2.3 Polynomial Functions & Their Graphs Objectives –Identify polynomial functions. –Recognize characteristics of graphs of polynomials. –Determine end behavior. –Use factoring to find zeros of polynomials. –Identify zeros & their multiplicities. –Use Intermediate Value Theorem. –Understand relationship between degree & turning points. –Graph polynomial functions. Pg.297 #18-62 (every other even), 74, 76 For #42-62 skip part d.

The highest degree in the polynomial is the degree of the polynomial. The leading coefficient is the coefficient of the highest degreed term. Even-degreed polynomials have both ends opening up or both opening down together. Odd-degreed polynomials open up on one end and down on the other end.

Odd Degree Polynomials point opposite directions: They fall to the left and rise to the right when the leading coefficient is positive. y = 6x 3 y = 10x 5 y = 3x 7 + 4x Leading Coefficient: Degree: 3 5 7

Odd Degree Functions point opposite directions: They rise to the left and fall to the right when the leading coefficient is negative. y = -2x 5 y = -7x 3 y = -x 9 Leading Coefficient: Degree: 5 3 9

Even Degree Polynomials point in the same direction. They rise to the left and the right when the leading coefficient is positive. y = 2x 6 + 7x y = 5x 8 y = x 4 +4x 3 +4x 2 Leading Coefficient: 25 1 Degree: 6 8 4

Even Degree Polynomials point in the same direction. They fall to the left and the right when the leading coefficient is negative. y = -3x 4 + 8x 3 y = -x x y = -5x 4 – 2 Leading Coefficient: Degree: 4 6 4

Which function could possibly coincide with this graph?

Use the Leading Coefficient Test to determine the end behavior of the following graphs: Left BehaviorRight Behavior 1.Y = 2x x 2.Y = -3x 9 – 8 3.Y = 17x 5 4.Y = -x 16

Zeros of Polynomials When f(x) crosses or “bounces off” the x-axis. How can you find them? –Let f(x)=0 and solve. –Graph f(x) and see where it touches the x-axis. What if f(x) just touches the x-axis, doesn’t cross it, then turns back up (or down) again? The zero stems from a square term (or some even power). We say this has a multiplicity of 2 (if squared) or 4 (if raised to the 4 th power).

Finding zeros by factoring A. Find all the zeros of f(x) = x 3 + 2x 2 -4x -8

B. Find the zeros of f(x) = x 4 -4x 2

Finding zeros of multiplicity If a multiple zero factor is raised to an even degree, the graph touches the x- axis and turns around. If a multiple zero factor is raised to an odd degree, the graph crosses through the x-axis. C. Find the zeros of f(x) = -4 (x+3) 2 (x-5) 3 D. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

This is the graph of f(x) = -4 (x+3) 2 (x-5) 3 Notice the graph only touches the x-axis at -3. It crosses the x-axis at 5.

Graph, State the Zeros & the End Behavior End Behavior: 3 rd degree equation and the leading coefficient is negative, so f(x) goes UP as you move to the left and f(x) goes DOWN as you move to the right. Zeros: x = 0, x = 3 of multiplicity 2

Intermediate Value Theorem If f(x) is positive (above the x-axis) at some point and f(x) is negative (below the x- axis) at another point, then f (x) = 0 (crosses through the x-axis) at some point in between. E. Show that the polynomial function f(x) = 3x 3 – 10x + 9 has a real zero between -3 and -2.

Turning Points of a Polynomial If a polynomial is of degree “n”, then it has at most n-1 turning points. Graph changes direction at a turning point. State the maximum number of turning points each graph could show F.Y = 2x x G.Y = -3x 9 – 8 H.Y = 17x 5 I.Y = -x 16

Graphing a Polynomial Function 1.Use the Leading Coefficient Test to determine the graph’s end behavior. 2.Find the x-intercepts by setting f(x) = 0 and solving. 3.Find the y-intercept by computing f(0). 4.Check to make sure the maximum number of turning points have not been exceeded. J. Use the above strategy to graph f(x) = x 3 – 3x 2 by hand.

K. Use the four step strategy to graph the function below by hand.