Download PC 2.2 Polynomials of Higher Degree

Chapter 2
Polynomial and Rational Functions
Warm Up 2.2
 A rancher has 200 feet of fencing to enclose
two adjacent rectangular corrals. Write the
area A of the corral as a function of x. Find
the dimensions that will produce the
maximum area.
y
x
x
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2.2 Polynomial Functions of
Higher
Degree
Objectives:
Use transformations to sketch graphs of
polynomial functions.
Use the Leading Coefficient Test to determine
the end behavior of graphs of polynomial
functions.
Find and use zeros of polynomial functions as
sketching aids.
Use the Intermediate Value Theorem to help
locate zeros of polynomial functions.
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Vocabulary
Continuous Function
Polynomial Function
Monomial, Binomial, Trinomial Functions
Leading Coefficient Test
Zeros of Polynomial Functions
Repeated Zero
Multiplicity
Intermediate Value Theorem
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Continuous Function
No breaks, holes or gaps.
Continuous or not?
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Polynomial Function Characteristics
Continuous
Smooth, rounded turns. No sharp corners.
Polynomial or not?
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Definition of a Polynomial Function
Polynomial Function of x with degree n
f (x) = anxn + an – 1 xn – 1 + … + a2x2 + a1x + a0
where:
n is a non-negative integer and
an, an – 1, … , a2, a1, a0 are real numbers
an ≠ 0
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Monomial Functions
Simplest type of polynomial.
Monomial form: f (x) = xn,
where n is an integer greater than zero.
n = Even Integer
Graph similar to
f (x) = x2
Touches x-axis at
x-intercept.
n = Odd Integer
Graph similar to
f (x) = x3
Crosses x-axis at
x-intercept.
As n increases, the graph gets flatter at the origin.
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Transformations of Monomials
Sketch the graph of each function.
1.
f ( x)   x
5
2. g ( x)  x 4  1
3. h( x)  x  1
4
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Leading Coefficient Test
Using the degree of the function (n) and the
leading coefficient (an), we can determine the
end behavior of a function.
n is even
n is odd
an > 0
an < 0
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Use the Leading Coefficient Test
Describe the end behavior of each function.
1.
f ( x)   x  4 x
2.
f ( x)  x 4  5 x 2  4
3.
f ( x)  x  x
3
5
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Zeros of Polynomial Functions
Zero = Root = x-intercept = Solution
A number x for which f (x) = 0.
For a polynomial function f of degree n:
The function f has at most n real zeros.
The graph of f has at most n – 1 relative
extrema (relative minima or maxima).
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Real Zeros of Polynomials
If f is a polynomial function and a is a real
number, then the following statements are
equivalent.
1. x = a is a zero of the function.
2. x = a is a solution of the polynomial equation
f (x) = 0.
3. (x – a) is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of f.
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Finding Zeros of Polynomials
Find all real zeros and relative extrema of the
polynomial functions.
1.
f ( x)  x  x  2 x
2.
f ( x)  2 x 4  2 x 2
3
2
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Warm Up 2.2.2
Use your graphing calculator to find the zeros and
relative extrema of the function
f ( x)  2 x  2 x
4
2
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Repeated Zeros
A repeated zero of multiplicity k occurs if a function
has a factor of the form (x – a)k, where k > 1.
If k is odd
If k is even
The graph crosses the
x-axis at x = a.
The graph touches
(but does not cross)
the x-axis at x = a.
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Example - Multiplicity
Find all the real zeros of each function and determine
the multiplicity of each zero. Solve algebraically and
graphically.
1.
f ( x)  x 2  10 x  25
2. g ( x)  2 x 3  6 x 2
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Find a Function Given the Zeros
Find a polynomial function that has the given zeros.
1. 0, 2, 5
2. 4, 2  7 , 2  7
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Basic Curve Sketching
1.
Apply the Leading Coefficient Test.
2. Find the zeros of the polynomial.
3. Plot a few additional points.
4. Draw the graph.
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Curve Sketching Example 1
Sketch the graph
f ( x)  3 x 4  4 x 3
of by hand.
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Curve Sketching Example 2
Sketch the graph
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f ( x)  2 x  6 x  x
2
3
2
of by hand.
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Homework 2.2
Worksheet 2.2
# 47, 49, 53, 59, 61, 65, 67, 69, 71, 85
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