5.8 Analyze Graphs of Polynomial Functions

Zeros, Factors, Solutions, and Intercepts. Let 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. 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.

Graph the function: Solution: Since (x+2) and (x-1) are factors… -2 and 1 are x-intercepts… Plotting a few more points will determine the complete graph.

Graph the function: Solution: Local maximum Local mimimum Since (x+2) and (x-1) are factors… -2 and 1 are x-intercepts… We will call these points turning points. The turning points of the graph of a polynomial function are another important characteristic of the graph.

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 it’s graph has exactly n – 1 turning points. You can use a graphing calculator to find the maximums and minimums of quadratic functions. If you take calculus, you will learn analytic techniques for finding maximums and minimums.

Use the graphing calculator to: *Identify the x-intercepts *Identify where the local maximum and minimum occur.