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2.3 Polynomial Functions & Their Graphs

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.

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2.3 Polynomial Functions & Their Graphs

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  1. 2.3Polynomial 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.

  2. 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.

  3. Odd Degree Polynomials point opposite directions: They fall to the left and rise to the right when the leading coefficient is positive. y = 6x3 y = 10x5 y = 3x7 + 4x Leading Coefficient: 6 10 3 Degree: 3 5 7

  4. Odd Degree Functions point opposite directions: They rise to the left and fall to the right when the leading coefficient is negative. y = -2x5 y = -7x3 y = -x9 Leading Coefficient: -2 -7 -1 Degree: 5 3 9

  5. Even Degree Polynomials point in the same direction. They rise to the left and the right when the leading coefficient is positive. y = 2x6 + 7x y = 5x8 y = x4 +4x3 +4x2 Leading Coefficient: 2 5 1 Degree: 6 8 4

  6. Even Degree Polynomials point in the same direction. They fall to the left and the right when the leading coefficient is negative. y = -3x4 + 8x3 y = -x6 + 12x y = -5x4 – 2 Leading Coefficient: -3 -1 -5 Degree: 4 6 4

  7. Which function could possibly coincide with this graph?

  8. Use the Leading Coefficient Test to determine the end behavior of the following graphs: • Left BehaviorRight Behavior • Y = 2x4 + -5x • Y = -3x9 – 8 • Y = 17x5 • Y = -x16

  9. 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 4th power).

  10. Finding zeros by factoring A. Find all the zeros of f(x) = x3 + 2x2 -4x -8

  11. B. Find the zeros of f(x) = x4 -4x2

  12. 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.

  13. 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.

  14. Graph, State the Zeros & the End Behavior • End Behavior: 3rd 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

  15. 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) = 3x3 – 10x + 9 has a real zero between -3 and -2.

  16. 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 = 2x4 + -5x G. Y = -3x9 – 8 H. Y = 17x5 I. Y = -x16

  17. Graphing a Polynomial Function • Use the Leading Coefficient Test to determine the graph’s end behavior. • Find the x-intercepts by setting f(x) = 0 and solving. • Find the y-intercept by computing f(0). • Check to make sure the maximum number of turning points have not been exceeded. J. Use the above strategy to graph f(x) = x3 – 3x2 by hand.

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

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