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) = x4 - 12x3 + 48x2 – 64x

1st: Find all your intercepts

X-intercept:0 = x4 - 12x3 + 48x2 – 64x0 = x(x - 4)3

x = 0x = 4

: (0, 0)Multiplicity 1Crosses: (4, 0)Multiplicity 3Crosses

Y-intercept:f(x)=0(0, 0)

2nd: Find all your critical numbersDerivative:f’(x) = 4x3 - 36x2 + 96x - 64

0 = 4x3 - 36x2 + 96x - 64

x = 1x = 4

: (1, -27): (4, 0)

0 = 4(x - 1)(x - 4)2

Interval (-∞, 1)

(1,4) (4, ∞)

Test Value x = 0 x = 2 x = 5

Sign of f’(x) f’(0) = -64 f’(2) = 16 f’(5) = 16

Conclusion Decreasing Increasing Increasing

Relative Min: (1, -27)

Relative Max: None

Example 1. Graph f(x) = x4 - 12x3 + 48x2 – 64x

3rd: Find all your inflection pointsSecond Derivative:f’’(x) = 12x2 - 72x + 96

0 = 12x2 - 72x + 96

x = 2x = 4

: (2, -16): (4, 0)

0 = 12(x - 2)(x - 4)

Interval (-∞, 2) (2, 4) (4, ∞)

Test Value x = 0 x = 3 x = 5

Sign of f’’(x)

f’’(0) = 96 f’’(3) = -12

f’’(5) = 36

Conclusion Concave Up

Concave Down

Concave Up

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: ,27

Example 2. Graph f(x) = x3 - 6x2 + 3x + 10

1st: Find all your intercepts

X-intercept:0 = x3 - 6x2 + 3x + 100 = (x + 1)(x – 2)(x - 5)x = -1x = 2

: (-1, 0)Multiplicity 1Crosses: (2, 0)Multiplicity 1Crosses

Y-intercept:f(x) = 10(0, 10)

2nd: Find all your critical numbersDerivative:f’(x) = 3x2 - 12x + 3

0 = 3(x2 - 4x + 1)x = 1x = 4

: (3.73, -10.39): (.27, 10.39)

Interval (-∞, .27) (.27, 3.73) (3.73, ∞)

Test Value x = 0 x = 2 x = 5

Sign of f’(x) f’(0) = 3 f’(2) = -9 f’(5) = 18

Conclusion Increasing Decreasing Increasing

Relative Min: (3.73, -10.39)

Relative Max: (.27, 10.39)

x = 5: (5, 0)Multiplicity 1Crosses

3rd: Find all your inflection pointsSecond Derivative:f’’(x) = 6x - 12

0 = 6x - 12

x = 2: (2, 0)

Interval (-∞, 2) (2, ∞)

Test Value x = 0 x = 3

Sign of f’’(x)

f’’(0) = -12 f’’(3) = 6

Conclusion Concave DownConcave Up

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:

Example 2. Graph f(x) = x3 - 6x2 + 3x + 10

,

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

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