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Section 2.3

Polynomial Functions andTheir Graphs

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Smooth, Continuous Graphs

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Polynomial functions of degree 2 or higher have graphs that

are smooth and continuous. By smooth, we mean that the

graphs contain only rounded curves with no sharp corners.

By continuous, we mean that the graphs have no breaks

and can be drawn without lifting your pencil from the

rectangular coordinate system.

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Notice the breaks and lack of smooth curves.

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End Behavior of Polynomial

Functions

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Odd-degr

eepolynom

ial

functions

havegrap

hswith

oppositeb

ehaviora

teachend

.

Even-d

egreepol

ynomial

functionshaveg

raphswit

hthe

samebeha

viorateachend

.

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Example

Use the Leading Coefficient Test todetermine the end behavior of the graph of

f(x)= - 3x3- 4x + 7

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Example

Use the Leading Coefficient Test to determine the endbehavior of the graph of f(x)= - .08x4- 9x3+7x2+4x + 7

This is the graph that you get with the standard viewing

window. How do you know that you need to change the

window to see the end behavior of the function? What

viewing window will allow you to see the end behavior?

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Zeros of Polynomial Functions

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If f is a polynomial function, then the values of x for

which f(x) is equal to 0 are called the zeros of f. These

values of x are the roots, orsolutions, of the

polynomial equation f(x)=0. Each real root of the

polynomial equation appears as an x-intercept of the

graph of the polynomial function.

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Find all zeros of f(x)= x3+4x2- 3x - 12

( ) ( )

3 2

2

2

2

By definition, the zeros are the values of x

for which f(x) is equal to 0. Thus we set

f(x) equal to 0 and solve for x as follows:

(x 4 ) (3 12) 0

x (x 4) 3(x 4) 0

x+4 x - 3 0

x+4=0 x - 3=0

x=-4

x x+ + + =

+ + =

=

2x 3

x = 3

=

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Example

Find all zeros of x3

+2x2

- 4x-8=0

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Multiplicity of x-Intercepts

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2 2For f(x)=-x ( 2) , notice that each

factor occurs twice. In factoring this equationfor the polynomial function f, if the same

factor x- occurs times, but not +1 times,

we call a zero with multip

x

r k k

r

licity . For thepolynomial above both 0 and 2 are zeros with

multiplicity 2.

k

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( ) ( )

( )

( ) ( )

3 2

3 2

2

2

Find the zeros of 2 4 8 0

2 4 8 0

x 2 4( 2) 0

2 4 0

x x x

x x x

x x

x x

+ =

+ + =

+ + =

+ =

( ) ( ) ( )2 2 2 0

2 has a multiplicity of 2, and 2 has a multiplicity of 1.

Notice how the graph touches at -2 (even multiplicity),

but crosses at 2 (odd multiplicity).

x x x+ + =

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Graphing Calculator- Finding the Zeros

x3+2x2- 4x-8=0

One of the

zeros

The other

zero

Other

zero

One zero

of thefunction

The x-intercepts are the zeros of the function. To

find the zeros, press 2nd Trace then #2. The zero -2

has multiplicity of 2.

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Example

Find the zeros of f(x)=(x- 3)2

(x-1)3

and give themultiplicity of each zero. State whether the

graph crosses the x-axis or touches the x-axis

and turns around at each zero.

Continued on the next slide.

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Example

Now graph this function on your calculator.f(x)=(x- 3)2(x-1)3

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

7

6

5

4

3

2

1

1

2

3

4

5

6

x

y

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The Intermediate Value

Theorem

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Show that the function y=x3- x+5 has a zero

between - 2 and -1.3

3

f(-2)=(-2) ( 2) 5 1

f(-1)=(-1) ( 1) 5 5

Since the signs of f(-1) and f(-2) are opposites then

by the Intermediate Value Theorem there is at least onezero between f(-2) and f(-1). You can also see th

+ =

+ =

ese values

on the table below. Press 2nd Graph to get the table below.

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Example

Show that the polynomial function f(x)=x3

- 2x+9has a real zero between - 3 and - 2.

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Turning Points of Polynomial

functions

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The graph of f(x)=x5- 6x3+8x+1 is shown below.

The graph has four smooth turning points. The

polynomial is of degree 5. Notice that the graph

has four turning points. In general, if the

function is a polynomial function of degree n,

then the graph has at most n-1 turning points.

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A Strategy for Graphing

Polynomial Functions

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Example

Graph f(x)=x4- 4x2 using what you have

learned in this section.

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

7

6

5

4

3

2

1

1

2

3

4

5

6

x

y

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Example

Graph f(x)=x3- 9x2 using what you have

learned in this section.

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

7

6

5

4

3

2

1

1

2

3

4

5

6

x

y

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(a)

(b)

(c)

(d)

Use the Leading Coefficient Test to

determine the end behavior of the graph ofthe polynomial function f(x)=x3- 9x2 +27

falls left, rises right

rises left, falls right

rises left, rises right

falls left, falls right

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(a)

(b)

(c)

(d)

-3 touches, 1 touches

-3 crosses, 1 crosses

-3 touches, 1 crosses

-3 crosses, 1 touches

State whether the graph crosses the x-axis, or

touches the x-axis and turns around at the

zeros of 1, and - 3.

f(x)=(x-1)2(x+3)3