Rational Functions and Asymptotes:
A rational function is a function of the form
where P(x) and Q(x) are polynomials (that we assume without common factors). When graphing a rational function we must pay special attention to the behavior of the graph near the x-values which make the denominator equal to zero.
)(
)()(
xQ
xPxr
Example 1:
Find the domain of each of the following rational functions:
4
3)(
2
23
1
x
xxxr
The domain well be all x’s such that x2 – 4 ≠ 0.
042 x
42 x4x 2 ,22,22,
Consequently, the domain is all x’s except -2 and 2. That is,
4
52)(
22
x
xxr The domain well be all x’s such that x2 + 4 ≠ 0.
042 x
42 x
This is impossible! Consequently, the domain is all real numbers.
xxxr
4
5)(
23
The domain well be all x’s such that x2 – 4x ≠ 0.
042 xx 04 xx
0x ,44,00,
Consequently, the domain is all x’s except 0 and 4. That is,
04 x4x
Example 2:
Sketch the graphs of:
xxf
1)(
x y
0 undefined
1 1
2 1/23 1/3
1/2 2
1/3 3
1/4 4
x y
-1 -1
-2 -1/2
-3 -1/3-1/2 -2
-1/3 -3
-1/4 -4
Definition of Vertical and Horizontal Asymptotes:
The line x = a is a vertical asymptote of y = r(x) if y → ∞ or y → −∞ as x → a+ or x → a−
Note: x → a− means that “x approaches a from the left”;x → a+ means that “x approaches a from the right”.
Asymptotes of Rational Functions:
The vertical asymptotes of r(x) are the lines x = a, where a is a zero of the denominator.
If n < m, then r(x) has horizontal asymptote y = 0.If n = m, then r(x) has horizontal asymptote y = .If n > m, then r(x) has no horizontal asymptote.
Let r(x) be the rational function:
011
1
011
1
...
...)(
bxbxbxb
axaxaxaxr
mm
mm
nn
nn
m
n
b
a
Example 3:
Find the horizontal and vertical asymptotes of each of the following functions:
65
26)(
2
xx
xxr
Vertical Asymptotes
0652 xx
061 xx
01 x 06 x
1x 6x
Horizontal Asymptotes
)26deg( x 1 )65deg( 2 xx 2
Consequently, r(x) has horizontal asymptote y = 0.
xx
xxt
23
3)(
2
Vertical Asymptotes
023 xx
03 x 02 x
3x x2
Horizontal Asymptotes
)3deg( 2x 2
2Consequently, r(x) has horizontal asymptote y =
xx 23 xxx 362 2 xx 62
)6deg( 2 xx
1
3
3y
4
62)(
2
24
x
xxxs
Vertical Asymptotes
042 x
42 x
4x 2
Horizontal Asymptotes
)62deg( 24 xx 4 )4deg( 2x 2
Consequently, r(x) has no horizontal asymptote.
Example 4:
Graph the rational function
65
6)(
2
xxxr
intercept -x
0y
65
60
2
xx
This is impossible so there are NO x-intercepts
intercept -y 0x
6050
6)0( 2 r
6
6
1
So the y-intercept is (0,-1)
Vertical Asymptotes
0652 xx
Horizontal Asymptotes
)6deg( 0 )65deg( 2 xx 2
Consequently, r(x) has horizontal asymptote y = 0.
65
6)(
2
xxxr
061 xx
01 x 06 x
1x 6x
65
6)(
2
xxxr
1. No x-intercepts2. y-intercept is (0,-1)3. Vertical Asymptotes x = – 1 and x = 64. Horizontal Asymptotes y = 0
6252
6)2( 2 r
6104
6
614
6
8
6
43,2
6858
6)8( 2 r
64064
6
624
6
18
6
31,8
Example 5:
Graph the rational function
31
21)(
xx
xxxr
intercept -x
0y
intercept -y 0x
3
2
So the y-intercept is
31
210
xx
xx
210 xx
10 x 20 x
x1 x 2
3010
2010)0(
r
32,0
0,1 0,2
31
21
3
2
Vertical Asymptotes
031 xx
Horizontal Asymptotes
2
Consequently, r(x) has horizontal asymptote y = .
01 x 03 x1x 3x
31
21)(
xx
xxxr
31 xx 332 xxx 322 xx
21 xx 222 xxx 22 xx
21deg xx
31deg xx 2
1
1