graphs of rational functions prepared for mth

8/9/2019 Graphs of Rational Functions Prepared for Mth

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Graphs of Rational Functions

Prepared for Mth 163: Precalculus 1 Online

By Richard Gill

Through funding provided by a

VCCS LearningWare Grant


2/24

Arational function is a function that can be expressed in the form

)(

)(

xg

xfy ! where both f(x) and g(x) are polynomial functions.

Examples of rational functions would be:

xx

xxg

x

xxf

xy

2

4)(

3

2)(

2

1

2

2

!

!

!

Over the next few frames we will look at the graphs of each of

the above functions.


3/24

First we will look at

This function has one value of x that is banned from the domain.

What value of x do you think that would be?And why?

If you guessed x = 2, congratulations. This is the value at which

the function is undefined because x = 2 generates 0 in thedenominator.

Consider the graph of the function. What impact do youthink this forbidden point will have on the graph?

.2

1

!x

y

Think before you click.


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Now just because we cannot use x = 2 in our x-y table, it does not

mean that we cannot use values of x that are close to 2. So before

you click again, fill in the values in the table below.

x

1.5

1.7

1.9

2.0

2

1

!x

y

-2

As we pick values of x that are smaller

than 2 but closer and closer to 2what do

you think is happening to y?

If you said that y is getting closer and

closer to negative infinity, nice job!

-3.33

-10

undefined


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Now fill in the values in the rest of the table.

x

1.5

1.7

1.9

2.0

2.1

2.3

2.5

2

1

!x

y

-2-3.33

-10

Und

What about the behavior of the function

on the other side of x = 2?As we pick

values of x that are larger than 2 but closer

and closer to 2what do you think is

happening to y?

If you said that y is getting closer and

closer to positive infinity, you are right

on the money!10

3.33

2


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Lets see what the points that we have calculated so farwould look

like on graph.

21

!x

y

The equation is

x = 2 becauseevery point on

the line has an

x coordinate of

2.

This dotted

vertical line is a

crucial visual aid

for the graph. Do

you knowwhatthe equation of

this dotted line is?

x

y

(2. , )

(2. , . )

(2. , 2)

( . , -2)

( . , - . )

( . , - )


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21

!x

y

x

y

(2. , )

(2. , . )

(2. , 2)

( . , -2)

( . , - . )

( . , - )

Hint: it is one of

the many great

and imaginative

words in

mathematics.

The line x = 2 is a

vertical asymptote.

Do you knowwhat this dotted

vertical line is

called?


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2

1

!

x

y

x

y

,

,

,

,

,

,

If fx)

approachespositiveor

negativeinfinity

as x approachesc

fromtherightor

theleft,thenthe

line x =cisa

vertical

asymptote ofthe

graphoff

Ourgraphwillget

closerandcloser

tothisverticalasymptotebut

nevertouchit


9/24

A horizontal asymptote is a horizontal line that the graph gets

closer and closer to but never touches. The official definition of a

horizontal asymptote:The line y = c is a horizontal asymptote for the graph of a

function f if f(x) approaches c as x approaches positive or

negative infinity.

Huh?!

Dont you just love official definitions?At any rate,

rational functions have a tendency to generate

asymptotes, so lets go back to the graph and see ifwe can

find a horizontal asymptote.


10/24

2

1

!x

y

x

y(2.1,10)

(2.3,3.33)

(2.5,2)

(1.5,-2)

(1.7,-3.33)

(1.9,-10)

Looking at the graph, as

the x values get larger

and larger in thenegative direction, the y

values of the graph

appear to get closer and

closer to what?

If you guessed that the

y values appear to get

closer and closer to 0,

you may be onto

something. Lets look

at a table of values for

confirmation.


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x

0

-5

-20

-100

2

1

!x

y

Before you click again, take a minute to calculate the y values in

the table below. What is your conclusion about the trend?

-(1/2)

-(1/7)

-(1/22)

-(1/102)

Conclusion: as the x values get closer and closer to negativeinfinity, the y values will get closer and closer to 0.

Question: will the same thing happen as x values get closer to

positive infinity?


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How about a guess? What do you think is going to happen to the y

values of our function as the x values get closer to positive infinity?

?2

1, p

!gpx

yxAs

By looking at the fraction analytically, you can hopefully see that

very large values of x will generate values of y very close to 0. If

you are uneasy about this,expand the table in the previous slide toinclude values like x = 10, 100, or 1000.

On the next frame then, is our final graph for this problem

02

1, p!gp xyxAs


13/24

x

yy = 1/(x-2)

Vertical Asymptote

at x = 2

Horizontal

Asymptote at y = 0.

Note how the graph is very much dominated by its asymptotes. You

can think of them as magnets for the graph. This problem was an

exploration but in the future, it will be very important to knowwhereyour asymptotes are before you start plotting points.


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Next up is the graph of one of the functions that was mentioned

back in frame #2.

xxxf

!

32)(

Lets see ifwe can pick out the asymptotes analytically before we

start plotting points in an x-y table.Do we have a vertical asymptote? If so, at what value of x?

We have a vertical asymptote at x = 3 because at that value of x, the

denominator is 0 but the numerator is not. Congratulations if you

picked this out on your own.

The horizontal asymptote is a little more challenging, but go ahead

and take a guess.


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Notice though that as values of x get larger and larger, the 3 in the

denominator carries less and less weight in the calculation.

x

xxf

!

3

2)( As the 3 disappears, the function looks

more and more like

xxxf

! 2)(

which reduces to y = -2.

This means that we should have a horizontal asymptote at y = -2.

We already have evidence of a vertical asymptote at x = 3. So we

are going to set up the x-y table then with a few values to the leftof x = 3 and a few values to the right of x = 3. To confirm the

horizontal asymptote we will also use a few large values of x just

to see if the corresponding values of y will be close to y = -2.


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Take a few minutes and work out the

y values for this table.

Dont be lazy now,work them out

yourself.

As expected, y values tend to explode

when they get close to the vertical

asymptote at x = 3.

Also, as x values get large, y values

get closer and closer to the horizontal

asymptote at y = -2.

The graph is a click away.

x

-5

0

2.5

3

3.5

5

10

50

x

xy

!

3

2

-10/8 = -1.25

0

5/.5 = 10

Undefined

7/-.5 = -14

10/-2 = -5

20/-7 = -2.86

100/-47= -2.13


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x

yy = 2x/(3-x)

Here is the graph

with most of the

points in our table.

Vertical asymptote

at x = 3.

Horizontal

asymptote at y = -2.


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Believe it or not, you are now sophisticated enough mathematically

to draw conclusions about the graph three ways:

Analytically:

finding

asymptotes with

algebra!!

Numerically:

supporting and

generating

conclusions

with the x-y

table!!

Graphically: a

visual look at the

behavior of the

function.

If your conclusions from the above areas do not agree, investigate

further to uncover the nature of the problem.


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We are going to finish this lesson with an analysis of the third

function that was mentioned in the very beginning:

xx

xxg

2

4)(

2

2

! This is a rational function so we have

potential for asymptotes and this is

what we should investigate first. Take

a minute to form your own opinion

before you continue.

Hopefully you began by setting the denominator equal to 0.

2,0

02

022

!!

!

!

xx

xx

xx It appears that we may have verticalasymptotes at x = 0 and at x = 2. We will

see if the table confirms this suspicion.


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xx

x

y 2

4

2

2

!x

-2

-1

-.5

-.1

0

11.5

1.9

2

0

-1

-3

-19

Und

32.33

2.05

und

See anything peculiar?

Notice that as x values get closer and

closer to 0, the y values get larger and

larger. This is appropriate behavior

near an asymptote.

But as x values get closer and closer

to 2, the y values do not get large. In

fact, the y values seem to get closer

and closer to 2.

Now, if x =2 creates 0 in thedenominatorwhy dont we have an

asymptote at x = 2?


21/24

We dont get a vertical asymptote at x = 2

because when x = 2 both the numerator and

the denominator are equal to 0. In fact, ifwehad thought to reduce the function in the

beginning,we could have saved ourselves a lot

of trouble. Check this out:

xx

xxxx

xxxy 2

222

24

2

2

!

!!

Does this mean thatxx

xy

2

42

2

! and

x

xy

2!

are identical functions?

Yes, at every value of x except x = 2where the former is undefined.

There will be a tiny hole in the graph where x = 2.


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As we look for horizontal asymptotes,we

look at y values as x approaches plus or

minus infinity. The denominatorwill getvery large but so will the numerator.

As was the case with

the previous function,

we concentrate on the

ratio of the term withthe largest power of x

in the numerator to

the term with the

largest power of x in

the denominator.As xgets large

xx

xxg

2

4)(

2

2

!

12

42

2

2

2

!}

x

x

xx

x

You can verify this in the table.

x

xxxy

24

2

2

!

10

100

1000

1.2

1.02

1.002

So,we have a horizontal asymptote at

y = 1.


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To summarize then,we have a vertical asymptote at x = 0, a hole in

the graph at x = 2 and a horizontal asymptote at y = 1. Here is the

graph with a few of the points that we have in our tables.

x

y

Horizontal asymptoteat y = 1.

Vertical asymptote at

x = 0.

xx

xxg

2

4)(

2

2

! Hole in the graph.


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Now you will get a chance to practice on exercises that use the

topics that were covered in this lesson:

Finding vertical and horizontal asymptotes in rational functions.

Graphing rational functions with asymptotes.

Good luck and watch out for those asymptotes!

graphs of rational functions prepared for mth

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