Section 4.1Polynomial Functions

A polynomial function is a function of the form

f x a x a x a x an

n

n

n( )

1

1

1 0

an , an-1 ,…, a1 , a0 are real numbers

n is a nonnegative integer

D: {x|x å real numbers}

Degree is the largest power of x

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

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

nn xaxf )(

where a is a real number

a = 0

n > 0 is an integer.

2 1 0 1 2

2

4

6

8

10

y x 4

y x 8

(1, 1)(-1, 1)

(0, 0)

Power Functions with Even Degree

Summary of Power Functions with Even Degree

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

2.) D: {x|x is a real number} R: {x|x is a non negative real number}

3.) Graph (0, 0); (1, 1); and (-1, 1).

4.) As the exponent increases, 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.

2 1 0 1 2

10

6

2

2

6

10

y x 5

y x 9

(1, 1)

(-1, -1)

(0, 0)

Power Functions with Odd Degree

Summary of Power Functions with Odd Degree

1.) Symmetric with respect to the origin.

2.) D: {x|x is a real number} R: {x|x is a real number}

3.) Graph contains (0, 0); (1, 1); and (-1, -1).

4.) As the exponent increases, 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.

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)

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

If r is a Zero of Even Multiplicity

Graph crosses x-axis at r.

If r is a Zero of Odd Multiplicity

Graph touches x-axis at r.

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)

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

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

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)

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.

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)

Sections 4.2 & 4.3Rational Functions

28

A rational function is a function of the form

Rxpxqx

()()()

• p and q are polynomial functions• q is not the zero polynomial. • D: {x|x å real numbers & q(x) = 0}.

Find the domain of the following rational functions.

(a) R xx

x x( )

1

8 122

x

x x1

6 2

All real numbers x except -6 and -2.

(b) R xx

x( )

4

162

x

x x4

4 4

All real numbers x except -4 and 4.

(c) R xx

( ) 5

92 All Real Numbers

30

Vertical Asymptotes.

Domain gives vertical asymptotes•Reduce rational function to lowest terms, to find vertical asymptote(s).

•The graph of a function will never intersect vertical asymptotes.

•Describes the behavior of the graph as x approaches some number c

Range gives horizontal asymptotes•The graph of a function may cross intersect

horizontal asymptote(s).

•Describes the behavior of the graph as x approaches infinity or negative infinity (end behavior) 31

Example: Find the vertical asymptotes, if any, of the graph of each rational function.

(a) R xx

( ) 3

12

3

1 1( )( )x x

Vertical asymptotes: x = -1 and x = 1

(b) R xxx

( )

512

No vertical asymptotes

(c) R xx

x x( )

3

122

x

x x3

3 4( )( )

1

4x

Vertical asymptote: x = -432

(3,2)

(1,0)

(2,0)(0,1)

12

1)(

xxf

In this example there is a vertical asymptote at x = 2 and a horizontal asymptote at y = 1.

Examples of Horizontal Asymptotes

y = L

y = R(x)

y

x

y = L

y = R(x)

y

x

LxRx

)(lim

LxRx

)(lim

LxRx

)(lim

Examples of Vertical Asymptotes

x = cy

x

x = c

y

x

If an asymptote is neither horizontal nor vertical it is called oblique.

y

x

Note: a graph may intersect it’s oblique asymptote. Describes end behavior. More on this in Section 3.4.

Recall that the graph of isxxf

1)(

(1,1)

(-1,-1)

37

Graph the function using transformations

12

1)(

xxf

(1,1)

(-1,-1)

xxf

1)(

(3,1)

(1,-1)(2,0)

2

1)(

xxf

(3,2)

(1,0)

(2,0)(0,1)

011

1

011

1

)(

)()(

bxbxbxb

axaxaxa

xq

xpxR

mm

mm

nn

nn

Consider the rational function

1. If n < m, then y = 0 is a horizontal asymptote

2. If n = m, then y = an / bm is a horizontal asymptote

3. If n = m + 1, then y = ax + b is an oblique asymptote, found using long division.

4. If n > m + 1, neither a horizontal nor oblique asymptote exists.

39

Example: Find the horizontal or oblique asymptotes, if any, of the graph of

(a) R xx x

x x x( )

3 4 15

4 7 1

2

3 2

Horizontal asymptote: y = 0

(b) R xx xx x

( )

2 4 13 5

2

2

Horizontal asymptote: y = 2/3

(c) R xx x

x( )

2 4 12

x x xx

x x

x

x

2 4 16

6 1

6 12

2

- 2

-

13

2

Oblique asymptote: y = x + 6

To analyze the graph of a rational function:

1) Find the Domain.

2) Locate the intercepts, if any.

3) Test for Symmetry. If R(-x) = R(x), there is symmetry with respect to the y-axis. If - R(x) = R(-x), there is symmetry with respect to the origin.

4) Find the vertical asymptotes.

5) Locate the horizontal or oblique asymptotes.

6) Determine where the graph is above the x-axis and where the graph is below the x-axis.

7) Use all found information to graph the function.

42

Example: Analyze the graph of 9

642)(

2

2

x

xxxR

R x

x x

x x( )

2 2 3

3 3

2

2 3 13 3

x xx x

2 1

33

xx

x,

Domain: x x x 3 3,

a.) x-intercept when x + 1 = 0: (-1,0)

b.) y-intercept when x = 0: 3

2

)30(

)10(2)0(

R

y - intercept: (0, 2/3)

3

12)(

x

xxR

c.) Test for Symmetry: R xx

x( )

( )( )

2 13

)()()( xRxRxR No symmetry

R xx

xx( ) ,

2 1

33

d.) Vertical asymptote: x = -3

Since the function isn’t defined at x = 3, there is a hole at that point.

e.) Horizontal asymptote: y = 2

f.) Divide the domain using the zeros and the vertical asymptotes. The intervals to test are:

x x x3 3 1 1

x x x3 3 1 1

Test at x = -4

R(-4) = 6

Above x-axis

Point: (-4, 6)

Test at x = -2

R(-2) = -2

Below x-axis

Point: (-2, -2)

Test at x = 1

R(1) = 1

Above x-axis

Point: (1, 1)

g.) Finally, graph the rational function R(x)

8 6 4 2 0 2 4 6

10

5

5

10

(-4, 6)

(-2, -2) (-1, 0) (0, 2/3)

(1, 1) (3, 4/3)

y = 2

x = - 3