Transcript Rational Functions II: Analyzing Graphs

Sullivan PreCalculus
Section 3.4
Rational Functions II:
Analyzing Graphs
Objectives
• Analyze the Graph of a Rational Function
• Solve Applied Problems Involving Rational Functions
To analyze the graph of a rational function:
a.) Find the Domain of the rational function.
b.) Locate the intercepts, if any, of the graph.
c.) Test for Symmetry. If R(-x) = R(x), there is
symmetry with respect to the y-axis. If - R(x) =
R(-x), there is symmetry with respect to the origin.
d.) Write R in lowest terms and find the real zeros of
the denominator, which are the vertical asymptotes.
e.) Locate the horizontal or oblique asymptotes.
f.) Determine where the graph is above the x-axis
and where the graph is below the x-axis.
g.) Use all found information to graph the function.
2x2  4x  6
Example: Analyze the graph of R ( x) 
2
x 9
2 x  2 x  3
R( x) 
 x  3 x  3
2
2 x  3 x  1

 x  3 x  3
2 x  1

, x3
 x  3
Domain:
 x x  3, x  3
2 x  1
R ( x) 
 x  3
a.) x-intercept when x + 1 = 0:
(-1,0)
2 ( 0  1) 2

b.) y-intercept when x = 0: R ( 0 ) 
( 0  3)
3
y - intercept: (0, 2/3)
c.) Test for Symmetry:
2(  x  1)
R(  x ) 
(  x  3)
R(x)  R(x)  R(x)
No symmetry
2 x  1
R( x) 
, x3
 x  3
d.) Vertical asymptote: x = -3
Since the function isn’t defined at x = 3, there is
a whole at that point.
e.) Horizontal asymptote: y = 2
f.) Divide the domain using the zeros and the
vertical asymptotes. The intervals to test are:
   x  3
 3  x  1
1 x  
   x  3  3  x  1  1  x  
Test at x = -4
Test at x = -2
Test at x = 1
R(-4) = 6
R(-2) = -2
R(1) = 1
Above x-axis
Below x-axis
Above x-axis
Point: (-4, 6)
Point: (-2, -2)
Point: (1, 1)
g.) Finally, graph the rational function R(x)
x=-3
10
(-4, 6)
5
(1, 1)
(3, 4/3)
y=2
8
6
(-2, -2)
4
2
0
5
10
2
(-1, 0)
4
(0, 2/3)
6
Example: The concentration C of a certain
drug in a patients bloodstream t minutes after
injection is given by:
50 t
C (t )  2
(t  25 )
a.) Find the horizontal asymptote of C(t)
Since the degree of the denomination is larger
than the degree of the numerator, the horizontal
asymptote of the graph of C is y = 0.
b.) What happens to the concentration of the drug
as t (time) increases?
The horizontal asymptote at y = 0 suggests that
the concentration of the drug will approach
zero as time increases.
c.) Use a graphing utility to graph C(t). According
the the graph, when is the concentration of the drug
at a maximum?
The concentration will be at a maximum
five minutes after injection.