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0301 ch 3 day 1

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0301 ch 3 day 1

  1. 1. Chapter 3 Polynomial & Rational FunctionsRomans 8:6 For to set the mind on the flesh isdeath, but to set the mind on the Spirit is lifeand peace.
  2. 2. 3.1 Polynomial Functions and Their Graphs
  3. 3. 3.1 Polynomial Functions and Their Graphs Standard Form of a polynomial
  4. 4. 3.1 Polynomial Functions and Their Graphs Standard Form of a polynomial n n−1 n−2 P(x) = an x + an−1 x + an−2 x + ... + a1 x + a0 an ≠ 0
  5. 5. Review the graphs of polynomial functions of increasing degree
  6. 6. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  7. 7. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  8. 8. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  9. 9. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  10. 10. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  11. 11. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  12. 12. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  13. 13. Review the graphs of polynomial functions of increasing degree Degree # Extrema 0 NA 1 NA 2 1 3 2 4 3 5 4 n n-1
  14. 14. End Behavior
  15. 15. End BehaviorHow does the graph behave as x getsvery large or very small ...
  16. 16. End BehaviorHow does the graph behave as x getsvery large or very small ... x→∞x approaches positive infinity (right end)
  17. 17. End BehaviorHow does the graph behave as x getsvery large or very small ... x→∞x approaches positive infinity (right end) x → −∞x approaches negative infinity (left end)
  18. 18. The end behavior of a polynomial functionis determined by the term that containsthe highest power of x (the variable)
  19. 19. The end behavior of a polynomial functionis determined by the term that containsthe highest power of x (the variable)The other terms become insignificant as x→∞
  20. 20. Use a grapher to show that 3 2 f (x) = 8x + 7x + 3x + 7and f (x) = 8x have the same end behavior 3
  21. 21. Use a grapher to show that 3 2 f (x) = 8x + 7x + 3x + 7and f (x) = 8x have the same end behavior 3 as x → ∞, y → ∞ as x → −∞, y → −∞
  22. 22. Real Zeros of Polynomials If P is a polynomial and c is a realnumber and c is a zero of P Then
  23. 23. Real Zeros of Polynomials If P is a polynomial and c is a realnumber and c is a zero of P Then1) x = c is a solution of P(x) = 0
  24. 24. Real Zeros of Polynomials If P is a polynomial and c is a realnumber and c is a zero of P Then1) x = c is a solution of P(x) = 02) (x − c) is a factor of P(x)
  25. 25. Real Zeros of Polynomials If P is a polynomial and c is a realnumber and c is a zero of P Then1) x = c is a solution of P(x) = 02) (x − c) is a factor of P(x)3) x = c is an x-intercept of P(x)
  26. 26. Graph: y = x(x − 4)(x + 2)
  27. 27. Graph: y = x(x − 4)(x + 2) 3 x-intercepts (3 zeros)
  28. 28. Graph: y = x(x − 4)(x + 2) 3 x-intercepts (3 zeros) 3 solutions to 0 = x(x − 4)(x + 2)
  29. 29. Graph: y = x(x − 4)(x + 2) 3 x-intercepts (3 zeros) 3 solutions to 0 = x(x − 4)(x + 2) This is a cubic function
  30. 30. Intermediate Value Theorem If P is a polynomial and P(a) and P(b)have opposite signs, then there is at leastone value c between a and b such thatP(c)=0.
  31. 31. Intermediate Value Theorem If P is a polynomial and P(a) and P(b)have opposite signs, then there is at leastone value c between a and b such thatP(c)=0.
  32. 32. Multiplicity of Roots
  33. 33. Multiplicity of RootsConsider: y = (x − 3)(x − 3)(x + 1)
  34. 34. Multiplicity of RootsConsider: y = (x − 3)(x − 3)(x + 1)Zeros are:
  35. 35. Multiplicity of RootsConsider: y = (x − 3)(x − 3)(x + 1)Zeros are: x = 3, x = 3, x = −1
  36. 36. Multiplicity of RootsConsider: y = (x − 3)(x − 3)(x + 1)Zeros are: x = 3, x = 3, x = −1 Multiplicity of 2
  37. 37. Multiplicity of RootsConsider: y = (x − 3)(x − 3)(x + 1)Zeros are: x = 3, x = 3, x = −1 Multiplicity of 2
  38. 38. Multiplicity of RootsConsider: y = (x − 3)(x − 3)(x + 1)Zeros are: x = 3, x = 3, x = −1 Multiplicity of 2Graph the function to “see” the multiplicity
  39. 39. Multiplicity of Roots
  40. 40. Multiplicity of RootsWhen a graph “bounces” ... even multiplicity (2, 4, 6, ... )When a graph flattens out and goesthrough the axis ... odd multiplicity (3, 5, 7, ...)
  41. 41. Multiplicity of RootsWhen a graph “bounces” ... even multiplicity (2, 4, 6, ... )When a graph flattens out and goesthrough the axis ... odd multiplicity (3, 5, 7, ...)Graph: y = (x + 3)(x + 3)(x + 3)(x − 4)
  42. 42. HW #1“If we are to go only halfway or reduce oursights in the face of difficulty ... it would bebetter to not go at all.” John F. Kennedy

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