Chapter 11 Polynomial Functions 11.1 Polynomials and Polynomial Functions.
Polynomial Functions
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Transcript of Polynomial Functions
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POLYNOMIAL FUNCTIONSFebruary 27,2012
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What is a polynomial function? What is a polynomial?
The sum or difference of two or more algebraic terms (monomials)
A polynomial equation used to represent a function is called a polynomial function
Example:f(x) = 4x2 – 3x + 2f(x) = 2x3 + 4x2 – 5x + 7
Quadratic Function
Cubic Function
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Do Now: What does it mean to evaluate a function?
Evaluate p(2) p(x) = x2 + 3x + 4
p(2) = 22 + 3(2) + 4 = 14 Evaluate p(b+1)
p(x) = x2 + 3x + 4p(b+1) = (b+1)2 + 3(b+1) + 4
= b2 + 2b + 1 + 3b + 3 + 4 = b2 + 5b + 8
Plug in or substitute what is in the parenthesis for x in
the function
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Vocabulary Degree - refers to the highest exponent
of a function Leading Coefficient - is the coefficient of
the term with the highest degreeExample:What is the degree and leading coefficient
of: 5x2 + 2x + 1
7x4 + 3x2 + x - 9
D: 2, LC: 5D: 4, LC: 7
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Graphs of polynomial functions
• What relationship is there between the degree of the function and the maximum number of zeros?
The degree tells you how many roots you have• What do you notice about the degree and number of humps/ curves in the graph? The degree is one more than the number of humps
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End Behavior What happens to the graph as x
approaches ±∞? This is represented by
x + ∞ and x - ∞x approaches positive infinityX approaches negative infinity
End behavior expressed f(x) –∞, as x –∞ f(x) + ∞, as x +∞
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• For any polynomial function, the domain is all real numbers• For any polynomial function of odd degree, the range is all real numbers• The graph of any odd degree function has at least one x intercept
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Put it all together
• f(x) - ∞, as x - ∞ and f(x) - ∞, as x + ∞• degree of 4 represents an even-degree polynomial function• 2 real zeros
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Exit Slip
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Agenda Do Now – Determine end behavior
through graph Review HW Notes – Determining end behavior
through equation Class work
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Do Now
a) f(x) +∞ , as x +∞ f(x) +∞ , as x -∞b) Evenc) No real zeros
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END BEHAVIOR
Degree: Even
Leading Coefficient: +
End Behavior: Up Up
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END BEHAVIOR
Degree: Even
End Behavior: Down Down
Leading Coefficient:
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END BEHAVIOR
Degree: Odd
Leading Coefficient: +
End Behavior: Down Up
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END BEHAVIOR
Degree: Odd
End Behavior: Up Down
Leading Coefficient:
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End Behavior
Function Degree
Leading Coeffici
ent
End Behavio
rGraph
f(x) = x2 Even + Up, Up
f(x) = – x2 Even — Down, Down
f(x) = x3 Odd + Down, Up
f(x) = – x3 Odd — Up, Down
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END BEHAVIOR
PRACTICE Give the End Behavior:
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END BEHAVIOR
PRACTICE Give the End Behavior:
Up Down
Up Up
Down Down
Down Up
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Do Now What will the end behavior of the
following polynomial functions be? (hint: find the degree and leading coefficient)
y = – x2 + 6x – 5
y = x5 – 3x2 + 2x – 6
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Agenda Do Now- Review Notes
Behavior on intervals Group Work Activity Explain Project Homework
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Today’s Objective: Identify increasing, decreasing, and
constant intervals
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Increasing, Decreasing, and Constant Intervals
An interval is the set of all real numbers between two given numbers
A function f is increasing on an interval if as x increases, then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
A function f is constant on an interval if as x increases, then f(x) remains the same.
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Increasing, Decreasing, Constant Intervals
Find the interval(s) over which the interval is increasing, decreasing and constant?
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Increasing, Decreasing, Constant Intervals
f(x) is decreasingin the interval (-1,1).
f(x) is increasingin the intervals
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Today’s Agenda Do Now Review Home work
Intervals Review Notes
What do they say about… Class work
End behavior worksheet
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Find the interval(s) over which the interval is increasing, decreasing and constant?
Do now
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f(x) is decreasing over the intervalsf(x) is increasingover the interval (3,5).
f(x) is constantover the interval (-1,3).