C7 – Asymptotes of Rational and
Other Functions
IB Math HL/SL - Santowski
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(A) Introduction
• To help make sense of any of the following
discussions, graph all equations and view
the resultant graphs as we discuss the
concepts
• It may also be helpful to use the graphing
technology to generate a table of values as
you view the graphs
• Use either WINPLOT or a GDC
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(A) Review of Rational Functions
• A rational number is a number that can be written in the
form of a fraction. So likewise, a rational function is
function that is presented in the form of a fraction.
• We have seen two examples of rational functions in this
course  we can generate a graph of polynomials when
we divide them in our work with the Factor Theorem 
i.e. Q(x) = (x3 - 2x + 1)/(x - 1). If x-1 was a factor of P(x),
then we observed a hole in the graph of Q(x). If x-1 was
not a factor of P(x), then we observed an asymptote in the
graph of Q(x)
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(A) Review of Rational Functions
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(A) Review of Rational Functions
•
We have seen several examples of
rational functions in this course, when
we investigated the reciprocal functions
of linear fcns, i.e. f(x) = 1/(x + 2) and
quadratic fcns, i.e. g(x) = 1/(x2 - 3x 10) and the tangent function y = tan(x).
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(B) Domain, Range, and Zeroes of
Rational Functions
• Given the rational function r(x) = n(x)/d(x) ,
• The domain of rational functions involve the fact that we cannot divide
by zero. Therefore, any value of x that creates a zero denominator is a
domain restriction. Thus in r(x), d(x) cannot equal zero.
• For the zeroes of a rational function, we simply consider where the
numerator is zero (i.e. 0/d(x) = 0). So we try to find out where n(x) = 0
• To find the range, we must look at the various sections of a rational
function graph and look for max/min values
• EXAMPLES: Graph and find the domain, range, zeroes of
• f(x) = 7/(x + 2),
• g(x) = x/(x2 - 3x - 4), and
• h(x) = (2x2 + x - 3)/(x2 - 4)
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(B) Domain, Range, and Zeroes of
Rational Functions
•
EXAMPLES: Graph and find the
domain, range, zeroes of
• f(x) = 7/(x + 2),
• g(x) = x/(x2 - 3x - 4), and
• h(x) = (2x2 + x - 3)/(x2 - 4)
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(C) Vertical and Horizontal
Asymptotes
• Illustrate with a graph of y = 1/x and draw several others (i.e. pg 348)
• A vertical asymptote occurs when the value of the function increases
or decreases without bound as the value of x approaches a from the
right and from the left.
• We symbolically present this as f(x)  + ∞ as x  a+ or x  a• We re-express this idea in limit notation  lim x  a+ f(x) = + ∞
• A horizontal asymptotes occurs when a value of the function
approaches a number, L, as x increases or decreases without bound.
• We symbolically present this as as f(x)  + a as x  + ∞ or x  - ∞
• We can re-express this idea in limit notation  lim x  ∞ f(x) = a
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(D) Finding the Equations of the
Asymptotes
• To find the equation of the vertical asymptotes, we simply find the
restrictions in the denominator and there is our equation of the
asymptote i.e. x = a
• To find the equation of the horizontal asymptotes, we can work
through it in two manners. First, we can prepare a table of values and
make the x value larger and larger positively and negatively and see
what function value is being approached
• The second approach, is to rearrange the equation to make it more
obvious as to what happens when x gets infinitely positively and
negatively.
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(D) Finding the Equations of the
Asymptotes
• ex. Find the asymptotes of y = (x+2)/(3x-2)
• So we take lim x  2/3+ f(x) = + ∞ and lim x  2/3-
f(x) = - ∞  thus we have an asymptote at x =
2/3
• To find the horizontal asymptote  a table of
values (or simply large values for x) returns the
following values:
• x = 109  f(109) = 0.3333333342 or close to 1/3
• x = -(109)  f(-(109)) = 0.3333333342 or close to
1/3
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(D) Finding the Equations of the
Asymptotes
• Alternatively, we can find the horizontal
asymptotes of y = (x+2)/(3x-2) using
algebraic methods  divide through by the
x term with the highest degree
• as x  + ∞, then 2/x  0
x 2
2

1

x 2
1
x
x
x
f ( x) 



3
x
2
2
3x  2
3
3


x
x
x
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(E) Examples
• Further examples to do  Find vertical and
horizontal asymptotes for:
• y = (4x)/(x2+1)
• y = (2-3x2)/(1-x2)
• y = (x2 - 3)/(x+5)
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(E) Examples
• y = (4x)/(x2+1)
• y = (2-3x2)/(1-x2)
• y = (x2 - 3)/(x+5)
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(F) Graphing Rational Functions
• If we want to graph rational functions (without graphing
technology), we must find out some critical information
about the rational function. If we could find the
asymptotes, the domain and the intercepts, we could get a
sketch of the graph
• ex => f(x) = (x2)/(x3-2x2 - 5x + 6)
• NOTE: after finding the asymptotes (at x = -2, 1,3) we find
the behaviour of the fcn on the left and the right of these
asymptotes by considering the sign of the ∞ of f(x).
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(F) Oblique Asymptotes
•
Some asymptotes that are neither vertical or horizontal => they are slanted.
These slanted asymptotes are called oblique asymptotes.
•
Ex. Graph the function f(x) = (x2 - x - 6)/(x - 2) (which brings us back to our
previous work on the Factor Theorem and polynomial division)
Recall, that we can do the division and rewrite f(x) = (x2 - x - 6)/(x - 2) as f(x)
= x + 1 - 4/(x - 2).
Again, all we have done is a simple algebraic manipulation to present the
original equation in another form.
So now, as x becomes infinitely large (positive or negative), the term 4/(x - 2)
becomes negligible i.e. = 0.
So we are left with the expression y = x + 1 as the equation of the oblique
asymptote.
•
•
•
•
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(G) Internet Links
• Rational Functions from WTAMU
• Calculus@UTK 2.5 - Limits Involving
Infinity
• Calculus I (Math 2413) - Limits - Limits
Involving Infinity from Paul Dawkins
• Limits Involving Infinity from P.K. Ving
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(G) Homework
• MCB4U:
• DAY 1; Nelson text, p356, Q1-4
• DAY 2; Nelson text, p357, Q10,11,12,14,15
•
•
•
•
IB Math HL/SL:
Stewart, 1989, Chap 5.1, p212, Q2,3
Stewart, 1989, Chap 5.2, p222, Q2-6
Stewart, 1989, Chap 5.6, p244, Q1,2
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