MAC 1105
Rational Functions
Section 3.5
Apply Notation Describing Infinite Behavior of a Function
Let p(x) and q(x) be polynomials where q(x) ≠ 0. A function f defined by
Rational Function:
f ( x) 
Notation
Meaning
x approaches c from the right (x comes in from the right side of c, x ≠ c).

xc
x  c
x approaches c from the left (x comes in from the left side of c, x ≠ c).
x 
x  
1.
p ( x)
is called a rational function.
q ( x)
x approaches infinity (x increases without bound: go right on the x-axis).
x approaches – infinity (x decreases without bound: go left on the x-axis).
The graph of f  x  
x 1
is given. Complete the statements.
x3
As x   , f  x   ______
As x   , f  x   ______
As x  3 , f  x   ______

As x  3 , f  x   ______
Identify Vertical Asymptotes
Vertical Asymptote:
As x  c  , f  x   
The line x = c is a vertical asymptote of the graph of a function f if
f (x) approaches infinity or negative infinity as x approaches c from either side.
As x  c  , f  x   
As x  c  , f  x   
As x  c  , f  x   
The function will not cross a vertical asymptote. Think of VA as walls the graph will not go through.
Since we cannot divide by zero, we want to take out any value for x that makes the denominator zero. This will be
a list of the form x = constant.
Let f ( x) 
p ( x)
where p(x) and q(x) have no common factors other than 1. To locate the vertical asymptotes of
q ( x)
f  x  , determine the real numbers x where the denominator is zero, but the numerator is nonzero.
Identify the vertical asymptotes (if any).
2.
f ( x) 
5
x 8
5.
f  x 
x 3
x  4x  5
3.
f ( x) 
2x
x  25
4.
2
2
6.
f  x 
f ( x) 
4
x 9
x 3
x  2x  3
2
Identify Horizontal Asymptotes
Horizontal Asymptote: The line y = d is a horizontal asymptote of the graph of a function f
if f (x) approaches d as x approaches infinity or negative infinity.
As x  , f  x   d
Think of these as horizontal walls – but these might be crossed.
As x  , f  x   d
2
Let f be a rational function defined by f ( x) 
an x n  an1 x n1  an 2 x n2  ...  a1 x  a0
bm x m  bm1 x m1  bm2 x m2  ...  b1 x  b0
where n is the degree of the numerator and m is the degree of the denominator.
one horizontal asymptote at y  0 (the x-axis) if n  m (top power smaller than bottom power)
an
(ratio of lead coefficients) if n  m (top power equals bottom power)
bm
no horizontal asymptote if n  m (top power bigger than bottom power)
one horizontal asymptote, y 
Identify the horizontal asymptotes (if any).
5.
f  x 
3x
2
x 9
6.
f  x 
3x 2
x2  9
7.
How do I know if a function crosses its horizontal asymptote?
8.
f  x 
3x
2
x 9
f  x 
3x 2
x2  9
f  x 
3x3
x2  9
Graph Rational Functions
f ( x) 
p ( x)
where p(x) and q(x) are polynomials with no common factors.
q ( x)
Asymptotes: Determine if the function has vertical asymptotes and graph them as dashed lines.
Determine if the function has a horizontal asymptote and graph as a dashed line.
Determine if the function crosses the horizontal asymptote.
Intercepts:
Determine the x- and y-intercepts (easy points)
Plot points:
Plot points (you pick) on the left and right of the x-intercepts, vertical asymptotes.
Find the point where the function crosses a horizontal asymptote (if any).
Graph. Identify asymptotes.
9.
y
1
x2
5
y
x
-5
5
-5
10. y 
1
( x  2) 2
5
y
x
-5
5
-5
11. y 
2x
x3
Will Casio help? Your Casio will make a table, but picking points might be easier.
Start? –1
End? –5
Step? 1