3.2 Graphs of Polynomial Functions of Higher Degree.

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3.2 Graphs of Polynomial Functions of Higher Degree

3.2 Graphs of Polynomial Functions of Higher DegreeContinuous 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 TheoremIf a