3.4 Analyze the Graph of a Rational Function
November 20, 2008
3.4 ­ Analyze the graph of a rational function
Objective: To analyze the graph of a rational function
Nov 14­8:46 AM
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Analyzing the graph of a rational function step 1: Find the domain of the rational function
step 2: Write R in lowest terms (simplify the rational function if possible)
step 3: Locate the intercepts of the graph.
step 4: Test for symmetry
step 5: Locate the vertical asymptotes
step 6: Locate the horizontal or oblique asymptotes
step 7: Determine points, if any, where the graph crosses the asymptotes
(horizontal or oblique)
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Analyzing the graph of a rational function R(x) = x ­ 1
(x + 2)(x ­ 2)
step 1: Find the domain of the rational function
{x| x = ­2, x = 2}
step 2: Write R in lowest terms (simplify the rational function if possible)
R is in lowest terms
Nov 14­8:46 AM
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Analyzing the graph of a rational function R(x) = x ­ 1
(x + 2)(x ­ 2)
step 3: Locate the intercepts of the graph.
x ­ 1
0 = (x + 2)(x ­ 2)
R(0) = Nov 14­8:46 AM
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Analyzing the graph of a rational function R(x) = x ­ 1
(x + 2)(x ­ 2)
step 4: Test for symmetry
R(­x) = ­x ­ 1
2
x ­ 4
=
No symmetry
­(x + 1)
2
x ­ 4
R(­x) = R(x) The function has symmetry about the y­axis.
R(­x) = ­R(x)
The function hs symmetry about the origin.
Nov 14­8:46 AM
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Analyzing the graph of a rational function R(x) = x ­ 1
(x + 2)(x ­ 2)
step 5: Locate the vertical asymptotes
x = 2
x = ­2
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Horizontal & Oblique Asymptote Reminder
(the degree of the numerator is n and the degree of the denominator is m)
1.
If n < m, then R is a proper fraction and will have the horizontal asymptote y = 0.
2.
If n > m, then R is improper and long division is used.
an
bm
, and the line y = an
bm
(a)
If n = m, the quotient obtained will be a number
is a horizontal asymptote.
(b)
If n = m + 1, the quotient obtained is of the form ax + b (a polynomial of degree 1), and the line y = ax + b is an oblique asymptote.
(c)
If n > m + 1, the quotient obtained is a polynomial of degree 2 or higher and R has neither a horizontal nor an oblique asymptote.
Nov 16­12:48 PM
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Analyzing the graph of a rational function R(x) = x ­ 1
(x + 2)(x ­ 2)
step 6: Locate the horizontal or oblique asymptotes
y = 0
step 7: Determine points, if any, where the graph crosses the asymptotes
x ­ 1 = 0
x = 1
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3.4 Analyze the Graph of a Rational Function
November 20, 2008
Homework: page: 208
#'s: 7­15
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