0301 ch 3 day 1
- 1. Chapter 3 Polynomial & Rational Functions Romans 8:6 For to set the mind on the flesh is death, but to set the mind on the Spirit is life and peace.
- 2. 3.1 Polynomial Functions and Their Graphs
- 3. 3.1 Polynomial Functions and Their Graphs Standard Form of a polynomial
- 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. Review the graphs of polynomial functions of increasing degree
- 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. 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. 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. 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. 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. 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. 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. 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. End Behavior
- 15. End Behavior How does the graph behave as x gets very large or very small ...
- 16. End Behavior How does the graph behave as x gets very large or very small ... x→∞ x approaches positive infinity (right end)
- 17. End Behavior How does the graph behave as x gets very large or very small ... x→∞ x approaches positive infinity (right end) x → −∞ x approaches negative infinity (left end)
- 18. The end behavior of a polynomial function is determined by the term that contains the highest power of x (the variable)
- 19. The end behavior of a polynomial function is determined by the term that contains the highest power of x (the variable) The other terms become insignificant as x→∞
- 20. Use a grapher to show that 3 2 f (x) = 8x + 7x + 3x + 7 and f (x) = 8x have the same end behavior 3
- 21. Use a grapher to show that 3 2 f (x) = 8x + 7x + 3x + 7 and f (x) = 8x have the same end behavior 3 as x → ∞, y → ∞ as x → −∞, y → −∞
- 22. Real Zeros of Polynomials If P is a polynomial and c is a real number and c is a zero of P Then
- 23. Real Zeros of Polynomials If P is a polynomial and c is a real number and c is a zero of P Then 1) x = c is a solution of P(x) = 0
- 24. Real Zeros of Polynomials If P is a polynomial and c is a real number and c is a zero of P Then 1) x = c is a solution of P(x) = 0 2) (x − c) is a factor of P(x)
- 25. Real Zeros of Polynomials If P is a polynomial and c is a real number and c is a zero of P Then 1) x = c is a solution of P(x) = 0 2) (x − c) is a factor of P(x) 3) x = c is an x-intercept of P(x)
- 26. Graph: y = x(x − 4)(x + 2)
- 27. Graph: y = x(x − 4)(x + 2) 3 x-intercepts (3 zeros)
- 28. Graph: y = x(x − 4)(x + 2) 3 x-intercepts (3 zeros) 3 solutions to 0 = x(x − 4)(x + 2)
- 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. Intermediate Value Theorem If P is a polynomial and P(a) and P(b) have opposite signs, then there is at least one value c between a and b such that P(c)=0.
- 31. Intermediate Value Theorem If P is a polynomial and P(a) and P(b) have opposite signs, then there is at least one value c between a and b such that P(c)=0.
- 33. Multiplicity of Roots Consider: y = (x − 3)(x − 3)(x + 1)
- 34. Multiplicity of Roots Consider: y = (x − 3)(x − 3)(x + 1) Zeros are:
- 35. Multiplicity of Roots Consider: y = (x − 3)(x − 3)(x + 1) Zeros are: x = 3, x = 3, x = −1
- 36. Multiplicity of Roots Consider: y = (x − 3)(x − 3)(x + 1) Zeros are: x = 3, x = 3, x = −1 Multiplicity of 2
- 37. Multiplicity of Roots Consider: y = (x − 3)(x − 3)(x + 1) Zeros are: x = 3, x = 3, x = −1 Multiplicity of 2
- 38. Multiplicity of Roots Consider: y = (x − 3)(x − 3)(x + 1) Zeros are: x = 3, x = 3, x = −1 Multiplicity of 2 Graph the function to “see” the multiplicity
- 40. Multiplicity of Roots When a graph “bounces” ... even multiplicity (2, 4, 6, ... ) When a graph flattens out and goes through the axis ... odd multiplicity (3, 5, 7, ...)
- 41. Multiplicity of Roots When a graph “bounces” ... even multiplicity (2, 4, 6, ... ) When a graph flattens out and goes through the axis ... odd multiplicity (3, 5, 7, ...) Graph: y = (x + 3)(x + 3)(x + 3)(x − 4)
- 42. HW #1 “If we are to go only halfway or reduce our sights in the face of difficulty ... it would be better to not go at all.” John F. Kennedy
Editor's Notes
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- #24: 1. Discuss how the calculator uses this theorem to find the zero in the Calc menu.\n2. Discuss and show the SGN ERR error on the calculator.\n
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