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Objectives: 1. Be able to graph a polynomial function using calculus.

Objectives: 1. Be able to graph a polynomial function using calculus. Critical Vocabulary: Polynomial Function, End Behavior, Intercepts, Multiplicity, Critical Numbers, Extrema, Inflection Points, Concavity, Domain, Range. Example 1. Graph f(x) = x 4 - 12x 3 + 48x 2 – 64x.

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Objectives: 1. Be able to graph a polynomial function using calculus.

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  1. Objectives: 1. Be able to graph a polynomial function using calculus. Critical Vocabulary: Polynomial Function, End Behavior, Intercepts, Multiplicity, Critical Numbers, Extrema, Inflection Points, Concavity, Domain, Range

  2. Example 1. Graph f(x) = x4 - 12x3 + 48x2 – 64x 1st: Find all your intercepts X-intercept: 0 = x4 - 12x3 + 48x2 – 64x Y-intercept: f(x)=0 0 = x(x - 4)3 (0, 0) x = 0 : (0, 0) Multiplicity 1 Crosses x = 4 : (4, 0) Multiplicity 3 Crosses 2nd: Find all your critical numbers Derivative: f’(x) = 4x3 - 36x2 + 96x - 64 0 = 4x3 - 36x2 + 96x - 64 Relative Min: (1, -27) 0 = 4(x - 1)(x - 4)2 Relative Max: None x = 1 : (1, -27) x = 4 : (4, 0) Interval (-∞, 1) (1,4) (4, ∞) Test Value x = 0 x = 2 x = 5 f’(0) = -64 f’(2) = 16 f’(5) = 16 Sign of f’(x) Conclusion Decreasing Increasing Increasing

  3. Example 1. Graph f(x) = x4 - 12x3 + 48x2 – 64x 3rd: Find all your inflection points Second Derivative: f’’(x) = 12x2 - 72x + 96 0 = 12x2 - 72x + 96 0 = 12(x - 2)(x - 4) x = 2 : (2, -16) x = 4 : (4, 0) Interval (-∞, 2) (2, 4) (4, ∞) Test Value x = 0 x = 3 x = 5 f’’(0) = 96 f’’(3) = -12 f’’(5) = 36 Sign of f’’(x) Conclusion Concave Up Concave Down Concave Up

  4. Example 1. Graph f(x) = x4 - 12x3 + 48x2 – 64x 4th: Draw your Graph Plot your points for your X-intercepts, Y-intercept, Relative Extrema, and Inflection Points Connect the points based on the End Behavior and your 2 tables (Increasing, decreasing and Concavity) 5th: State your Domain and Range of the graph Domain: Range:

  5. Example 2. Graph f(x) = x3 - 6x2 + 3x+ 10 1st: Find all your intercepts X-intercept: 0 = x3 - 6x2 + 3x+ 10 Y-intercept: f(x) = 10 0 = (x + 1)(x – 2)(x - 5) (0, 10) x = -1 : (-1, 0) Multiplicity 1 Crosses x = 2 : (2, 0) Multiplicity 1 Crosses x = 5 : (5, 0) Multiplicity 1 Crosses 2nd: Find all your critical numbers Derivative: f’(x) = 3x2 - 12x+ 3 Relative Min: (3.73, -10.39) 0 = 3(x2 - 4x+ 1) x = 1 : (3.73, -10.39) Relative Max: (.27, 10.39) x = 4 : (.27, 10.39) Interval (-∞, .27) (.27, 3.73) (3.73, ∞) Test Value x = 0 x = 2 x = 5 f’(0) = 3 f’(2) = -9 f’(5) = 18 Sign of f’(x) Conclusion Increasing Decreasing Increasing

  6. Example 2. Graph f(x) = x3 - 6x2 + 3x+ 10 3rd: Find all your inflection points Second Derivative: f’’(x) = 6x - 12 0 = 6x - 12 x = 2 : (2, 0) Interval (-∞, 2) (2, ∞) Test Value x = 0 x = 3 f’’(0) = -12 f’’(3) = 6 Sign of f’’(x) Conclusion Concave Down Concave Up

  7. Example 2. Graph f(x) = x3 - 6x2 + 3x+ 10 4th: Draw your Graph Plot your points for your X-intercepts, Y-intercept, Relative Extrema, and Inflection Points Connect the points based on the End Behavior and your 2 tables (Increasing, decreasing and Concavity) 5th: State your Domain and Range of the graph Domain: Range:

  8. Directions: Graph the following polynomial function. Use your notes as a template as to what is expected 1. f(x) = x4 - 4x3 - 13x2 + 28x + 60

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