Let's organize all of the given data into lists on the graphing calculator. By pushing $STAT$ and choosing the first option, EDIT, we can fill out the first two lists.

Is It A Good Model?

We can transform an exponential function into a linear function by taking the logarithm of each side. In general, a linear function is easier to recognize than an exponential function. If the logarithm of the function's $y -$ values exhibits a straight line, an exponential model will be a good fit.

To calculate the logarithm of our $y -$ values, place the cursor on L3. Push $LOG,$ $2nd,$ and then 2 to select L2.

Once the values have been entered, we can plot them by pushing $2nd$ and $Y =,$ then choosing one of the plots in the list. Make sure you turn the plot ON, choose scatter plot as the type, and set L1 and L3 as XList and YList. Finally, you can pick whatever mark you want.

By pushing $GRAPH$ the calculator will plot L1 against L3. However, first we should change the window settings. The $y -$ values of L3 are all very close to $0,$ so to get a better picture we limit the $y -$ axis to $0 < y < 3 .$ Also, the $x -$ values of L1 are all in the first quadrant, so we will limit our $y -$ axis to $x > 0 .$ We can change the settings in $WINDOW .$

The points do look like they follow a straight line. Therefore, we can conclude that an exponential function is a good model.

Finding Our Exponential Function

To perform a regression on an exponential function, we push $STAT$ and scroll right to the second option, CALC. In this menu we see all of the available regressions the calculator can perform. To perform an exponential regression, choose ExpReg.

Using the results we got, we can say the exponential function is $y = 5.58 (1.48)^{x} .$

Exercise