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College Algebrapolynomial and Rational Functions

(Chapter3)L:15

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University of PalestineIT-College

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ObjectivesAfter completing this Section, you should be able to:

1. Graph the Functions in the Library of Functions2. Graph Piecewise-defined Functions3. Identify Polynomials and Their Degree4. Graph Polynomial Functions Using Transformations5. Identify the Zeros of a Polynomial and Their Multiplicity6. Analyze the Graph of a Polynomial Function

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The following library of functions will be used throughout the text. Be able to recognize the shape of each graph and associate that shape with the given function.

The Constant Functioncxf )( x

y (0,c)

The Identity Function

xxf )(x

y

(0,0)

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The Square Function

x

y

(0,0)

The Cube Function

2)( xxf

3)( xxf x

y

(0,0)

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The Square Root Function

x

y

(0,0)xxf )(

x

y

(1,1)

(-1,-1)

The Reciprocal Function

xxf

1)(

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x

y

(0,0)

The Absolute Value Function

xxf )(

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When functions are defined by more than one equation, they are called piece-wise defined functions.

Example: The function f is defined as:

1 3

1 3

12- 3

)(

xx

x

xx

xf

a.) Find f (1) = 3Find f (-1) = (-1) + 3 = 2

Find f (4) = - (4) + 3 = -1

piece-wise defined functions.

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1 3

1 3

12- 3

)(

xx

x

xx

xf

b.) Determine the domain of f

Domain: in interval notation),2[

or in set builder notation}2|{ xx

c.) Graph f

x

y

1 2 3

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1

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d.) Find the range of f from the graph found in part c.

Range: in interval notation)4,(

or in set builder notation}4|{ yy

x

y

1 2 3

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1

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A polynomial function is a function of the form

f x a x a x a x an

n

n

n( )

1

1

1 0where an , an-1 ,…, a1 , a0 are real numbers and n is a nonnegative integer.

The domain consists of all real numbers. The degree of the polynomial is the largest power of x that appears.

polynomial function

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Example: Determine which of the following are polynomials. For those that are, state the degree.

(a) f x x x( ) 3 4 52

Polynomial of degree 2

(b) h x x( ) 3 5

Not a polynomial

(c) F xx

x( )

3

5 2

5

Not a polynomial

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A power function of degree n is a function of the form

nn xaxf )(

where a is a real number, a 0, and n > 0 is an integer.

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Summary of Power Functions with Even Degree

1.) Symmetric with respect to the y-axis.

2.) Domain is the set of all real numbers. Range is the set of nonnegative real numbers.

3.) The graph always contains the points (0, 0); (1, 1); and (-1, 1).

4.) As the exponent increases in magnitude, the graph increases very rapidly as x increases, but for x near the origin the graph tends to flatten out and lie closer to the x-axis.

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2 1 0 1 2

2

4

6

8

10

y x 4

y x 8

(1, 1)(-1, 1)

(0, 0)

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Summary of Power Functions with Odd Degree

1.) Symmetric with respect to the origin.

2.) Domain is the set of all real numbers. Range is the set of all real numbers.

3.) The graph always contains the points (0, 0); (1, 1); and (-1, -1).

4.) As the exponent increases in magnitude, the graph becomes more vertical when x > 1 or x < -1, but for -1 < x < 1, the graphs tends to flatten out and lie closer to the x-axis.

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2 1 0 1 2

10

6

2

2

6

10

y x 5

y x 9

(1, 1)

(-1, -1)

(0, 0)

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Graph the following function using transformations.

4)1(2124)( 44 xxxf

y x 4

5 0 5

15

15

(0,0)

(1,1)

5 0 5

15

15

y x 2 4

(0,0)

(1, -2)

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5 0 5

15

15

(1,0)

(2,-2)

y x 2 1 4

5 0 5

15

15

(1, 4)

(2, 2)

412 4 xy

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Consider the polynomial: f x x x( ) ( )( ) 1 4 2

Solve the equation f (x) = 0

f x x x( ) ( )( ) 1 4 2 = 0

x + 1 = 0 OR x - 4 = 0

x = - 1 OR x = 4

If f is a polynomial function and r is a real number for which f (r) = 0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then

a.) (r,0) is an x-intercept of the graph of f.

b.) (x - r) is a factor of f.

Zeros (or Roots) of Polynomial Functions

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If is a factor of a polynomial f and is not a factor of f, then r is called a zero of multiplicity m of f.

mrx 1 mrx

In other words, when a polynomial function is set equal to zero and has been completely factored and each different factor is written with the highest appropriate exponent, depending on the number of times that factor occurs in the product, the exponent on the factor that the zero is a solution for, gives the multiplicity of that zero.

The exponent indicates how many times that factor would be written out in the product, this gives us a multiplicity.

Zero of Multiplicity m

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Example: Find all real zeros of the following function and their multiplicity.

f x x x x( )

3 7

12

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x = 3 is a zero with multiplicity 2.

x = - 7 is a zero with multiplicity 1.

x = 1/2 is a zero with multiplicity 5.

Zero of Multiplicity m

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If r is a Zero of Even Multiplicity

Sign of f (x) does not change from one side to the other side of r.

Graph touches x-axis at r and turns around. .

If r is a Zero of Odd Multiplicity

Sign of f (x) changes from one side to the other side of r.

Graph crosses x-axis at r.

Multiplicity of Zeros and the x-intercept

There are two cases that go with this concept

Case 1:

Case 2:

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Theorem: If f is a polynomial function of degree n, then f has at most n - 1 turning points.

Theorem: For large values of x, either positive or negative, the graph of the polynomial

resembles the graph of the power function n

n xaxf )(

f x a x a x a x an

n

n

n( )

1

1

1 0

Turning Points

A turning point is a point at which the graph changes direction.

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For the polynomial f x x x x( ) 1 5 42

(a) Find the x- and y-intercepts of the graph of f.

The x intercepts (zeros) are (-1, 0), (5,0), and (-4,0)

To find the y - intercept, evaluate f(0)

20)40)(50)(10()0( f

So, the y-intercept is (0,-20)

Graphing a Polynomial Function

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For the polynomial f x x x x( ) 1 5 42

b.) Determine whether the graph crosses or touches the x-axis at each x-intercept.

x = -4 is a zero of multiplicity 1 (crosses the x-axis)

x = -1 is a zero of multiplicity 2 (touches the x-axis)

x = 5 is a zero of multiplicity 1 (crosses the x-axis)

c.) Find the power function that the graph of f resembles for large values of x.

4)( xxf

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d.) Determine the maximum number of turning points on the graph of f.

At most 3 turning points.

e.) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis.

On the interval x 4

Test number: x = -5

f (-5) = 160

Graph of f: Above x-axis

Point on graph: (-5, 160)

For the polynomial f x x x x( ) 1 5 42

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For the polynomial f x x x x( ) 1 5 42

On the interval 4 1x

Test number: x = -2

f (-2) = -14

Graph of f: Below x-axisPoint on graph: (-2, -14)

On the interval 1 5x

Test number: x = 0f (0) = -20

Graph of f: Below x-axis

Point on graph: (0, -20)

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For the polynomial f x x x x( ) 1 5 42

On the interval 5 x

Test number: x = 6

f (6) = 490

Graph of f: Above x-axis

Point on graph: (6, 490)

f.) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

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8 6 4 2 0 2 4 6 8

300

100

100

300

500(6, 490)

(5, 0)(0, -20)

(-1, 0)

(-2, -14)(-4, 0)

(-5, 160)

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Important Note: Determine if there is any symmetry.

 

y-axis symmetry:

Recall that your function is symmetric about the y-axis if it is an even function.  In other words, if f(x)= f(-x, (then your function is symmetric about the y-axis.

Origin symmetry: Recall that your function is symmetric about the origin if it is an odd function.  In other words, if  - f(x)= f(-x), then your function is symmetric about the origin.

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Example : Given the polynomial function

a) find the x-intercepts (or zeros) and state whether the graph crosses the x-axis or touches the x-axis and turns around at each x-intercept.

b) find the y-intercept.c) determine the symmetry of the graph.d) indicate the maximum possible turning points.e) graph.

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a) x-intercepts or zeros

First Factor: Second Factor Third Factor

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b) y-intercept

The y-intercept is (0, 0).

c) symmetry.

y-axis symmetry:

It is not symmetric about the y-axis.

Origin symmetry:

It is not symmetric about the origin.

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d) Find the number of maximum turning points.

Since the degree of the function is 4, then there is at most 4 - 1 = 3 turning points.

e) Draw the graph.

Find extra points

x                                             (x,y)

-0.5

                                                                                      

(-0.5,-0.4375)

1

                                                              

(1,-4)

2

                                                                   

(2,12)

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