6.
Graphing Rational Functions
Functions of the form
where a(x) and b(x) are polynomial functions.
Examples
7.
Graphing Rational Functions
Appearance
Where n is odd, the
Where n is even, the
graph looks like this:
graph looks like this:
8.
Graphing Rational Functions
Sketching (7 steps)
Step 1: Find the y-intercept (let x = 0)
Step 2: Factor everything. (Use rational roots theorem if necessary.)
Step 3: Find the roots of the function by ﬁnding the roots of the
numerator a(x).
Step 4: Find the vertical asymptotes by ﬁnding the roots of the
denominator b(x).
9.
Graphing Rational Functions
Step 5: Find the horizontal asymptotes by dividing each term in the
function by the highest power of x, and take the limit as x goes to inﬁnity.
(Use the UNfactored form.)
You will ﬁnd that, in general, there are three possible results:
i When [degree of numerator < degree of denominator]
the horizontal asymptote is y = 0.
ii When [degree of numerator = degree of denominator]
the H.A. is the ratio leading coefﬁcient of a(x)
leading coefﬁcient of b(x)
iii When [degree of numerator > degree of denominator] there is
no horizontal asymptote; however there may be a slant asymptote or
a hole in the graph.
10.
Graphing Rational Functions
Sketching (7 steps)
Step 6: Determine the sign of the function over the intervals deﬁned by
the roots and vertical asymptotes. (Use the factored form.)
Step 7: Sketch the graph.
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