Model: y=-5.867x^3+125.533x^2-629.267x+1421 Population in 2008: 1097
Practice makes perfect
There are two parts to this question. First, we need to use a cubic regression to get a function model for the data points, then we need to use that model to predict what the population was in 2008. Let's begin by plotting the values using our graphic calculator. Let's treat 2000 as x=0 in our model and the population as y.
year
x
y
2004
4
457
2007
7
910
2009
9
1244
2010
10
1315
Now, to enter our values into the calculator, push STAT, choose Edit, and enter your values.
Once the values have been entered, we can plot them by pushing 2nd and Y= and choosing one of the plots in the list. Make sure you turn the plot ON, choose scatterplot as the type, and use L1 and L2 as XList and YList. Finally, you can pick whatever mark you want.
By pushing GRAPH the calculator will plot the dataset. A standard viewing window might not show the data points, so we can change it if necessary.
We will use the cubic regression to find the polynomial of best fit.
Performing the Regression
By pressing STAT you can find the cubic regression under the CALC menu. If we choose CubicReg the calculator performs a cubic regression test.
Now we can write the cubic function.
y=-5.867x^3+125.533x^2-629.267x+1421
Using the Model to Predict Population
To find the population in 2008, we need to find the value of the function when x=8. We can use the trace or the table function on the calculator. First, let's plot the function. We can press Y= and enter our function.
Next, press the TRACE button and move the cursor until you see x=8. Use the arrow buttons to move the trace and the up and down buttons to toggle between the plotted points and the function.
To get a more precise answer, we can press 2nd then GRAPH to see the table of values.
Scroll down until you see x=8, then look at the y-value on that row. We can see that y=1097.2 when x=8. This means that, in 2008, the model predicts the population as approximately 1097.
