3.2 Graphs of Polynomial Functions of Higher Degree

Continuous Polynomial Function- no breaks, holes, or gaps; only smooth rounded turns (no sharp turns like )

Sketching by hand… *** Must be in Standard Form*** There will be 2 cases

Case 1: If n is even (therefore even degree), the graph has a shape similar to The Right and Left Hand Behavior: If the leading coefficient is positive, the graph rises to the left and right. If the leading coefficient is negative, the graph falls to the left and right.

Case 2: If n is odd (therefore odd degree), the graph has a shape similar to The Right and Left Hand Behavior: If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right.

Describe the right-hand and left-hand behavior.

x - a is a factor of If x = a is a zero, then: x = a is a solution for Zeros of Polynomial Functions (x values): For a polynomial of degree n, f has at most n -1 turning points (where the graph goes from increasing to decreasing and vice versa a.k.a EXTREMA) and f has at most n real zeros. (a, 0) is an x intercept of the graph of f.

Find all zeros.

Given a factor of, there is a repeated zero at x = a, of multiplicity k. If k is odd, the graph crosses the x axis at x = a If k is even, the graph touches the x axis at x = a (bounces)

Steps: Determine the right and left hand behavior. Factor to determine the zeros. Use the multiplicity factor to determine how the zeros affects the graph (crosses through or touches x axis). Sketch the graph.

Graph Without a Calculator

Intermediate Value Theorem If a<b and, on [a,b], f takes on every value between. This can be used to approximate the real zero. Example: