In Lesson 5A, you considered characteristics of polynomial functions and their corresponding graphs. This lesson introduces polynomial functions as mathematical models. Starting with a realistic situation, you will use a polynomial function to construct a regression model. Then, you will decide whether the model is appropriate for the given data. If it is, you will use the model to answer questions about the situation.

The next two examples show how to graph a scatter plot and corresponding regression model using the graphing calculator. As you read the examples, complete the steps on your calculator. Contact your teacher if you have any difficulty following the examples with your calculator.

Note: It is not necessary to graph a scatter plot before finding a regression equation. This is done in in this lesson to give you some practice plotting points and to give you an idea of the general shape of the graph.

Triangular numbers can be formed by counting the number of dots used to make a triangle. The first five triangular numbers are shown in the table below. Determine the quadratic regression model for this data. Explain how you know that your model represents the data accurately.

*Note: To find a regression equation only, omit steps 1, 2, 3, and 7.

Step 1: Turn on the STAT PLOT; press 2nd, Y=, ENTER, ENTER.

Step 2: Clear the functions; press Y= then, CLEAR for each function that needs deleting.

Step 3: Select values for the viewing window; press WINDOW. Use the keypad to type the values from the table. Use the up and down arrow keys to scroll through the list.

When you graph manually, Xscl and Yscl are the values assigned to each square on graph paper. They are often left as 1, but they can be adjusted for very small or very large data values.

Step 4: Go to the lists; press STAT, ENTER.

Step 5: Clear the lists; press the up arrow until the column heading is highlighted then, CLEAR, ENTER.

Step 6: Enter the data.

Step 7: View the data; press GRAPH.

Step 8: Choose the regression model; press STAT, right arrow.

Step 9: Select the quadratic regression; press 5.

Step 10: Place the quadratic regression into the function area; press VARS, right arrow, ENTER, ENTER, ENTER.

Step 11: View the scatter plot and regression model; press GRAPH.

The quadratic regression model in standard form is y = 0.5x2 + 0.5x.

In Example 1, the regression model went through all the data points exactly. It was a perfect fit! In many real-life situations, you cannot find a model to fit data points exactly. Instead, you can find the model that best fits the data by using the regression feature on the graphing calculator.