Rational Functions and Their Characteristics (DAY 1):
Definition of a Rational Function
A rational function is a quotient of polynomials that has the form h  x  
f ( 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} )
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.
Example 1: Determine the domain and intercepts of each of the following:
(a) f ( x ) 
7
x2
(b) f  x  
x
x  3x  4
(c) f  x  
2 x2  x  3
x2  4
(d) f ( x) 
x2
x 2  2x
2
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.
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 x  restriction . A
graph never crosses a vertical asymptote because that x – value can never occur in the
domain of the function.
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.
Example 2: Determine the vertical asymptotes and or holes of the following rational
functions AND sketch on a Cartesian plane.
a) f  x  
2 x2  x  3
x2  4
b) f ( x) 
x2
x 2  2x
Horizontal Asymptotes (H.A.):
Horizontal asymptotes are imaginary horizontal lines that the graph will approach as x
approaches very large positive values  x    and/or very large negative values  x    .
The following are general rules for finding a horizontal asymptote:
Given that the numerator and denominator in the rational function are polynomials in x of
degree n and m , respectively.
1. If n  m, then the horizontal asymptote is y  0 .


coefficient of x n term
2. If n  m, then the horizontal asymptote is y 
.
coefficient of x m term
 3. If n  m, there is no horizontal asymptote.


NOTE: Since a horizontal asymptote isonly a barrier as x gets to be very large positive
values  x    and/or very large negative values  x    , 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.
Example 3: Determine the equation of the horizontal asymptote, if it exists.
x
x3  7 x
f
x

a)  
b) h  x  
x 2  3x  4
x2
c) g  x 
 x  3 2 x  1

4  3x 2  5 x3
2
x2
. Find the domain, intercepts, and vertical and
x3  2 x 2  5 x  6
horizontal asymptotes. Then use this information to sketch what you now know about your
function.
Example 4: Given f  x  
ASSIGNED EXERCISES:
Do # 1 – 5, 7, 8, 11 –
15
Knowledge
For question 1 – 6, refer to the following functions. Answer questions 1 to 5 without graphing
technology.
(a)
f ( x) 
2 x
x7
(b)
g( x) 
x2
x  5x  6
(c)
h( x ) 
x2  x  6
x2
(d)
k ( x) 
x2  9
x 3  4x 2  x  4
2
1.
Determine the x- and y-intercepts of each function.
2.
State the domain for each function.
3.
Will the graph of each function have a vertical asymptote?
If so, determine the equation for each vertical asymptote.
4.
Will the graph of each function have a horizontal asymptote?
Give reasons for you answers. Determine the equation of each asymptote.
5.
Use the information from questions 1 to 5 to sketch what you know about the graph of
each function.
6.
Confirm your answers to questions 1 to 5 by graphing each function using graphing
technology.
Application
7.
Functions R(x) = -2x2 + 8x and C(x) = 3x + 2 are the estimated revenue and cost functions
for the manufacture of a new product. Determine the average profit function
AP(x) =
P( x)
. Express this function in two different forms. Explain what can be
x
determined from each form. Restrict the domain of the function to represent the
context. What are the break-even quantities?
8.
Repeat question 7 for R(x) = -x2 + 30x and C(x) = 17x + 36.
9.
The model for the concentration y of a drug in the bloodstream, x hours after it is taken
orally, is y =
7x
. What is the domain of y in this context? What do you know about
x 2
2
the graph of y just by looking at the equation?
Graph the function. Describe what happens to the concentration of the drug over 24
consecutive hours. Does the model seem reasonable?
10.
A rectangular garden, 21 m2 in area, will be fenced to keep out rabbits and skunks. Find
the dimensions that will require the least amount of fencing if a barn already protects
one side of the garden.
Communication/Thinking & Inquiry
11.
What is a rational function? How is the graph of a rational function different from the
graph of a polynomial function?
12.
Use specific examples of your own to describe the condition for the graph of a rational
function to have vertical and horizontal asymptotes.
13.
For each case, create a function that has a graph with the given features.
(a) a vertical asymptote x = 1 and a horizontal asymptote y = 0
(b) two vertical asymptotes x = -1 and x = 3, horizontal asymptote y = -1, and x-intercepts
– 2 and 4.
Explain the difference between the graph of f (x) 
15.
Find constants a and b that guarantee that the graph of the function defined by
h(x) 
3
ax 2  7

will have a vertical asymptote
at * x   and a horizontal asymptote at

2
9  bx
5
y  2
.

Answers:

1.
(a) x-int (-2,0), y-int (0,-2/7)

(c) x-int: (2,0), (-3,0), y-int: (0,-3) (d)
(b)
x-int (2,0), y-int: (0,-1/3)
x-int(3,0),(-3,0), y-int: (0,9/4)
2.
(a) x  7
(b) x  -3,-2
(c) x  -2
(d) x  -4,-1,1
3.
(a) x = 7
(b) x = 4
(c) x = -2
(d) x = -4,-1,1
4.
(a) y = 1
5.
Graphs
6.
Graphing Calculator
7.
AP(x) =
 (2 x  1)( x  2)
2
or  2 x  5  , D: x>0. Break even: x = ½,2 (zeroes)
x
x
8.
AP(x) =
 ( x  9)( x  4)
36
or  x  13 
. D: x>0, Break even: x= 4 or 9 (zeroes)
x
x
9.
D: x  0. Function increases to a maximum of (1.4, 2.5) Model is reasonable.
10.
6.5m X 3.2 m
13.
(a)
14.
The function f (x) is not defined at the value b which is represented by a hollow dot.
The function g(x) is a linear graph.
(b)
y
15. a  50, b 25


(x  a)(x  b)
and that of g(x)  x  a .
(x  b)
14.

1
x 1
y=0
(b)
(d)
y
y=0
 x 2  2x  8
x 2  2x  3