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

2. 5.4 - Analyzing Graphs of Polynomial Functions
Day 1

3. Location Principle:

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

5. Maximum & Minimum Points
Relative Maximum – a point on the graph of a function where no other nearby points have a greater y-coordinate.
Relative Minimum - a point on the graph of a function where no other nearby points have a lesser y-coordinate.

6. Maximum & Minimum Points
Extrema – max. and min. values of a function.
Turning Point rules to remember
The graph of a polynomial function of degree n has at most n – 1 turning points.
If 2+ turning points are between roots, imaginary numbers exist

7. Example:
Graph f(x) = x3 – 3x Find the x-coordinates at which the relative maxima and relative minima occur.

8. Find Extrema on Calculator:
Enter equation into y =.
2nd Calc
Choose 3: minimum or 4: maximum.
Curser on left of min/max, enter.
Curser on right of min/max, enter.
Enter.

9. 5.4 - Analyzing Graphs of Polynomial Functions
Day 2: Real-World Problems

10. Example:
a. 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 creating a table of values.
Describe the turning points of the graph and its end behavior.
Is it reasonable to assume the trend will continue indefinitely?

11. There is a relative minimum at week 4.
As n → ∞, w(n) → ∞ .

12. Example:
The graph models the cross section of Mount Rushmore. What is the smallest degree possible for the equation that corresponds with this graph?