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Math 160. 3.2 – Polynomial Functions and Their Graphs. A polynomial function of degree is a function that can be written in the form :. Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners .

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math 160

Math 160

3.2 – Polynomial Functions and Their Graphs

slide11

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.

slide12

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.

slide13

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.

slide14

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

slide15

Ex 1.

Determine the end behavior of the polynomial .

slide16

Ex 1.

Determine the end behavior of the polynomial .

slide17

Ex 2.

Determine the end behavior of the polynomial .

slide18

Ex 2.

Determine the end behavior of the polynomial .

slide19

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 .

slide20

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 .

slide21

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 .

slide22

Graphing Polynomial Functions

Factor to find zeros and plot -intercepts.

Plot test points (before smallest -intercept, between -intercepts, and after largest -intercept).

Determine end behavior.

4. Graph.

slide23

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.

slide24

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.

slide25

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.

slide26

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.

slide27

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.

slide28

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.

slide29

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.

slide30

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.

slide31

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.

slide32

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.

slide33

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.

slide34

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.

slide35

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.

slide36

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.

slide37

Multiplicity

For the polynomial , the factor has multiplicity ___, and the factor has multiplicity ___.

slide38

Multiplicity

For the polynomial , the factor has multiplicity ___, and the factor has multiplicity ___.

slide39

Multiplicity

For the polynomial , the factor has multiplicity ___, and the factor has multiplicity ___.

slide40

Multiplicity

If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :

slide41

Multiplicity

If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :

odd

slide42

Multiplicity

If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :

odd

pass through

slide43

Multiplicity

If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :

slide44

Multiplicity

If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :

even

slide45

Multiplicity

If a factor has an ______ multiplicity, then the curve will ______________ the -axis at :

even

“bounce” off

slide46

Ex 5.

Based on the graph below, determine if the multiplicities of each zero of are even or odd.

slide47

Ex 5.

Based on the graph below, determine if the multiplicities of each zero of are even or odd.

slide48

Ex 5.

Based on the graph below, determine if the multiplicities of each zero of are even or odd.

slide49

Ex 5.

Based on the graph below, determine if the multiplicities of each zero of are even or odd.

slide50

Ex 5.

Based on the graph below, determine if the multiplicities of each zero of are even or odd.

slide51

Ex 5.

Based on the graph below, determine if the multiplicities of each zero of are even or odd.

slide54

Note: Since polynomials are continuous (can be drawn without picking up your pencil), if you find two function values, and , that have opposite signs, then must cross the -axis at some -value between and . This is called the Intermediate Value Theorem (for Polynomials). The same thing is true for all continuous functions.

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