6.2 Graphs of Polynomials

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6.2 Graphs of Polynomials
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The Degree of Polynomials
The degree of a polynomial is the value of the
largest exponent.
y 5x4 3x2 7 Degree 4 y -7x5 2x9 3x2
5x4 2 Degree 9
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The Leading Coefficient of a Polynomial
The leading coefficient of a polynomial is the
value of coefficient of the term with the largest
exponent.
y 5x4 3x2 7 Leading Coefficient 5 y
-7x5 2x9 3x2 5x4 2 Leading Coefficient 2
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End Behavior of Polynomials
Use your calculator to graph the following
y x
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End Behavior of Polynomials
Use your calculator to graph the following
y x2
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End Behavior of Polynomials
Use your calculator to graph the following
y x3
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End Behavior of Polynomials
Use your calculator to graph the following
y x4
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End Behavior of Polynomials
Use your calculator to graph the following
y x5
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End Behavior of Polynomials
Use your calculator to graph the following
y x6
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Is there a pattern to the graphs?
y x3
y x5
y x
y x6
y x2
y x4
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Graphs of Polynomials with Odd Degrees
y x3
y x5
y x
Odd degree polynomials are like lines. They start
low and end high
Unless they have a negative leading coefficient.
Then they start high and end low.
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Graphs of Polynomials with Even Degrees
Even degree polynomials are like parabolas. They
start high and end high
Unless they have a negative leading coefficient.
Then they start low and end low.
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Roots of Polynomials
What are the roots of y 2 (x - 5)(x - 2)?
What is its degree?
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Roots of Polynomials
What are the roots of y 2 (x - 5)(x - 2)(x
4)?
What is its degree?
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Roots of Polynomials
What are the roots of y -5 (x - 5)(x - 2)(x
4)(x 7)?
What is its degree? What about the leading
coefficient?
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Roots of Polynomials
What are the roots of y 3 (x - 5)(x - 2)2?
What is its degree? What about the leading
coefficient?
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Roots of Polynomials
What are the roots of y (x - 5)2(x - 2)2?
What is its degree? What about the leading
coefficient?
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Roots of Polynomials
What are the roots of y (x 3)(x - 5)2(x - 2)2?
What is its degree? What about the leading
coefficient?
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Roots of Polynomials
What are the roots of y -(x 3)3(x - 5)?
What is its degree? What about the leading
coefficient?
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Roots of Polynomials
What are the roots of y -(x 3)3(x - 5)2(x
7)3?
What is its degree? What about the leading
coefficient?
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Graphs of Polynomials
To graph a polynomial in factored form 1.)
Determine the degree 2.) Check the sign of the
leading coefficient. 3.) Mark on your graph where
the curve begins and ends (the end behavior) 4.)
Plot the position of the roots. 5.) Connect the
dots.
When connecting the dots, dont forget to check
the degree of each root. Single Roots
Straight through the x-axis (like a line). Double
Roots Bounce off the x-axis (like a
parabola). Triple Roots The line bends as it
passes through the x-axis (like a cubic).
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Graphs of Polynomials
Graph y -x(x 3)5(5x - 10)2(x 7)3(x - 6)
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Graphs of Polynomials
Graph y 15x6(x 8)5(2x - 4)2(3x 6)3(x - 2)
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Graphs of Polynomials
Graph y -x2(x 6)2(x - 4)2(3x 6)167(x - 8)
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Can you figure out a possible equations for this
graph?
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Can you figure out a possible equations for this
graph?
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Can you figure out a possible equations for this
graph?
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Can you figure out a possible equations for this
graph?