Section 4.1 Polynomial Functions

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Section 4.1Polynomial Functions
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A polynomial function is a function of the form
an , an-1 ,, a1 , a0 are real numbers n is a
nonnegative integer D xx å real
numbers Degree is the largest power of x
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Example Determine which of the following are
polynomials. For those that are, state the
degree.
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A power function of degree n is a function of the
form
where a is a real number a 0 n gt 0 is an
integer.
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Power Functions with Even Degree
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Summary of Power Functions with Even Degree
1.) Symmetric with respect to the y-axis.
2.) D xx is a real number R xx is a
non negative real number
3.) Graph (0, 0) (1, 1) and (-1, 1).
4.) As the exponent increases, the graph
increases very rapidly as x increases, but for x
near the origin the graph tends to flatten out
and lie closer to the x-axis.
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Power Functions with Odd Degree
(1, 1)
(0, 0)
(-1, -1)
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Summary of Power Functions with Odd Degree
1.) Symmetric with respect to the origin.
2.) D xx is a real number R xx is a
real number
3.) Graph contains (0, 0) (1, 1) and (-1, -1).
4.) As the exponent increases, the graph becomes
more vertical when x gt 1 or x lt -1, but for -1
lt x lt 1, the graphs tends to flatten out and lie
closer to the x-axis.
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Graph the following function using
transformations.
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(No Transcript)
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If r is a Zero of Even Multiplicity
Graph touches x-axis at r.
Graph crosses x-axis at r.
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(a) Find the x- and y-intercepts of the graph of
f.
The x intercepts (zeros) are (-1, 0), (5,0), and
(-4,0)
To find the y - intercept, evaluate f(0)
So, the y-intercept is (0,-20)
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b.) Determine whether the graph crosses or
touches the x-axis at each x-intercept.
x -4 is a zero of multiplicity 1 (crosses the
x-axis) x -1 is a zero of multiplicity 2
(touches the x-axis) x 5 is a zero of
multiplicity 1 (crosses the x-axis)
c.) Find the power function that the graph of f
resembles for large values of x.
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d.) Determine the maximum number of turning
points on the graph of f.
At most 3 turning points.
e.) Use the x-intercepts and test numbers to find
the intervals on which the graph of f is above
the x-axis and the intervals on which the graph
is below the x-axis.
Test number x -5
f (-5) 160
Graph of f Above x-axis
Point on graph (-5, 160)
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Test number x -2
f (-2) -14
Graph of f Below x-axis
Point on graph (-2, -14)
Test number x 0
f (0) -20
Graph of f Below x-axis
Point on graph (0, -20)
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Test number x 6
f (6) 490
Graph of f Above x-axis
Point on graph (6, 490)
f.) Put all the information together, and connect
the points with a smooth, continuous curve to
obtain the graph of f.
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(6, 490)
(-1, 0)
(-5, 160)
(0, -20)
(5, 0)
(-4, 0)
(-2, -14)
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Sections 4.2 4.3Rational Functions
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A rational function is a function of the form

• p and q are polynomial functions
• q is not the zero polynomial.
• D xx å real numbers q(x) 0.

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Find the domain of the following rational
functions.
All real numbers x except -6 and -2.
All real numbers x except -4 and 4.
All Real Numbers
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Vertical Asymptotes.

• Domain gives vertical asymptotes
• Reduce rational function to lowest terms, to find
  vertical asymptote(s).
• The graph of a function will never intersect
  vertical asymptotes.
• Describes the behavior of the graph as x
  approaches some number c

• Range gives horizontal asymptotes
• The graph of a function may cross intersect
  horizontal asymptote(s).
• Describes the behavior of the graph as x
  approaches infinity or negative infinity (end
  behavior)

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Example Find the vertical asymptotes, if any, of
the graph of each rational function.
Vertical asymptotes x -1 and x 1
No vertical asymptotes
Vertical asymptote x -4
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In this example there is a vertical asymptote at
x 2 and a horizontal asymptote at y 1.
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Examples of Horizontal Asymptotes
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Examples of Vertical Asymptotes
x c
y
x c
y
x
x
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If an asymptote is neither horizontal nor
vertical it is called oblique.
Note a graph may intersect its oblique
asymptote. Describes end behavior. More on this
in Section 3.4.
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Recall that the graph of is
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Graph the function
using transformations
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Consider the rational function
1. If n lt m, then y 0 is a horizontal asymptote
2. If n m, then y an / bm is a horizontal
asymptote
3. If n m 1, then y ax b is an oblique
asymptote, found using long division.
4. If n gt m 1, neither a horizontal nor
oblique asymptote exists.
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Example Find the horizontal or oblique
asymptotes, if any, of the graph of
Horizontal asymptote y 0
Horizontal asymptote y 2/3
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Oblique asymptote y x 6
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To analyze the graph of a rational function
1) Find the Domain.
2) Locate the intercepts, if any.
3) Test for Symmetry. If R(-x) R(x), there is
symmetry with respect to the y-axis. If - R(x)
R(-x), there is symmetry with respect to the
origin.
4) Find the vertical asymptotes.
5) Locate the horizontal or oblique asymptotes.
6) Determine where the graph is above the x-axis
and where the graph is below the x-axis.
7) Use all found information to graph the
function.
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Example Analyze the graph of
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a.) x-intercept when x 1 0 (-1,0)
y - intercept (0, 2/3)
c.) Test for Symmetry
No symmetry
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d.) Vertical asymptote x -3
Since the function isnt defined at x 3, there
is a hole at that point.
e.) Horizontal asymptote y 2
f.) Divide the domain using the zeros and the
vertical asymptotes. The intervals to test are
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Test at x -4
Test at x -2
Test at x 1
R(-4) 6
R(-2) -2
R(1) 1
Above x-axis
Below x-axis
Above x-axis
Point (-4, 6)
Point (-2, -2)
Point (1, 1)
g.) Finally, graph the rational function R(x)
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x - 3
(-4, 6)
(1, 1)
(3, 4/3)
y 2
(-2, -2)
(-1, 0)
(0, 2/3)