1. 6-7: Investigating Graphs of Polynomial Functions.
In - CLass

2. Example 4: Determine Maxima and Minima with a Calculator
Graph f(x) = 2x3 – 18x + 1 on a calculator, and estimate the local maxima and minima. –5
–25 25 5
Step 1 Graph.
Step 2 Find the maximum.
Press to access the CALC menu. Choose 4:maximum.
The local maximum is approximately

3. Graph f(x) = 2x3 – 18x + 1 on a calculator, and estimate the local maxima and minima.
Step 3 Find the minimum.
Press to access the CALC menu. Choose 3:minimum.
The local minimum is approximately –

4. Example 4: Now You Try
Graph g(x) = x3 – 2x – 3 on a calculator, and estimate the local maxima and minima.
The local maximum is approximately –
The local minimum is approximately –

5. 6-9: Curve Fitting with Polynomial Functions.
In - CLass

7. Example 1: Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data. x 4 6 8 10 12 14 y –2
4.3
8.3
10.5
11.4
11.5
Step 1: Determine if x-values are evenly spaced x 4 6 8 10 12 14

8. Example 1: Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data.
Step 2: Find the difference of the y-values until the differences are constant y –2
4.3
8.3
10.5
11.4
11.5
First differences: Not constant
Second differences: –2.3 –1.8 –1.3 – Not constant
Third differences: Constant
Step 3: Identify the type of graph that best describes the data.
A cubic polynomial best describes the data.

9. Example 1: Now You Try
Use finite differences to determine the degree of the polynomial that best describes the data. x –6 –3 3 6 9 y –9 16 26 41 78
151
The fourth differences are constant. A quartic polynomial best describes the data.

10. Example 2: Using Finite Differences to Write a Function
The table below shows the population of a city from 1960 to Write a polynomial function for the data.
Year
1960
1970
1980
1990
2000
Population (thousands)
4,267
5,185
6,166
7,830
10,812
Step 1 Find the finite differences of the y-values.
First differences:
Second differences:
Third differences: Close

11. Step 2 Determine the degree of the polynomial.
Because the third differences are relatively close, a cubic function should be a good model.
Step 3 Use the cubic regression feature on your calculator.
f(x) ≈ 0.10x3 – 2.84x x

12. Example 2: Now You Try Speed 25 30 35 40 45 50 55 60 Gas (gal) 23.8
The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data.
Speed 25 30 35 40 45 50 55 60
Gas (gal)
23.8
25.2
25.4 27
30.6 37
f(x) ≈ 0.001x3 – 0.113x x

13. Example 3: Curve Fitting with Polynomial Functions
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2000.
Year
1994
1995
1996
1997
1998
1999
Price ($)
683
652
948
1306
863
901
Step 1 Choose the degree of the polynomial model.
Let x represent the number of years since 1994.
Use the regression feature to check the R2-values.
quadratic: R2 ≈ , cubic: R2 ≈ , quartic: R2 ≈
The quartic function is more appropriate choice.

14. Step 2 Write the polynomial model. The data can be modeled by f(x) = 32.23x4 – x x2 – x
Step 3 Find the value of the model corresponding to 2000.
2000 is 6 years after Substitute 6 for x in the quartic model.
f(6) = 32.23(6)4 – (6) (6)2 – (6)
Based on the model, the opening value was about $ in 2000.

15. Example 3: Now You Try
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 1999.
Year
1994
1995
1996
2000
2003
2004
Price ($)
3754
3835
5117
11,497
8342
10,454
Based on the model, the opening value was about $11, in 1999.