3.7 Graphing Rational Functions

Obj: graph rational functions with asymptotes and holes and evaluate limits of rational functions

Graph

f(x) =

g(x) =

Infinite versus Removable Discontinuity (asymptote) (point)

9

22

x

x

9

32

x

x

2

( 3)( 3)

x

x x

3

( 3)( 3)

x

x x

DefinitionsLet y =

y has an infinite discontinuity at x = a if n > m

(because the factors cancelled, and there are only

factors left over in the denominator)

y has a removable (point) discontinuity at x = a if m > n

(because the factors cancelled, and there are only

factors left over in the numerator)

)()(

)()(

xqax

xpaxn

m

Examplesf(x) =

Factored:

Since x – 3 cancels in the numerator and denominator,the graph has removable discontinuity at x = 3.

2 6 9

3

x x

x

( 3)( 3)

3

x x

x

TheoremLet h be a rational function,

If h has an infinite discontinuity at x = a, then the graph of h has a vertical asymptote at x = a.

If h has a removable discontinuity at x = a, then the graph of h has a hole at x = a.

Horizontal Asymptotes

Use limits to find any horizontal asymptotes of the graph of each function.

To find limits as x approaches infinity,

substitute ∞ for x, and evaluate.

74159

3

3

lim

xxx

x

93

2lim

xx

x

xxx

x

312

lim

Practice

Calculate each limit.

413

2

2

lim

xx

x

53710

3

3

lim

xx

x

1031

3

2

lim

xx

x

xxx

x 34510

2

3

lim

Find AsymptotesFind the vertical asymptote(s) and

horizontal asymptote, if any.

1) Factor, cancel any like factors.2) To find vertical asymptotes, set denominator = 0,

solve for x.3) To find horizontal asymptotes, find the limit as x

approaches infinity.

Does the function have any holes?

8243

2

2

)(xxxxxf

Sketch a graph

Sketch the graph using the vertical and horizontal asymptotes.

f(x)=1) Factor and cancel to find

holes

2) Locate asymptotes

3) Locate x- and y-intercepts

4) Graph

1

42

2

x

x

Your Turn

Round Table Practice

Assignment page 215 - 21710 – 12, 15, 18, 19a, b,

22a, b, c