Consider the rational function below. 2

2

2( )1

xf xx

We know that since d = n, f has a horizontal asymptote at y = 2.

Since a rational function is telling us to divide, let’s do so.

2 21 2 0 0x x x 2

22x 2−−2

2

21x

Consider the rational function below. 2

2

2( )1

xf xx

We know that since d = n, f has a horizontal asymptote at y = 2.

2

2( ) 21

f xx

So f (x) can be rewritten as:

Approaches 0 as x → ∞

And our graph is trying to look like y = 2 at large values of x.

Consider the rational function below. 3

2

2( )1

xf xx

We know that since d < n, f has no horizontal asymptotes.

Since a rational function is telling us to divide, let’s do so.

2 3 21 2 0 0 0x x x x 2x

32x 2x−−2x

2

21

xx

Consider the rational function below. We know that since d < n,

f has no horizontal asymptotes.

2

2( ) 21

xf x xx

So f (x) can be rewritten as:

Approaches 0 as x → ∞

And our graph is trying to look like y = 2x at large values of x. This is called a slant asymptote.

3

2

2( )1

xf xx

Slant AsymptotesA rational function has a slant asymptote if

n = d + 1– The degree of the numerator is one more than

the degree of the denominator

To find the equation of a slant asymptote, use long division and forget about the remainder.– At large values of x, the remainder approaches 0

anyway.

Exercise 1Find all asymptotes of the rational function.

22 15 8( )3

x xf xx

To Graph:To graph a rational function:1. Factor N(x) and D(x).2. Find vertical asymptotes (where D(x) = 0) and plot as

dashed lines.– If a factor cancels, it is not an asymptote (A Hole)

3. Find horizontal asymptote (comparing d and n) and plot as a dashed line.

4. Find slant asymptote (by long division w/o the remainder) and plot as a dashed line.

5. Plot x- and y-intercepts.– If a factor cancels, it is not a zero (A Hole)

6. Use smooth curves to finish the graph.

More on AsymptotesVertical Asymptotes:• Your graph can never cross one!• If x = a is a vertical asymptote, then

interesting things happen really close to a:– f (x) could approach +∞ or −∞– Think of vertical asymptotes as black holes

that attract values near a

More on AsymptotesVertical Asymptotes:• The end behavior around a vertical

asymptote is similar to that of polynomials:

V.A. at x = 1 (multiplicity of 1) V.A. at x = 1 (multiplicity of 1)

More on AsymptotesVertical Asymptotes:• The end behavior around a vertical

asymptote is similar to that of polynomials:

V.A. at x = 1 (multiplicity of 2) V.A. at x = 1 (multiplicity of 2)

More on AsymptotesHorizontal

Asymptotes:• Your graph can

cross one!• Attracts values

approaching +∞ or −∞

More on AsymptotesSlant Asymptotes:• Your graph can

cross one of these, too!

• Attracts values approaching +∞ or −∞

Exercise 2Graph:

1( )3

f xx

Exercise 3Graph:

2 1( ) xf xx

Exercise 4Graph:

2

3( )2

xf xx x

Exercise 5Graph:

2

3( )( 2)( 1)

xf xx x

Exercise 6Graph:

2 2

3( )( 2) ( 1)

xf xx x

Exercise 7Graph:

2

2

4( )4 4

xf xx x

Exercise 8Graph:

2

( )1

x xf xx