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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 – 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
0 = 4(x - 1)(x - 4)2
x = 1 : (1, -27)
x = 4 : (4, 0)
Relative Min: (1, -27)
Relative Max: None
Interval
(-∞, 1)
(1,4)
(4, ∞)
Test Value
x=0
x=2
x=5
Sign of f’(x) f’(0) = -64
Conclusion
Decreasing
f’(2) = 16
Increasing
f’(5) = 16
Increasing
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
Sign of f’’(x) f’’(0) = 96
f’’(3) = -12
Conclusion
Concave Down
Concave Up
f’’(5) = 36
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 + 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
0 = 3(x2 - 4x + 1)
x = 1 : (3.73, -10.39)
x = 4 : (.27, 10.39)
Relative Min: (3.73, -10.39)
Relative Max: (.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
Conclusion
Increasing
Decreasing
f’(5) = 18
Increasing
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)
Test Value
x=0
Sign of f’’(x) f’’(0) = -12
Conclusion
(2, ∞)
x=3
f’’(3) = 6
Concave Down Concave 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:
 , 
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