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# 2.1: Modeling Data

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Calculus, Chapter 1, Lesson 3.

## Problem 1 – Exponential Growth

When Connor was born, his parents put \$1,000\begin{align*}\1,000\end{align*} dollars into an account to give him as a present on his 21st birthday. However, his parents forgot the yearly interest rate on the account.

Enter the data to the right in L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*} by pressing STAT ENTER.

Create a scatter plot of the data by pressing 2nd [Y=]\begin{align*}2^{nd} \ [Y=]\end{align*} ENTER and matching the screen to the right.

To view the scatter plot, press ZOOM and select 9:ZoomStat.

Using your knowledge of compound interest, study the data and the graph to determine a function for the growth of Connor’s money.

Enter your equation in Y1\begin{align*}Y1\end{align*} and press GRAPH to check your result.

Note: the regressions can be found by pressing STAT and scrolling over to the CALC menu. Select ExpReg, ENTER 2nd\begin{align*}2^{nd}\end{align*} [1] [,] 2nd\begin{align*}2^{nd}\end{align*} [2] ENTER.

The equation for the data is: ______________

## Problem 2 – Logarithmic Growth

Scientists are testing the amount of greenhouse gases present at a research site near the north pole to determine the effect on polar ice melting. Enter in the results in L3\begin{align*}L3\end{align*} and L4\begin{align*}L4\end{align*} of your graphing calculator.

Create a scatter plot of the data by pressing 2nd [Y=]\begin{align*}2^{nd} \ [Y=]\end{align*}, selecting 1:Plot1, and matching the screen to the right.

To view the scatter plot, press ZOOM and select 9:ZoomStat.

Note: L3\begin{align*}L3\end{align*} and L4\begin{align*}L4\end{align*} are found by pressing 2nd\begin{align*}2^{nd}\end{align*} [3] and 2nd\begin{align*}2^{nd}\end{align*} [4] respectively.

Determine a natural log equation to model the data by pressing STAT, scrolling over to CALC and selecting 9:LnReg.

To finish the regression, enter, L3,L4,Y1\begin{align*}L3, L4, Y1\end{align*} so the screen reads LnReg L3,L4,Y1\begin{align*}L3, L4, Y1\end{align*}

Press GRAPH to view the scatter plot and regression equation both plotted.

The equation for the data is: ______________

## Extension – Exponential Decay

Due to an environmental spill, a farmer is losing the amount of land on which he can plant crops. The data in L5\begin{align*}L5\end{align*} and L6\begin{align*}L6\end{align*} of your graphing calculator show the year and amount of useable land for each year. Determine an equation that models the amount of land the farmer can use each year.

After creating a scatter plot of the data, use your graphing calculator to perform an exponential regression that models the data and plot the function by pressing GRAPH.

The equation for the data is: ______________

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