This document covers the concepts of rational functions, including their definitions, domains, vertical and horizontal asymptotes, and graphing techniques. It highlights the steps to identify and determine asymptotes through various examples, as well as the importance of simplifying rational functions before analysis. Additionally, it provides guidance on writing rational functions based on graphical information.

1. 5.6 Rational Functions
Chapter 5 Polynomial and Rational Functions

2. Concepts and Objectives
⚫ Objectives for this section are
⚫ Find the domains of rational functions.
⚫ Identify vertical asymptotes.
⚫ Identify horizontal asymptotes.
⚫ Graph rational functions.

3. Rational Functions
⚫ A rational function is a function of the form
where p(x) and q(x) are polynomials, with q(x)  0.
⚫ The simplest rational function with a variable
denominator is the reciprocal function, defined by
( )
( )
( )
=
p x
f x
q x
( )= 
1
, 0
f x x
x

4. The Reciprocal Function
⚫ As x gets closer and closer to 0, the value of f(x) gets
larger and larger (or smaller and smaller)
x y
1 1
2
1
2
0.1 10
0.01 100

5. The Reciprocal Function (cont.)
⚫ The graph of f(x) will never intersect the vertical line
x = 0, which is called a vertical asymptote.
⚫ As |x| gets larger and larger, the values of f (x) get closer
and closer to 0. The line y = 0 is called a horizontal
asymptote.

6. Translating Functions
⚫ Looking at translations of the reciprocal function, f:
shifted 3 units to the right
shifted 2 units to the left, up 1
( ) ( )
= = −
−
1
1. 3
3
g x f x
x
( ) ( )
= + = + +
+
1
2. 1 2 1
2
h x f x
x

7. Rational Functions
⚫ As we can see from the table,
has range values that are all positive,
and like the reciprocal function, get
larger and larger, the closer x gets to
zero.
⚫ The graph looks like this:
( )= 2
1
f x
x

8. Determining Asymptotes
⚫ Vertical Asymptotes
⚫ To find vertical asymptotes, set the denominator
equal to 0 and solve for x. If a is a zero of the
denominator, then the line x = a is a vertical
asymptote.
⚫ Example: Find the vertical asymptote(s) of
( )=
−
3
2
x
f x
x

9. Determining Asymptotes
⚫ Vertical Asymptotes
⚫ To find vertical asymptotes, set the denominator
equal to 0 and solve for x. If a is a zero of the
denominator, then the line x = a is a vertical
asymptote.
⚫ Example: Find the vertical asymptote(s) of
asymptote is at x = 2
( )=
−
3
2
x
f x
x
− =
=
2 0
2
x
x

10. Determining Asymptotes
⚫ Horizontal Asymptotes
⚫ If the numerator has lower degree than denominator,
then there is a horizontal asymptote at y = 0.
⚫ If the numerator and denominator have the same
degree, then the horizontal asymptote is at the ratio
of the coefficients of the first terms.

11. Determining Asymptotes
⚫ Example: Find the horizontal asymptotes of
a) b)
( )
+
=
−
2
3
16
x
f x
x
( )
−
=
+
3 4
2 1
x
f x
x

12. Determining Asymptotes
⚫ Example: Find the horizontal asymptotes of
a) b)
a) The numerator has a lower degree (1) than the
denominator (2), so there is a H.A. at y = 0.
b) Since the numerator and denominator have the
same degree, the H.A. is at
( )
+
=
−
2
3
16
x
f x
x
( )
−
=
+
3 4
2 1
x
f x
x
=
3
2
y

13. Determining Asymptotes
⚫ Oblique Asymptotes
⚫ If the numerator is exactly one degree more than the
denominator, then the function has an oblique (slant)
asymptote.
⚫ To find it, divide the numerator by the denominator
and disregard the remainder. Set the rest of the
quotient equal to y for the equation of the asymptote.
⚫ The graph cannot intersect any vertical asymptote.
While you can have more than one vertical asymptote,
there can be at most one nonvertical asymptote, and the
graph can intersect that asymptote.

14. Determining Asymptotes
⚫ Example: Find the asymptotes of
( )
+
=
−
2
2 5
3
x
f x
x

15. ⚫ Example: Find the asymptotes of
V.A.: O.A.:
Note: If the denominator’s degree is greater than 1, you will need to use
long division to find the asymptote.
3 2 0 5
6 18
2 6 23
Determining Asymptotes
( )
+
=
−
2
2 5
3
x
f x
x
− =
=
3 0
3
x
x
= +
2 6
y x

16. Graphing a Rational Function
To graph a rational function in Desmos:
1. Make sure the function is in lowest terms and type the
function into the input line. You will need to use
parentheses to force an expression in the numerator.
2. Find any vertical, horizontal, or oblique asymptotes and
plot them with dashed lines.

17. Graphing a Rational Function
⚫ Example: Graph ( )
−
=
− −
2
2
6
x
f x
x x

18. Graphing a Rational Function
⚫ Example: Graph ( )
−
=
− −
2
2
6
x
f x
x x

19. Graphing a Rational Function
⚫ Example: Graph
V.A.:
H.A.:
( )
−
=
− −
2
2
6
x
f x
x x
− − =
2
6 0
x x
( )( )
− + =
3 2 0
x x
= −2, 3
x
y = 0
To make dashed lines, click
on settings, then on each
line, select

20. Graphing a Rational Function
⚫ Example: Graph ( )
−
=
− −
2
2
6
x
f x
x x

21. Graphing a Rational Function
⚫ Example: Graph ( )
+
=
−
2
1
2
x
f x
x

22. Graphing a Rational Function
⚫ Example: Graph
V.A.: x – 2 = 0
x = 2
( )
+
=
−
2
1
2
x
f x
x
O.A.:
= +2
y x
2 1 0 1
2 4
1 2 5

23. Graphing a Rational Function
⚫ Example: Graph ( )
+
=
−
2
1
2
x
f x
x

24. Graphing Rational Functions
⚫ As mentioned earlier, a rational function must be
defined by an expression in lowest terms before we can
determine asymptotes or anything else about the graph.
⚫ A rational function that is not in lowest terms usually
has a “hole”, or removable discontinuity, in its graph.
⚫ To graph these in Desmos, you will need to add the
discontinuity as a circle, instead of a dot.

25. Graphing Rational Functions
⚫ Example: Graph ( )
−
=
−
2
4
2
x
f x
x

26. Graphing Rational Functions
⚫ Example: Graph
⚫ When you simplify a rational function, you have to
take into account any values of x for which the
function is not defined.
( )
−
=
−
2
4
2
x
f x
x
( )
( )( )
( )
− +
=
−
2 2
2
x x
f x
x
= + 
2 ( 2)
x x

27. Graphing Rational Functions
⚫ Example: Graph ( )
−
=
−
2
4
2
x
f x
x

28. Writing Rational Functions
⚫ Now that we have analyzed the equations for rational
functions and how they relate to a graph of the function,
we can use information from the graph to write the
function.
⚫ A rational function written in factored form will have an
x-intercept where each factor of the numerator is equal
to zero (unless there is a removable discontinuity).
⚫ Likewise, because the function will have a vertical
asymptote where each factor of the denominator is
equal to zero, we can form a denominator that will
produce the vertical asymptotes by introducing a
corresponding set of factors.

29. Writing Rational Functions
⚫ Example: Write an equation for the rational function
shown below.

30. Writing Rational Functions
⚫ Example: Write an equation for the rational function
shown below.
First, we identify the
x-intercepts at ‒2 and 3.
This gives us
( ) ( )( )
2 3
f x x x
= + −

31. Writing Rational Functions
⚫ Example: Write an equation for the rational function
shown below.
Next, we identify two
vertical asymptotes.
• At x = ‒1, the graph
seems to exhibit the
same behavior as 1
x
( )
( )( )
( )
2 3
1
x x
f x
x
+ −
=
+

32. Writing Rational Functions
⚫ Example: Write an equation for the rational function
shown below.
• At x = 2, the graph is
exhibiting behavior
similar to .
2
1
x
( )
( )( )
( )( )
2
2 3
1 2
x x
f x
x x
+ −
=
+ −

33. Writing Rational Functions
⚫ Example: Write an equation for the rational function
shown below.
• Now we can use the
y-intercept to find
the stretch factor.
( )
( )( )
( )( )
2
0 2 0 3
0 2
0 1 0 2
6
2
4
4 4
2
6 3
f a
a
a
+ −
= = −
+ −
−
 
= −
 
 
 
= − =
 
−
 

34. Writing Rational Functions
⚫ Example: Write an equation for the rational function
shown below.
• So the final equation
looks like
( )
( )( )
( )( )
( )( )
( )( )
2
2
2 3
4
3 1 2
4 2 3
3 1 2
x x
f x
x x
x x
x x
+ −
= 
+ −
+ −
=
+ −

35. Classwork
⚫ College Algebra 2e
⚫ 5.6: 6-18 (even); 5.5: 24-52 (4); 5.4: 60-72 (even)
⚫ 5.6 Classwork Check
⚫ Quiz 5.5