Sullivan Algebra and Trigonometry: Section 4.4

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Sullivan Algebra and Trigonometry Section 4.4

• Objectives
• Analyze the Graph of a Rational Function
• Solve Applied Problems Involving Rational
  Functions

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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.
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Example Analyze the graph of
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a.) x-intercept when x 1 0 (-1,0)
y - intercept (0, 2/3)
c.) Test for Symmetry
No symmetry
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d.) Vertical asymptote x -3
Since the function isnt 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
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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)
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x - 3
(-4, 6)
(1, 1)
(3, 4/3)
y 2
(-2, -2)
(-1, 0)
(0, 2/3)
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Example The concentration C of a certain drug in
a patients bloodstream t minutes after injection
is given by
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.
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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.