3.4 Analyze the Graph of a Rational Function 2011
October 05, 2011
3.4 ­ Analyze the Graph of a Rational Function
Objective: Analyze the graph of a rational function.
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3.4 Analyze the Graph of a Rational Function 2011
October 05, 2011
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 2011
October 05, 2011
Analyzing the graph of a rational function R(x) = x ­ 1
x2 ­ 4
Step 1: Find the domain of the rational function.
D: x ≠ ­2, x ≠ 2
Step 2: Write R in lowest terms (simplify the rational function if possible).
R is in lowest terms
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3.4 Analyze the Graph of a Rational Function 2011
October 05, 2011
Analyzing the graph of a rational function R(x) = x ­ 1
(x + 2)(x ­ 2)
Step 3: Locate the intercepts of the graph.
R(0) = x ­ 1
0 = (x + 2)(x ­ 2)
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3.4 Analyze the Graph of a Rational Function 2011
October 05, 2011
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.
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3.4 Analyze the Graph of a Rational Function 2011
October 05, 2011
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 2011
October 05, 2011
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.
(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.
an
bm
, and the line y = an
bm
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3.4 Analyze the Graph of a Rational Function 2011
October 05, 2011
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 2011
October 05, 2011
Homework: page: 208
#'s: 7­15
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