3-3 Polynomial Graphs (Presentation)

8/8/2019 3-3 Polynomial Graphs (Presentation)

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3-3 Polynomial Graphs

Unit 3 Quadratic and Polynomial Functions


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Concepts and Objectives

Objective #11

Identify and interpret vertical and horizontal

translations

Identify the end behavior of a function Identify the number of turning points of a function

Use the Intermediate Value Theorem and the

Boundedness Theorem to locate zeros of a function

Use the calculator to approximate real zeros


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Graphing Polynomial Functions

If we look at graphs of functions of the form ,

we can see a definite pattern: n f x ax

2 f x x 3 g x x

4h x x 5 j x x


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For a polynomial function of degree n

Ifn is even, the function is an even function.

An even function has a range of the form , k] or

[k, for some real number k. Ifn is odd, the function is an odd function.

The range of an odd function is the set of all real

numbers, , .

For odd functions, the graph will have at least onereal zero (x-intercept).


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Compare the graphs of the two functions:

2 f x x

2 2 g x x

2h x x

2

1 j x x


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Vertical translation

The graph of is shifted kunits up if

k> 0 and |k| units down ifk< 0.

Horizontal translation

The graph of is shifted h units to the

right ifh > 0 and |h| units to the left ifh < 0.

n f x ax k

n

f x a x h


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Example: Write the equation of the function of degree 3

graphed below.

This is an odd function.The vertex has been shifted up 3

units and to the right2 units.

So, its going to be something like:

3

2 3 f x a x


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Example (cont.):

To determine whata is, we can picka point and plug in values:

3 4f

3

3 3 2 3 4f a

3 4a

1a

3

2 3 f x x


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Multiplicity and Graphs

What is the multiplicity of ?

The zero 4 has multiplicity 5

The multiplicity of a zero and whether the function iseven or odd determines what the graph does at a zero.

A zero of multiplicity one crosses thex-axis.

A zero of even multiplicity turns or bounces at the

x-axis . A zero of odd multiplicity greater than one crosses

thex-axis and wiggles.

5

4 g x x


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Turning Points and End Behavior

The point where a graph changes direction (bounces

or wiggles) is called a turning pointof the function.

A function of degree n will have at mostn 1 turning

points, with at least one turning point between each

pair of adjacent zeros.

The end behaviorof a polynomial graph is determined by

the term with the largest exponent (the dominating

term).

For example, has the same end

behavior as . 32 8 9 f x x x

32 f x x


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End Behavior

Example: Use symbols for end behavior to describe the

end behavior of the graph of each function.

1.

2.

3.

4 22 8

f x x x x even functionopens downward

3 23 5 g x x x x odd functionincreases

5 3

1h x x x odd functiondecreases


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Intermediate Value Theorem

This means that if we plug in two numbers and the

answers have different signs (one positive and one

negative), the function has to have crossed thex-axisbetween the two values.

Iffx defines a polynomial function with only real

coefficients, and if for real numbers a and b, the

valuesfa andfb are opposite in sign, then there

exists at least one real zero between a and b.


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Example: Show that has a real

zero between 2 and 3.

You can either plug the values in, or you can usesynthetic division to evaluate each value.

Since the sign changes, there must be a real zero

between 2 and 3.

3 22 1 f x x x x

2 1 2 1 1

1

2

0

0

1

2

1

3 1 2 1 1

1

3

1

3

2

6

7


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Iffa andfb are not opposite in sign, it does not

necessarily mean that there is no zero between a and b.

Consider the function, , at1 and 3: 2 2 1 f x x x

f1 = 2 > 0 andf3= 2 >0

This would imply that there is no

zero between 1 and 3, but we cansee thatfhas two zeros between

those points.


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Boundedness Theorem

Letfx be a polynomial function of degree n 1 with

real coefficients and with a positive leading coefficient.

Iffx is divided synthetically byx c, and

(a) if c > 0 and all numbers in the bottom row are

nonnegative, thenfx has no zeros greater than c;

(b) if c < 0 and the numbers in the bottom row

alternate in sign, thenfx has no zero less than c.


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Boundedness Theorem

Example: Show that the real zeros of

satisfy the following conditions,

a) No real zero is greater than 1b) No real zero is less than 2

4 25 3 7 f x x x x


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Boundedness Theorem



a) No real zero is greater than 1

Since the bottom row numbers are all 0,fx has

no zero greater than 1.

4 25 3 7 f x x x x

1 1 0 5 3 7

9

9

6

6

1

1

1

1 2


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Boundedness Theorem



b) No real zero is less than 2

Since the signs of the bottom numbers alternate,fx

has no zero less than 2.

4 25 3 7 f x x x x

2 1 0 5 3 7

30

15

18

9

4

2

2

1 23


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Approximating Real Zeros

Example: Approximate the real zeros of

Step 1: Enter the function into

3 28 4 10 f x x x x


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Step 2: Press and then

3 28 4 10 f x x x x


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Step 3: Position the cursor at the farleft above thex-axis and press

Step 4: Move the cursor below the

x-axis and press

3 28 4 10 f x x x x


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Step 5: Our first zero is at8.33594

3 28 4 10 f x x x x


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Repeat steps 1-5 to find the next twozeros

#2: 0.9401088

3 28 4 10 f x x x x


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Repeat steps 1-5 to find the next twozeros

#2: 0.9401088

#3: 1.2760488

3 28 4 10 f x x x x


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Homework

College Algebra

Page 352: 21-27 (3), 48-69 (3), 81

Turn In: 24, 48, 54, 60, 63, 66

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3-3 Polynomial Graphs (Presentation)