Definition of a Rational Function
A rational function is a quotient of polynomials
that has the form
f ( x)
h( x ) 
g ( x)
The domain of a rational function consists of all
real numbers except the zeroes of the polynomial
in the denominator. g ( x)  0, (i.e. D :  x  g ( x)  0)
S. Evans
Rational Functions and Their Characteristics
(DAY 1):
Over the next three days, you will examine the basic
characteristics of a rational function. Understanding these
functions will enable you to draw an accurate sketch of a
rational function. These characteristics are:
 Domain
 Intercepts (both x and y)
 Holes in the function
 Asymptotes (Vertical, Horizontal, & Linear Oblique)
 Behaviour of the function close to the asymptote(s)
Let’s start with familiar concepts, domain and intercepts…..
S. Evans
Rational Functions and Their Characteristics
(DAY 1):
Example 1: Determine the domain and intercepts of each
of the following:
a)
7
f ( x) 
x2
b) f  x  
x
x 2  3x  4
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
The graph of a rational function usually has at
least one asymptote, which may be vertical,
horizontal, or oblique. An oblique asymptote
is neither vertical nor horizontal.
S. Evans
Rational Functions and Their Characteristics
(DAY 1):
Vertical Asymptotes (V.A.):
Vertical asymptotes are imaginary vertical lines that form
boundaries in the graph. Vertical asymptotes can be
found where the function is undefined. However, not all
restrictions produce a vertical asymptote. To make sure
you have a vertical asymptote, simplify the rational
function fully. The restrictions that are left in the reduced
function will produce the vertical asymptotes. The
equation of a vertical asymptote will then be . A graph
never crosses a vertical asymptote because that x –
value can never occur in the domain of the function.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Holes:
 Sometimes a rational function has a hole in
the graph. This is a point in the function where
the graph gets really close to but never equals
it. This will occur at the x – value of a
restriction that is cancelled out during
simplification. To get the y – value of the hole,
substitute the x – value into the simplified
rational function. Note: The x – value you are
substituting in is a restriction and does not
exist in the domain of the function. This is why
the co-ordinates that you are finding are for a
hole in the function.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Example 2: Determine the vertical asymptotes
and or holes of the following rational functions
AND sketch on a Cartesian plane.
a)
2 x2  x  3
f  x 
x2  4
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
b) f ( x) 
x2
x 2  2x
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Horizontal Asymptotes (H.A.):
 Horizontal asymptotes are imaginary horizontal
lines that the graph will approach as x
approaches very large positive values and/or
very large negative values .
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Horizontal Asymptotes (H.A.):
The following are general rules for finding a
horizontal asymptote:
 Given that the numerator and denominator in
the rational function are polynomials in of
degree and , respectively.
1. If
2. If
3. If
n  m , then the horizontal asymptote is .
n  m , then the horizontal asymptote is .
n  m , there is no horizontal asymptote.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Horizontal Asymptotes (H.A.):
NOTE: Since a horizontal asymptote is only a
barrier as x gets to be very large positive values
and/or very large negative values , the graph
may cross the horizontal asymptote in the central
area of the graph. You will examine the
behaviour of the curve in the next lesson.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Example 3: Determine the equation of the
horizontal asymptote, if it exists.
a)
x
f  x 
x 2  3x  4
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
3
x
b) h x   7 x
 
x2
c) g  x 
x  3 2 x  1


2
4  3x 2  5 x3
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
2
x
Example 4: Given f x 
  3
x  2 x2  5x  6
Find the domain, intercepts, and vertical and
horizontal asymptotes. Then use this information
to sketch what you now know about your
function.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
Use this slide for work
x2
f  x  3
x  2 x2  5x  6
S. Evans
.
Rational Functions and Their Characteristics
(DAY 1):
x2
f  x  3
x  2 x2  5x  6
S. Evans
Example 1: Determine the intercepts and the
vertical and horizontal asymptotes for
3x  2
f  x 
x 1
Then sketch what you know about the function so
far.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
f  x 
3x  2
x 1
Sketch what we
know so far
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
To be able to sketch the curve in the boundaries created by the
asymptotes, you need to understand the behaviour of the graph as it
approaches the asymptotes.
Behaviour of the Function as it approaches the Vertical
Asymptote (VA):
(Does the curve point toward positive or negative infinity on either
side of the VA?)
You need to examine the sign of the function as x “approaches” the
VA from the left side ( ) and the sign of the function as x
“approaches” the VA from the right side ( ). To do this for the left
side, you substitute a value smaller than the VA but really close to
it and determine whether the output would be positive or negative.
Then for the right side, you substitute a value larger than the VA but
really close to it and determine whether the output would be
positive or negative. A positive output would imply that the graph is
pointing upwards towards positive infinity on that side of the VA and
a negative output would imply that the graph is pointing downwards
towards negative infinity on that side of the VA.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
Example 2: Examine the behaviour of the
3x  2
function
(From ex 1) around
f  x 
x 1
the vertical asymptote. Then add that information
to your sketch of f  x 
S. Evans
.
Rational Functions and Their Characteristics
3x  2
(DAY 2):
f  x 
x 1
Behaviour of the Function as it approaches the Horizontal Asymptote:
(Is the function above or below the HA as x is a very large positive (as x   )
or as x is a very large negative (as x  )?
When is the function above the HA? Solve: ___________________________
When is the function below the HA? Solve: ____________________________
Does the function cross the HA? Solve: ______________________________
As , x   is the function above or below the HA? You should substitute a
very ________ _________ x – value and if the output is
_____________________ the function is above the HA and if the output is
_______________________ the function is below the HA.
As x   is the function above or below the HA? You should substitute a
very _________ ___________ x – value and if the output is
_____________________ the function is above the HA and if the output is
_______________________ the function is below the HA.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
Example 3:
3x  2
f  x 
a) Does the function
x  1 from ex 1)
cross its horizontal asymptote? (Justify)
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
c) As x   is the function, f  x   3 x  2 ,
x 1
above or below the HA?
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
b) As x   is the function, f  x   3 x  2 ,
x 1
above or below the HA?
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
d) Add the above information to your sketch of
3x  2
(from Ex 1)
f  x 
x 1
e) Given that curves usually “hug” the
asymptotes, fill-in the possible sketch of the
function.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
(Linear) Oblique Asymptotes:
For rational functions, linear oblique asymptotes
occur when the degree of the numerator is
exactly one more than the degree of the
denominator. The equation of the linear oblique
asymptote can be found by dividing the
numerator by the denominator using long
division. (or synthetic division where appropriate)
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
2 x3  x 2  3
f  x 
x2
Example 4: Let
a) Determine the oblique asymptote.
.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
2 x3 .x 2  3
f  x 
x2
b) Determine the intercepts
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
2 x3 .x 2  3
f  x 
x2
b) Determine the intercepts
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
2 x3 .x 2  3
f  x 
x2
c) Determine the vertical asymptote(s) and
examine the behaviour around the VA.
S. Evans
.
Rational Functions and Their Characteristics
(DAY 2):
3x  2
d) Use your analysis to sketch the graph of . f  x   x  1
S. Evans
.
Rational Functions and Their Characteristics
(DAY 3):
Example 1: Use domain, intercepts, asymptotes
and behaviour to sketch the function
x  5  x  4 

f  x 
x4  x  2
2
S. Evans
.
Rational Functions and Their Characteristics
(DAY 3):
 x  5 x  4 
f  x 
x4  x  2
2
S. Evans
.
Rational Functions and Rates of Change
(DAY 4):
Concepts from the previous chapter are now
revisited using rational functions:
• Average rate of change (use given range)
• Instantaneous rate of change (Check three
following intervals that get close to the independent
value)
(work on the assigned worksheet)
S. Evans