math 160 notes packet #7 polynomial functions 1. 2
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Math 160
Notes Packet #7Polynomial Functions
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A polynomial function of degree is a function that can be written in the form:
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.
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The end behavior of a function means how the function behaves when or .
For non-constant polynomial functions, the end behavior is either or . The highest degree term of a polynomial, called the ___________, determines its end behavior.
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The end behavior of a function means how the function behaves when or .
For non-constant polynomial functions, the end behavior is either or . The highest degree term of a polynomial, called the ___________, determines its end behavior.
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The end behavior of a function means how the function behaves when or .
For non-constant polynomial functions, the end behavior is either or . The highest degree term of a polynomial, called the ___________, determines its end behavior.
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The end behavior of a function means how the function behaves when or .
For non-constant polynomial functions, the end behavior is either or . The highest degree term of a polynomial, called the ___________, determines its end behavior.leading term
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Ex 1.Determine the end behavior of the polynomial .
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Ex 1.Determine the end behavior of the polynomial .
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Ex 2.Determine the end behavior of the polynomial .
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Ex 2.Determine the end behavior of the polynomial .
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Note: Zeros of a polynomial correspond with factors, and visually mean -intercepts.
ex: If , then since , we must have a factor of . Also, there will be an -intercept at .
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Note: Zeros of a polynomial correspond with factors, and visually mean -intercepts.
ex: If , then since , we must have a factor of . Also, there will be an -intercept at .
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Note: Zeros of a polynomial correspond with factors, and visually mean -intercepts.
ex: If , then since , we must have a factor of . Also, there will be an -intercept at .
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1. Factor to find zeros and plot -intercepts.
2. Plot test points (before smallest -intercept, between -intercepts, and after largest -intercept).
3. Determine end behavior.
4. Graph.
Graphing Polynomial Functions
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 3.Sketch the graph of .Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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Ex 4.Sketch the graph of. Be sure to show intercepts, test points (before smallest -intercept, between -intercepts, and after largest -intercept), and end behavior.
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For the polynomial , the factor has multiplicity ___, and the factor has multiplicity ___.
Multiplicity
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For the polynomial , the factor has multiplicity ___, and the factor has multiplicity ___.
Multiplicity
𝟑
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For the polynomial , the factor has multiplicity ___, and the factor has multiplicity ___.
Multiplicity
𝟑𝟒
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If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :
Multiplicity
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If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :
Multiplicity
odd
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If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :
Multiplicity
oddpass through
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If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :
Multiplicity
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If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :
Multiplicity
even
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If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :
Multiplicity
even“bounce” off
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Ex 5.Based on the graph below, determine if the multiplicities of each zero of are even or odd.
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Ex 5.Based on the graph below, determine if the multiplicities of each zero of are even or odd.
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Ex 5.Based on the graph below, determine if the multiplicities of each zero of are even or odd.
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Ex 5.Based on the graph below, determine if the multiplicities of each zero of are even or odd.
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Ex 5.Based on the graph below, determine if the multiplicities of each zero of are even or odd.
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Ex 5.Based on the graph below, determine if the multiplicities of each zero of are even or odd.
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Note: Every polynomial of degree has at most turning points.
ex: has at most 3 turning points. Also, if a polynomial’s graph has turning points, then the polynomials’ degree is at least .
ex: A polynomial with 5 turning points is at least degree 6.
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Note: Every polynomial of degree has at most turning points.
ex: has at most 3 turning points. Also, if a polynomial’s graph has turning points, then the polynomials’ degree is at least .
ex: A polynomial with 5 turning points is at least degree 6.
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Note: Every polynomial of degree has at most turning points.
ex: has at most 3 turning points. Also, if a polynomial’s graph has turning points, then the polynomials’ degree is at least .
ex: A polynomial with 5 turning points is at least degree 6.
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Note: Every polynomial of degree has at most turning points.
ex: has at most 3 turning points. Also, if a polynomial’s graph has turning points, then the polynomials’ degree is at least .
ex: A polynomial with 5 turning points is at least degree 6.
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Ex 6.What is the least degree that the polynomial shown below could have?
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Ex 6.What is the least degree that the polynomial shown below could have?
3 turning points, so…
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Ex 6.What is the least degree that the polynomial shown below could have?
3 turning points, so…
Least degree: 4