Rational Rational FunctionsFunctions

Sec. 2.7aSec. 2.7a

Definition: Rational Definition: Rational FunctionsFunctions

f x

y xg x

Let f and g be polynomial functions with g (x ) = 0.

Then the function given by

is a rational function.

• The domain of a rational function is all reals except the zeros of its denominator.

• Every rational function is continuous on its domain.

Finding the Domain of aFinding the Domain of aRational FunctionRational Function

1

2f x

x

Find the domain of the given function and use limits to describeits behavior at value(s) of x not in its domain.

: , 2 2,D Domain?Now, sketchthe graph…

2

limx

f x

2

limx

f x

What does the function approach as x approaches 2?

A Reminder about A Reminder about AsymptotesAsymptotes

limx

f x b

The line y = b is a horizontal asymptote if

or limx

f x b

The line x = a is a vertical asymptote if

limx a

f x

or limx a

f x

Does this make sense with our previous example???Does this make sense with our previous example???

Now, Let’s Now, Let’s AnalyzeAnalyze the Reciprocal the Reciprocal Function…Function… 1

f xx

,0 0, Domain:

,0 0, Range:

Continuity: Continuous on D

Inc/Dec: ,0 , 0, Dec. on

Symmetry: Origin (odd function)

Boundedness: Unbounded

Local Extrema: None

0y H.A.:

0x V.A.:

lim lim 0x x

f x f x

End Behavior:

Transforming theTransforming theReciprocal FunctionReciprocal Function

Describe how the graph of the given function can be obtained bytransforming the graph of the reciprocal function. Identify thehorizontal and vertical asymptotes and use limits to describe thecorresponding behavior. Sketch the graph of the function.

2

3g x

x

1f x

x

12

3x

2 3f x

Translate f (x) left 3 units, then vertically stretch by 2

H.A: 0y lim lim 0x x

g x g x

Transforming theTransforming theReciprocal FunctionReciprocal Function

Describe how the graph of the given function can be obtained bytransforming the graph of the reciprocal function. Identify thehorizontal and vertical asymptotes and use limits to describe thecorresponding behavior. Sketch the graph of the function.

2

3g x

x

1f x

x

12

3x

2 3f x

Translate f (x) left 3 units, then vertically stretch by 2

V.A: 3x 3

limx

g x

3

limx

g x

Transforming theTransforming theReciprocal FunctionReciprocal FunctionLet’s do the same thing with a new function: 3 7

2

xh x

x

1f x

x

2 3f x 13

2h x

x

Begin with polynomial division:

Translate f (x) right 2, reflect across x-axis, translate up 3

H.A: 3y lim lim 3x x

h x h x

Transforming theTransforming theReciprocal FunctionReciprocal FunctionLet’s do the same thing with a new function: 3 7

2

xh x

x

1f x

x

2 3f x 13

2h x

x

Begin with polynomial division:

Translate f (x) right 2, reflect across x-axis, translate up 3

V.A: 2x 2

limx

h x

2

limx

h x

Find the horizontal and vertical asymptotes of the given function.Use limits to describe the corresponding behavior of the function.

2

2

2

1

xf x

x

: ,D What’s the Domain?

So there are no vertical asymptotes!!! Why not???

First, let’s solvethis algebraically…

Now…Limits and Asymptotes of Rational Functions

Find the horizontal and vertical asymptotes of the given function.Use limits to describe the corresponding behavior of the function.

2

2

2

1

xf x

x

First, let’s solve

this algebraically…

2

11

1f x

x

To find horizontal asymptotes, first use polynomial division:

As x becomes very large or verysmall, this last term approacheszero… Why?

So, the horizontal asymptote is the line y = 1

Using limit notation: lim lim 1x x

f x f x

Verifygraphically?

The graphs of

nnm

m

f x a xy

g x b x

Graphs of Rational FunctionsGraphs of Rational Functions

have the following characteristics:

1. End Behavior Asymptote:

If n < m, the end behavior asymptote is the horizontalasymptote of y = 0.

If n = m, the end behavior asymptote is the horizontalasymptote .

n my a bIf n > m, the end behavior asymptote is the quotientpolynomial function y = q(x), where f (x) = g(x)q(x) + r(x).There is no horizontal asymptote.

The graphs of

nnm

m

f x a xy

g x b x

Graphs of Rational FunctionsGraphs of Rational Functions

have the following characteristics:

2. Vertical Asymptotes:These occur at the zeros of the denominator, provided thatthe zeros are not also zeros of the numerator of equal orgreater multiplicity.

3. x-intercepts:These occur at the zeros of the numerator, which are notalso zeros of the denominator.

4. y-intercept:This is the value of f (0), if defined.

Find the asymptotes and intercepts of the given function, andthen graph the function.

3

2 9

xf x

x

Degree of Numerator > Degree of Denominator long division!

2

9

9

xf x x

x

The quotient q(x) = x

is our slant asymptote

Factor the denominator:

2 9 3 3x x x Vertical Asymptotesare at x = 3 and x = –3

x-intercept = 0, y-intercept = f (0) = 0

Verify all of this graphically???Verify all of this graphically???

Guided PracticeGuided Practice

2

2

3

1

xf x

x

Find the horizontal and vertical asymptotes of the given function.Use limits to describe the corresponding behavior.

No vertical asymptotes

H.A.: y = 3

lim lim 3x x

f x f x

Guided PracticeGuided Practice

2

3

3

xf x

x x

Find the horizontal and vertical asymptotes of the given function.Use limits to describe the corresponding behavior.

V.A.: x = 0, x = –3

H.A.: y = 0

lim lim 0x x

f x f x

3

limx

f x

3

limx

f x

0

limx

f x

0

limx

f x

Whiteboard Whiteboard PracticePractice

2

2

2 3

xf x

x x

Find the asymptotes and intercepts of the given function, thengraph the function.

Intercepts: (0, –2/3), (–2, 0)

Asymptotes: x = –3, x = 1, y = 0

Whiteboard Whiteboard PracticePractice

3

3

4f x

x x

Find the asymptotes and intercepts of the given function, thengraph the function.

No Intercepts

Asymptotes: x = –2, x = 0, x = 2, y = 0

Whiteboard Whiteboard PracticePractice

2

2

3 12

4

x xf x

x

Find the asymptotes and intercepts of the given function, thengraph the function.

Intercepts: (0, –3), (–1.840, 0), (2.174, 0)

Asymptotes: x = –2, x = 2, y = –3

Whiteboard Whiteboard PracticePractice

2 3 7

3

x xf x

x

Find the asymptotes and intercepts of the given function, thengraph the function.

Intercepts: (0, –7/3), (–1.541, 0), (4.541, 0)

Asymptotes: x = –3, y = x – 6