SAT Question of the Day
8.2 Rational Functions and Their Graphs
Objectives:
•Identify and evaluate rational functions
•Graph a rational function, find its domain, write equations for its
asymptotes, and identify any holes in its graph
Example 1
William begins with 75 milliliters of a 15% acid
solution. He adds x milliliters of distilled water to
the container holding the acid solution.
a) Write a function, C, that represents the acid
concentration of the solution in terms of x.
acid
11.25
15% of 75 = 11.25
solution
75
11.25
75  x
add x milliliters of distilled water
11.25
C ( x) 
75  x
Example 1
William begins with 75 milliliters of a 15% acid
solution. He adds x milliliters of distilled water to
the container holding the acid solution.
b) What is the acid concentration of the solution if
35 milliliters of distilled water is added?
11.25
C ( x) 
75  x
11.25
11.25
 0.102  10.2%
C (35) 

75  35
110
Example 2
Find the domain of
x 2  4 x  21
h( x )  2
x  9 x  36
Find the values of x for which the denominator equals 0.
x2 – 9x – 36 = 0
(x – 12)(x + 3) = 0
x = 12 or -3
The domain is all real numbers except 12 and -3.
Vertical Asymptote
In a rational function R, if x – a is a factor of the
denominator but not a factor of the numerator,
x = a is vertical asymptote of the graph of R.
Hole
In a rational function R, if x – a is a factor of the
denominator AND a factor of the numerator,
x = a is a hole in the graph of R.
Example 3
Identify all vertical asymptotes of
3x
r ( x)  2
x  3x  2
Factor the denominator.
3x
r ( x) 
( x  2)( x  1)
Equations for the vertical asymptotes are x = 2 and x = 1.
Horizontal Asymptote
P
R(x) = is a rational function;
Q
P and Q are polynomials
• If degree of P < degree of Q, then
the horizontal asymptote of R is y = 0.
x
f ( x)  2
x  2x  3
Horizontal Asymptote
P
R(x) = is a rational function;
Q
P and Q are polynomials
• If degree of P = degree of Q and a and b are
the leading coefficients of P and Q, then
a
the horizontal asymptote of R is y = .
b
x 2  16
f (x) 
2
4  5x  x
Horizontal Asymptote
P
R(x) = is a rational function;
Q
P and Q are polynomials
• If degree of P > degree of Q, then
there is no horizontal asymptote
x 7
f ( x)  2
x  4x  3
3
Homework
Lesson 8.2 exercises 11-39 odd
SAT Problem of the Day
8.2.2 Rational Functions and Their Graphs
Objectives:
•Identify and evaluate rational functions
•Graph a rational function, find its domain, write equations for its
asymptotes, and identify any holes in its graph
Example 1
x3
Let R( x)  2
. Identify all vertical
x  x  20
asymptotes and all horizontal asymptotes.
x3
R( x) 
( x  5)( x  4)
Equations for the vertical asymptotes are x = -5 and x = 4.
Because the degree of the numerator is greater than the
degree of the denominator, the graph has no horizontal
asymptotes.
Example 2
2x 1
Let R( x)  2
. Identify all vertical asymptotes
1x 9
2
and all horizontal asymptotes.
2 x2  1
R( x) 
( x  3)( x  3)
Vertical asymptotes: x = -3 and x = 3
Horizontal asymptotes:
numerator and denominator
have the same degree
leading
coefficients
2
1
y=2
Holes in Graphs
In a rational function R, if x – b is a factor of
the numerator and the denominator, there is a
hole in the graph of R when x = b (unless x = b is
a vertical asymptote).
Example 3
Identify all asymptotes and holes in the graph of the
rational function.
2x2 + 2x
f(x) =
x2 – 1
factor:
f(x) =
2x(x + 1)
(x + 1)(x – 1)
hole in the graph:
x = –1
vertical asymptote:
x=1
horizontal asymptote:
y=2
Example 4
Use asymptotes to graph the rational function.
2x 1
f ( x) 
x4
Write equations for the asymptotes, and graph them as
dashed lines.
horizontal asymptote:
vertical asymptote:
y=2
x = -4
Use a table to help obtain an accurate plot.
Collins Type 2
Explain how to use the asymptotes of the graph of
g(x) =
x-5
x-3
to sketch the graph of the function.
Homework
Worksheet “More problems on rational function
graphing”