1. 5.3
Polynomial Functions, Graphs, and Composition

2. Evaluating Polynomial Functions
Example 1
Evaluating Polynomial Functions
Answer

3. Using a Polynomial Model to Approximate Data
Example 2
The number of U.S. households estimated to see and pay at least one bill on-line each month during the years 2000 through 2006 can be modeled by the polynomial function
where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and P(x) is in millions. Use this function to approximate the number of households that paid at least one bill on-line each month in 2006.

4. Using a Polynomial Model to Approximate Data (continued)
Example 2
Answer
Since x = 6 corresponds to 2006, we must find P(6).
According to this model, in 2006, about million households paid at least one bill on-line each month.

5. Adding and Subtracting Functions
Example 3
Adding and Subtracting Functions
Let and Find each function.
(a) (f + g)(x) (b) (f – g)(x)
Answer
(a)

6. Adding and Subtracting Functions (continued)
Example 3
Adding and Subtracting Functions (continued)
(b)

7. Adding and Subtracting Functions
Example 4
Adding and Subtracting Functions
For and find each of the following.
(a) (f + g)(x) and (f + g)(–1) (b) (f – g)(x) and (f – g)(1)
Answer
(a)

8. Adding and Subtracting Functions (continued)
Example 4
Adding and Subtracting Functions (continued)
(b)

9. Finding a Composite Function
Example 5
Finding a Composite Function
Answer

10. Finding Composite Functions
Example 6
Finding Composite Functions
Find the following.
(a) (b)
Answer
(a)
(b)

11. Graphing Variations of the Identity, Squaring, and Cubing Functions
Example 7
Graph Give the domain and range.
Answer
Create a table of values. x
f(x) = –2x2 –2 –8 –1 1 2

12. Graphing Variations of the Identity, Squaring, and Cubing Functions (continued)
Example 7
Plot the points and join them with a curve.

13. Graphing Variations of the Identity, Squaring, and Cubing Functions (continued)
Example 7
Any value of x can be used, so the domain is
The maximum y-value is 0 and there is no minimum y-value, so the range is