4.9 Modeling with Polynomial Functions

Example 1 Write the cubic function whose graph is shown. Use the three x-intercepts to write the function in factored form.  

Example 2 Write the cubic function of whose graph is shown Use the three x-intercepts to write the function in factored form. Double root  

Example 3 Write the cubic function whose graph passes through the given points.   Use the three x-intercepts to write the function in factored form. Graph  

Finite Differences are the differences of consecutive y-values when the x-values in a data set are equally spaced. Properties   If the nth differences of equally-spaced data are nonzero and constant, then the data can be represented by a polynomial function of degree n.

Example 4 Use finite differences to determine the degree of the polynomial function that fits the data x 1 2 3 4 5 6 7 10 20 35 56 84 1 4 10 20 35 56 84 Write function values for equally spaced x values. 3 6 10 15 21 28 First differences 3 4 5 6 7 Second differences 1 Third differences Because the third differences are nonzero and constant, you can model the data exactly with a cubic function. Use Calculator: Enter the data into a graphing calculator and use cubic regression to obtain a polynomial function  

x 1 2 3 4 5 6 7 10 20 35 56 84    

Scatter Plot on TI-84  

Curve of the best fit using Scatter Plot After you entered the table, go to Y= , PLOT 1, Graph Remember to clear any equations that are already in Y= We want to turn the diagnostic feature ON, so we can see the coefficient of Determination, or This will give us the idea how strong the model is. The closer it to 1, the better the model is. Press 2nd, 0 for Catalog and scroll down for Diagnostic ON, Enter, Enter, Stat, Calc, 6 for Cubic Regression. For most TI-84 calculators: Go down to Store RegEQ line. Press Vars and go to Y-Vars. Select Function and then Y1. Press Calculate or press Enter twice. For TI-84 Plus Silver ONLY: cubReg should now be on the screen. Press Vars, Y-Vars, Function, Y1, and then press Enter again. Graph. You will see the curve passing through your scatter plot. To find f(3): VARS, right arrow, ENTER, ENTER, then type (3), ENTER.

Example 5 Use finite differences to determine the degree of the polynomial function that fits the data x 1 2 6 15 22 21 Write function values for equally spaced x values. 9 7   First differences   Second differences   Third differences Because the third differences are nonzero and constant, you can model the data exactly with a cubic function. Enter the data into a graphing calculator and use cubic regression to obtain a polynomial function  

x 1 2 6 15 22 21    

x 1 2 6 15 22 21  

After you entered the table, go to Y= , PLOT 1, Graph x 1 2 6 15 22 21 After you entered the table, go to Y= , PLOT 1, Graph Remember to clear any equations that are already in Y= We want to turn the diagnostic feature ON, so we can see the coefficient of Determination, or This will give us the idea how strong the model is. The closer it to 1, the better the model is. Press 2nd, 0 for Catalog and scroll down for Diagnostic ON, Enter, Enter, Stat, Calc, 6 for Cubic Regression. For most TI-84 calculators: Go down to Store RegEQ line. Press Vars and go to Y-Vars. Select Function and then Y1. Press Calculate or press Enter twice. For TI-84 Plus Silver ONLY: cubReg should now be on the screen. Press Vars, Y-Vars, Function, Y1, and then press Enter again. Graph. You will see the curve passing through your scatter plot.

  t 1 2 3 4 5 6 y 2622 2701 2807 3010 3117 3267 t 7 8 9 10 11 12 y 3493 3866 3951 4286 4421 4316 Step 1 Enter the data into a graphing calculator and make a scatter plot. Degree of polynomial function =number of turning points +1 The data suggest a cubic model because it has two turning points

Step 2 Use the cubic regression feature to obtain a cubic function.   t 1 2 3 4 5 6 y 2622 2701 2807 3010 3117 3267 t 7 8 9 10 11 12 y 3493 3866 3951 4286 4421 4316 Step 2 Use the cubic regression feature to obtain a cubic function. For most TI-84 calculators: Go down to Store RegEQ line. Press Vars and go to Y-Vars. Select Function and then Y1. Press Calculate or press Enter twice. For TI-84 Plus Silver ONLY: cubReg should now be on the screen. Press Vars, Y-Vars, Function, Y1, and then press Enter again.  

  t 1 2 3 4 5 6 y 2622 2701 2807 3010 3117 3267 t 7 8 9 10 11 12 y 3493 3866 3951 4286 4421 4316   For most TI-84 calculators: Go down to Store RegEQ line. Press Vars and go to Y-Vars. Select Function and then Y1. Press Calculate or press Enter twice. For TI-84 Plus Silver ONLY: cubReg should now be on the screen. Press Vars, Y-Vars, Function, Y1, and then press Enter again.

  t 1 2 3 4 5 6 y 2622 2701 2807 3010 3117 3267 t 7 8 9 10 11 12 y 3493 3866 3951 4286 4421 4316 4385   The approximate total U.S. biomass energy consumption in 2013 was about 4385 trillion Btus. For most TI-84 calculators: Go down to Store RegEQ line. Press Vars and go to Y-Vars. Select Function and then Y1. Press Calculate or press Enter twice. For TI-84 Plus Silver ONLY: cubReg should now be on the screen. Press Vars, Y-Vars, Function, Y1, and then press Enter again. 13