MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Summer 2016
Ex 1: From a rectangular piece of cardboard having dimensions 20 inches by 30 inches, an open box is to be
made by cutting out identical squares of area x2 from each corner and, turning up the sides and taping.
x
x
20
x
20 − 2𝑥
30
30 − 2𝑥
(a)
Find all positive values of x such that 𝑉(𝑥) > 0, and sketch the graph of V for 𝑥 > 0.
𝑉(𝑥) = 𝑥(20 − 2𝑥)(30 − 2𝑥)
Hint: Find the zeros and make a sign chart.
V
x V
0
2
4
5
6
8
10
12
14
16
20
x
(b)
What is the ‘reasonable’ domain of this volume function?
(c)
(d)
Can there exist a volume of 0?
Estimate a maximum volume for the open box.
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MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Summer 2016
Asymptotes and Holes
Consider the following function:
x
x
1)
Let x→3 or −3 What is the value of y?
y 2

x  9 ( x  3)( x  3)
As x gets very, very close to −3 𝑜𝑟 3, the denominator becomes a very, very small decimal.
x
The value of
  or  
very, very small decimal
The domain of the function above does not include −3 𝑜𝑟 3. 𝐷 = (−∞, −3) ∪ (−3,3) ∪ (3, ∞).
In fact, from the argument above, at x  3 or x  3 , the function is approaching ∞ or -∞. The graph of the
function would be getting greater and greater or lesser and lesser. The graph would be going toward infinity or
negative infinity. This indicates there would be vertical asymptotes at 𝑥 = −3 𝑎𝑛𝑑 𝑥 = 3.
An asymptote is a line that the graph of a function will approach, but probably not intersect. **There is an
exception. Some graphs of functions may intersect horizontal asymptotes. However, a graph of a function will
never, never intersect vertical asymptotes.
Ex 2: Usually vertical asymptotes of rational functions may be found by determining what values of x made
zero denominators.
As x  2, y   or  
x9
g ( x) 
As x  5, y   or  
( x  2)( x  5)
( x  2)( x  5)  0
x  2  0 or
x 5  0
x  2
x5
Above are the equations for the vertical asymptotes of g.
Below is a graph of function g with the asymptotes shown.
y
5
4
3
2
1
-4
-2
2
-1
4
6
8
10
x
-2
-3
-4
-5
-6
2
MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Summer 2016
Ex 3: As seen in example 2, usually vertical asymptotes may be found by setting the denominator of a rational
function equal to zero. However, if there is a factor of the denominator that cancels with a factor in the
numerator, that factor does not yield a vertical asymptote; it yields the location of a ‘hole’ in the graph.
x2  2x
x( x  2)
x
f ( x)  2


x  4 ( x  2)( x  2) x  2
The factor 𝑥 − 2 cancels from the denominator and numerator. At x = 2, there is a hole in the graph.
As x  2, y   or  .
However, as x  2, y  a hole in the graph at
The graph of f is the same as the graph of g ( x) 
[ g ( x) 
x
2
1
 g (2) 
 ]
x2
22 2
1
(2, )
2
x
x2
2 1
when x  2, f (2)  
4 2
simplified f ( x) 
x
1
, except for a hole when 𝑥 = 2 or at the point (2, 2).
x2
𝟏
There is a vertical asymptote of 𝒙 = −𝟐 and a hole at (𝟐, 𝟐).
To determine when a denominator indicates vertical asymptotes or holes (or neither), let’s examine the
following (1 – 3).
2x
(1)
As x gets very, very large (𝑥 → ∞), in the denominator the 5𝑥 does not add very
y 2
x  5x
2x
much onto the 𝑥 2 . Basically, we are comparing 2 . As 𝑥 → ∞, the denominator 𝑥 2 is ‘growing’ or
x
‘increasing’ much faster than the numerator, 2𝑥. The result is a value approaching zero.
As x  , y  0
As x  , y  0
2 x3  x 2
(2)
As x becomes very large, x→∞; the denominator and numerator are growing or
y 3
3x  12 x
increasing at about the same rate. In the numerator, subtracting 𝑥 2 does not change the 2𝑥 3 much. In the
2 x3
denominator, subtracting the 12𝑥 does not change the 3𝑥 3 much. Therefore the resulting ratio,
, yields a
3x3
2
resulting value of 3.
As x  , y 
2
3
As x  , y 
(3)
y
3x3
x2  2
2
3
As x gets very, very large, x→∞; the numerator is growing or increasing much
faster than the denominator. The resulting ratio is
As x  , y   or 
As x  , y   or 
3x 3
, which will approach infinity.
x2
3
MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Summer 2016
The logical arguments on the previous page lead to the following conclusions about how to find
horizontal asymptotes’ equations of rational functions.
Compare the ratio of the term of the highest degree in the numerator to the term of the highest degree in the
ax n
denominator. (Let’s let this ratio be k .) The horizontal asymptote, if it exists, is given by:
bx
(1)
y  0 (x-axis) if n  k In other words, if the degree of the numerator is less than the degree of
the denominator, the horizontal asymptote is the x-axis.
a
(2)
In other words, if the degree of the numerator equals the degree of the
y  if n  k
b
a
denominator, the horizontal asymptote is the line y  (the ratio of the leading terms in
b
numerator and denominator).
(3)
There is no horizontal asymptote, if the degree of the numerator is greater than the degree of the
denominator. As x gets very large or very small, the function values get very large or very small
as well.
One remark about horizontal asymptotes: Vertical asymptotes are ‘sacred’; they will never intersect
with the graph of the function. Horizontal asymptotes are not ‘sacred’; they may intersect with the
graph of the function, usually ‘close to the origin’.
3x
. You might also need a few points and determine if
x2
the graph will intersect the horizontal asymptote, if it exists. As sign chart might also help determine when the
function is positive and when it is negative.
Ex 4: Use asymptotes to sketch the graph of f ( x) 
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MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Summer 2016
x2
. You might also make a sign
x  x6
chart to determine when the graph is above or below the x-axis. You might need to determine if the graph
intersects the horizontal asymptote and where.
Ex 5: Use asymptotes and a few points to sketch the graph of g ( x) 
2
5
MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 6: Use asymptotes, a few points, and a sign chart to help sketch the graph of f ( x) 
Summer 2016
4
. Does the
( x  2)2
graph intersect the horizontal asymptote?
6
MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 7: Use asymptotes, a few points, and a sign chart to help sketch the graph of h( x) 
Summer 2016
x 3
. Does the graph
x2 1
intersect the horizontal asymptote?
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MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 8: Use asymptotes, a few point, and a sign chart to help sketch the graph of y 
Summer 2016
3x 2  3x  36
. Does the
x2  x  2
graph intersect the horizontal asymptote? If so, where?
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MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 9: Use asymptotes, a few points, and a sign chart to help sketch the graph of f ( x) 
Summer 2016
x2  x  6
. Does the
x2  2 x  3
graph intersect the horizontal asymptote?
9
MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 10: Use asymptotes, a few points, and a sign chart to help sketch the graph of g ( x) 
Summer 2016
x2  x  2
.
x2
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MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 11: Use all of the information asked for in previous examples to sketch the graph of y 
Summer 2016
x 1
. Does the
1  x2
graph intersect a horizontal asymptote?
11
MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Ex 12: Use all of the information you can find to sketch the graph of f ( x) 
Summer 2016
x2  4 x  4
. Does the graph
x 2  3x  2
intersect a horizontal asymptote?
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MA 15800
Lesson 11
Rational Functions: Asymptotes, Holes, Intercepts
Summer 2016
Ex 13: Write an equation for a rational function f that satisfies the given conditions.
Vertical asymptote: 𝑥 = 2
Horizontal asymptote: 𝑦 = 0
y-intercept: 𝑓(0) = −4
Ex 14: Write an equation for a rational function g that satisfies the given conditions.
Vertical asymptotes: 𝑥 = −2, 𝑥 = 3
Horizontal asymptote: 𝑦 = −2
x-intercepts: (−3,0), (2,0)
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