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(2.3) Polynomial Functions and Their Graphs

(2.3) Polynomial Functions and Their Graphs. Identifying End Behavior Finding x-intercepts Using the Intermediate Value Theorem Graphing Polynomial Functions. End Behavior. Leading Coefficient. End Behavior. Best Graph Description. End Behavior. Example. Degree. Up Right.

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(2.3) Polynomial Functions and Their Graphs

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  1. (2.3) Polynomial Functions and Their Graphs Identifying End Behavior Finding x-intercepts Using the Intermediate Value Theorem Graphing Polynomial Functions

  2. End Behavior Leading Coefficient End Behavior Best Graph Description End Behavior Example Degree Up Right Line with a Positive Slope Odd Positive Down Left Down Right Line with a Negative Slope Odd Negative Up Left

  3. End Behavior Leading Coefficient End Behavior Best Graph Description End Behavior Example Degree Quadratic with a Positive Coefficient Up Right Even Positive Up Left Down Right Quadratic with a Negative Coefficient Even Negative Down Left

  4. Identifying End Behavior Odd Positive Up Right; Down Left Even Negative Down Right; Down Left Even Positive Up Right; Up Left Down Right; Up Left Odd Negative

  5. Analyzing Zeros (x-intercepts) x-intercept behavior Graphic Representation Multiplicity Graph goes THROUGH the x-intercept Odd Multiplicity Graph TURNS at the x-intercept Even Multiplicity

  6. Finding and Analyzing x-intercepts Multiplicity of Two Goes Through the x-axis at x-intercept -2 Turns at the x-axis at 2

  7. Finding and Analyzing x-intercepts Multiplicity of Two Goes Through the x-axis at x-intercept -5 Turns at the x-axis at x-intercepts -5 and -1

  8. Intermediate Value Theorem for Polynomials Take f(a) and f(b). If there is a sign change between the two values then there exists a zero between a and b.

  9. Using the Intermediate Value Theorem Sign changes so there IS a zero (x-intercepts) between x = 0 and x = 1

  10. Using the Intermediate Value Theorem Sign changes so there IS a zero (x-intercepts) between x = 1 and x = 2

  11. Graphing Polynomial Functions • Leading Coefficient Test:Analyze the end behavior. (Ex: #’s 1-4) • Find the x-intercepts:Solve the polynomial and analyze the zeros. (Ex: #’s 5-6) • Find the y-intercept:Substitute 0 in for x • Determine the symmetry:Is the function even (y-axis symmetry), odd (origin) symmetry, or neither. • Find a few addition points and turning points:Use a table of values if needed. The maximum number of turning points will be 1 LESS THAN THE DEGREE of the polynomial.

  12. Odd Negative Down Right; Up Left Graph goes THROUGH x-intercepts 0 and 3

  13. The y-intercept is 0

  14. y 2 -1 x 1 -2

  15. Odd Negative Down Right; Up Left Graphgoes THROUGH x-intercept 2and TURNS at x-intercept -2

  16. The y-intercept is 4

  17. y 1 -1 x 1 -1

  18. Even Positive Up Right; Up Left Graphgoes THROUGH x-intercepts -5 and 3and TURNS at x-intercept -1

  19. y 1 -1 x 1 -1

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