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Section 3.5
The Graph of a Rational
Function
Analyzing the Graph of a Rational Function
STEP 1: Factor the numerator and denominator of R. Find the domain of the
rational function.
STEP 2: Write R in lowest terms. A hole in the graph exists where the
numerator and denominator have common factors.
STEP 3: Locate the intercepts of the graph. When R is in lowest terms, set the
numerator equal to 0 for x-intercepts. The y-intercept is R(0).
STEP 4: 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.
STEP 5: Locate the vertical asymptotes. When R is in lowest terms, the vertical
asymptotes are found by setting the denominator to 0.
STEP 6: Locate the horizontal or oblique asymptotes.
STEP 7: Graph the rational function.
x 2  x  12
R  x 
2
x 1
x 2  3x  2
R  x 
x
x2  9
R  x  2
x  9 x  18
CLASSWORK
Find an equation of a rational function f that satisfies the
following conditions:
x–intercept: 4
vertical asymptote: x = – 2
horizontal asymptote: y = – 3/5
hole at x = 1