the entries in \begin{align*}L2\end{align*}. Press \begin{align*}2^{nd}\ [L1]\end{align*} for \begin{align*}L1\end{align*} and press \begin{align*}2^{nd}\ [L2]\end{align*} for \begin{align*}L2\end{align*}. \begin{align*}L3 = L1*L2\end{align*}

Press **ENTER** to execute the formula. The product in each case is \begin{align*}24\end{align*}. So, \begin{align*}L1 \cdot L2 = 24\end{align*} or \begin{align*}x \cdot y = 24\end{align*}. This relationship, when \begin{align*}x\end{align*} and \begin{align*}y\end{align*} 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*} 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.