1. Warm Up
What do you know about the graph of
f(x) = (x – 2)(x – 4)2 ?

2. Analyzing Graphs of Polynomial Functions
5.4
Analyzing Graphs of Polynomial Functions

3. Zeros, Factors, Solutions, and Intercepts
Let f(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x1 + a0 be a polynomial function. The following statements are equivalent.
Zero: k is a zero of the polynomial function f.
Factor: x – k is a factor of the polynomial f(x).
Solution: k is a solution of the polynomial equation f(x) = 0.
x-intercept: k (if it is real) is an x-intercept of the graph of the polynomial function f.

4. Find approximate real solutions for polynomial equations by graphing
1. Put polynomial equation into standard form.
2. Set equal to y
3. Graph
4. Find the zeros/ real solutions
table – y-value of 0
calc – zero

5. Example 1:
x3 + 6x2 – 4x – 24 = 0
b. x4 – 9x2 = 0

7. Example 2
Determine consecutive values of x between which each real zero of the function f(x) = x4 – x3 – 4x2 + 1 is located using the table below.

8. To identify key points on a graph of a polynomial function
Turning points are called local maximums and local minimums because they are the highs or lows for the area.
If there are two turning points without a zero between them  there are imaginary zeros.

9. Example 3:
Estimate the coordinates of each turning point and state whether each corresponds to a local max or local min. Then list all real zeros and determine the least degree the function can have.

10. Example 4
Graph each function. Identify the x-intercepts (zeros), local maximums, and local minimums.
a. f(x) = x3 + 2x2 – 5x + 1 x y

11. b. f(x) = 2x4 – 5x3 – 4x2 – 6 x y

12. Example 5
The weight w, in pounds, of a patient during a 7-week illness is modeled by the function
w(n) = 0.1n3 – 0.6n , where n is the number of weeks since the patient became ill.
Graph the equation by making a table of values for weeks 0 through 7. Plot the points and connect with a smooth curve.
Describe the turning points of the graph and its end behavior.

13. There is a relative minimum at week 4.
w(n) → ∞ as n → ∞.