1. 6.8 Analyzing Graphs of Polynomial Functions

2. Zeros, Factors, Solutions, and Intercepts
Let f(x) = anxn + an-1xn-1 + … + a1x + a0 be a polynomial function. Zero: k is a zero of the polynomial f(x) Factor: x – k is a factor of the polynomial equation f(x) = 0 Solution: k is a solution of the polynomial equation f(x) If k is a real number, then the following is also equivalent. x-intercept: k is an x-intercept of the graph of the polynomial function f.

3. Zeros, Factors, Solutions, and Intercepts
Graph: f(x) = -2(x2 – 9)(x + 4)
Graph: f(x) = x(2x – 5)(x + 5)
Graph: f(x) = (x + 3)(x – 4)(x + 1)
Graph: f(x) = 2(x – 1)(x + 2)3

4. Turning Points of Polynomial Functions
Local maximum- point higher than all nearby points
Local minimum- point lower that all nearby points
Turning Points of Polynomial Functions
The graph of every polynomial function of degree n has at most n – 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly n – 1 turning points

5. Examples
Graph each function. Identify the x-intercepts, local maximums, and local minimums
f(x) = x3 + 2x2 – 5x + 1
f(x) = 2x4 – 5x3 – 4x2 – 6
f(x) = x3 – 5x2 + 4x + 3
f(x) = 3x4 – 4x3 – x2 + 2x – 5

6. Examples
You want to make a rectangular box that is x cm high, (x + 5) cm long, and (10 – x) cm wide. What is the greatest volume possible? What will the dimensions of the box be?

7. Examples
A rectangular box is to be x in. by (12 – x) in. by (15 –x) in. What is the greatest volume possible? What will the dimensions of the box be?