3.7 Notes

Graphing Rational Functions

3.7 NotesUnlike polynomial functions which are continuous, rational functions have discontinuities.

types of discontinuities:

jump – associated with piece-wise functions

point

infiniteassociated with rational functions

3.7 Notes

Holes in a graph are point discontinuities. A hole is the “absence of a point” in a line or curve.

3.7 NotesAsymptotes are infinite discontinuities. Rational functions may have vertical, horizontal, and/or slant asymptotes.

3.7 Notes

1. Find and plot x and y intercepts.2. Use limit theorems to find and graph the

discontinuities.a. Check for holes. b. Check for horizontal asymptotes.c. Check for slant asymptotes.d. Check for vertical asymptotes.

3. Use limits to determine the behavior of the graph between discontinuities.

4. Sketch a smooth curve.

Finding the discontinuities:

a. Check for holes. The function may have a hole if there is a common factor in the numerator and denominator. If so, apply the theorem to find the coordinates of the hole:

If is a common factor of the numerator and denominator of f(x), then

is a hole.

x a

, lim ( )x a

a f x

Finding the discontinuities

b. Check for horizontal asymptotes. The function will have a horizontal asymptote if the degree of the numerator is less than or equal to the degree of the denominator. If so, apply the theorem to find the equation of the horizontal asymptote:

is a horizontal asymptote of f(x) if

or if

y b

lim ( )x

f x b

lim ( )x

f x b

Finding the discontinuities

c. Check for slant asymptotes. The function may have a slant asymptote if the degree of the numerator is one more than the degree of the denominator. If so, apply the theorem to find the equation of the slant asymptote:

The oblique line is a slant asymptote of f(x) if

or if

when f(x) is in quotient form.

y mx b

lim ( )x

f x mx b

lim ( )x

f x mx b

Finding the discontinuities

d. Check for vertical asymptotes. The function may have vertical asymptotes if the denominator is zero for some value(s) of x. If so, apply the theorem to find the equation of the vertical asymptote(s):

is a vertical asymptote of f(x) if

or if

from the left or the right.

lim ( )x a

f x

x a

lim ( )x a

f x

3.7 Notes

Example #1: Find the discontinuities of

Check for holes:

is a hole.

2 3 10

2

x xf x

x

2

2, lim 5xx

2 5

2

x xf x

x

2, 7

3.7 Notes

Example #1: Find the discontinuities of

Check for horizontal asymptotes:

The degree of the numerator is greater than the degree of the denominator; this rational function has no horizontal asymptotes.

2 3 10

2

x xf x

x

3.7 Notes

Example #1: Find the discontinuities of

Check for slant asymptotes:

The degree of the numerator is one more than the degree of the denominator. This rational function may have a slant asymptote.

2 3 10

2

x xf x

x

3.7 Notes

Example #1: Find the discontinuities of

Check for slant asymptotes:

Divide to put into quotient form:

2 3 10

2

x xf x

x

( ) 5f x x

3.7 Notes

Check for slant asymptotes:

Take the limit as x approaches infinity:

may be a slant asymptote. (It’s not, it is actually the graph of the function.)

lim 5xx

5x

5y x

3.7 Notes

Example #1: Find the discontinuities of

Check for vertical asymptotes:

If x = 2, the denominator is zero.

2 3 10

2

x xf x

x

3.7 Notes

Check for vertical asymptotes:

x = 2 is not a vertical asymptote.

This function does not have any vertical asymptotes.

7 2

2 5lim

2x

x x

x

3.7 Notes

Example #1: Find the discontinuities of

This function has a hole at . It does not have any horizontal or vertical asymptotes. It may have a slant asymptote at y = x + 5, but it doesn’t.

2 3 10

2

x xf x

x

2, 7

3.7 Notes

Example #2: Find the discontinuities of

Check for holes:

There are no common factors in the numerator and denominator; this function has no holes.

3 2

3

8 16

xf x

x x x

3

4 4

xf x

x x x

3.7 Notes

Example #2: Find the discontinuities of

Check for horizontal asymptotes:

The degree of the numerator is less than the degree of the denominator; the function has a horizontal asymptote.

3 2

3

8 16

xf x

x x x

3.7 NotesCheck for horizontal asymptotes:

y = 0 is a horizontal asymptote.

3 2

3lim

8 16x

x

x x x

0

3.7 Notes

Example #2: Find the discontinuities of

Check for slant asymptotes:

The degree of the numerator is less than the degree of the denominator; the function has no slant asymptote.

3 2

3

8 16

xf x

x x x

3.7 Notes

Example #2: Find the discontinuities of

Check for vertical asymptotes:

If x = 0 or x = 4, the denominator is zero.

3

4 4

xf x

x x x

3.7 Notes

Check for vertical asymptotes:

x = 0 is a vertical asymptote.

0

3lim

4 4x

x

x x x

0

3lim

4 4x

x

x x x

3.7 Notes

Check for vertical asymptotes:

x = 4 is a vertical asymptote.

4

3lim

4 4x

x

x x x

4

3lim

4 4x

x

x x x

3.7 Notes

Example #2: Find the discontinuities of

This function does not have any holes. It has a horizontal asymptote whose equation is y = 0, no slant asymptote, and two vertical asymptotes whose equations are x = 0 and x = 4.

3 2

3

8 16

xf x

x x x

Practice:

Find the discontinuities of the following rational functions.

1.

2.

2

2

4

12

x xf x

x x

2 2 1x x

f xx

3.7 NotesPractice #1: Find the discontinuities of

Check for holes:

is a hole.

2

2

4

12

x xf x

x x

44, lim

3x

x

x

4

3 4

x xf x

x x

44,

7

3.7 Notes

Practice #1: Find the discontinuities of

Check for horizontal asymptotes:

The degree of the numerator is equal to the degree of the denominator; the function has a horizontal asymptote.

2

2

4

12

x xf x

x x

3.7 NotesCheck for horizontal asymptotes:

y = 1 is a horizontal asymptote.

2

2

4lim

12x

x x

x x

1

3.7 Notes

Practice #1: Find the discontinuities of

Check for slant asymptotes:

The degree of the numerator is equal to the degree of the denominator; the function has no slant asymptote.

2

2

4

12

x xf x

x x

3.7 Notes

Practice #1: Find the discontinuities of

Check for vertical asymptotes:

If x = -3 or x = 4, the denominator is zero.

4

3 4

x xf x

x x

3.7 Notes

Check for vertical asymptotes:

x = 4 is not a vertical asymptote.

4

4lim

3 4x

x x

x x

4

7

4lim

3x

x

x

3.7 Notes

Check for vertical asymptotes:

x = -3 is a vertical asymptote.

3

3

4lim

3 4

4lim

3 4

x

x

x x

x x

x x

x x

3lim

3x

x

x

3

lim3x

x

x

3.7 NotesPractice #1: Find the discontinuities of

This function has a hole at .

It has a horizontal asymptote, y = 1.

It does not have a slant asymptote.

It has a vertical asymptote, x = -3.

2

2

4

12

x xf x

x x

44,

7

3.7 Notes

Practice #2: Find the discontinuities of

Check for holes:

There are no common factors in the numerator and denominator; this function has no holes.

2 2 1x x

f xx

1 1x xf x

x

3.7 Notes

Practice #2: Find the discontinuities of

Check for horizontal asymptotes:

The degree of the numerator is greater than the degree of the denominator; this rational function has no horizontal asymptotes.

2 2 1x x

f xx

3.7 Notes

Practice #2: Find the discontinuities of

Check for slant asymptotes:

The degree of the numerator is one more than the degree of the denominator. This rational function may have a slant asymptote.

2 2 1x x

f xx

3.7 Notes

Practice #2: Find the discontinuities of

Check for slant asymptotes:

Divide to put into quotient form:

2 2 1x x

f xx

1( ) 2f x x

x

3.7 Notes

Check for slant asymptotes:

Take the limit as x approaches infinity:

may be a slant asymptote.

1lim 2xx

x 2x

2y x

0

3.7 Notes

Practice #2: Find the discontinuities of

Check for vertical asymptotes:

If x = 0 the denominator is zero.

2 2 1x x

f xx

3.7 Notes

Check for vertical asymptotes:

x = 0 is a vertical asymptote.

0

1 1limx

x x

x

0

1 1limx

x x

x

3.7 Notes

Practice #2: Find the discontinuities of

This function does not have a hole or a horizontal asymptote.

It has a slant asymptote, y = x – 2.

It has a vertical asymptote, x = 0.

2 2 1x x

f xx