PreCalculus Class Notes RF4 End Behavior of Rational Functions
Rational
function (single
fraction)
Sum of proper
rational function
and polynomial
End behavior
asymptote
Graph
y
5
4
3
2
1
y=
2x
x −1
y = 0+
2
2 x2 − 8
y= 2
x − 2x +1
x2 − 4
y=
2x − 2
y = 2+
y=
2x
x2 −1
4 x − 10
x − 2x +1
2
1
1
−3
x+ +
2
2 2x − 2
y=0
horizontal
y=2
horizontal
(and look, the
function
crosses its
horizontal
asymptote!)
1
1
x+
2
2
slant
y=
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
1
2
3
4
5
x
1
2
3
4
5
x
1
2
3
4
5
y
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
−6
−7
−8
−9
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
y
140
130
120
110
100
90
80
70
60
50
40
3
x
y=
x−3
27
y = x + 3x + 9 +
x−3
2
30
2
y = x + 3x + 9
non-linear
20
10
−9
−8
−7
−6
−5
−4
−3
−2
−1−10
−20
−30
−40
−50
−60
−70
−80
−90
−100
−110
−120
−130
−140
x
1
2
3
4
5
6
7
8
9
x
Finding end behavior asymptotes: rewrite as the sum of a polynomial and a proper rational expression
Proper rational expression: the degree of the numerator is less than the degree of the denominator
Examples
y=
3x
x −x+4
2
x2 − 4x − 5
y=
x2 + x
y=
y=
3x − 4 x 2
x2 + 5
x2
( x − 2 )( x + 3)
2
y=
2 x 2 + 3x
x−2
Write the equation of the rational function whose graph has
a zero (root) at x = 4;
a zero at x = 2;
an odd vertical asymptote at x = 1;
an even vertical asymptote at x = –3;
and a horizontal asymptote at y = 5.