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Strategy For Graphing Polynomials and Rational Functions

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Strategy For Graphing
Polynomials And
Rational Functions
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1. State the domain.
2. Find the Y-Intercept (x=0) and the XIntercept (y=0) the easy ones in
particular.
– You can use the synthetic division to find
the easy zeros for the given polynomial
function.
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3. For rational functions ONLY, find the
asymptotes.
4. Find dy/dx and perform the sign
analysis.
5. Graph the function.
Asymptotes
For Rational Functions
•
1. Vertical Asymptote:
Whatever makes the denominator zero
is your vertical asymptote as long as you
do not have 0/0. Remember 0/0 means
that you have a hole in the graph.
•
2. Horizontal And Slant Asymptote:
Is the limit of the rational function as
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Horizontal & Slant
Asymptotes
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Consider the following rational function:
f ( x) пЂЅ
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n пЂ­1
an x
n
пЂ« a n пЂ­1 x
bm x
m
пЂ« b m пЂ­1 x
m пЂ­1
пЂ« ... пЂ« a 1 x пЂ« a 0
пЂ« ... пЂ« b1 x пЂ« b 0
If the power of the numerator is the same as the power of the denominator,
(n=m) then the horizontal asymptote is y = the ratio of the leading coefficients,
y пЂЅ
an
bm
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If the power of the numerator is less than the power of the denominator
(n<m), then the horizontal asymptote is y=0.
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If the power of the numerator is greater than the power of the denominator by
one degree (n=m+1), then the slant asymptote is y = the quotient of the
division.
Here the synthetic division can prove useful.
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Notice that for rational functions you can not have horizontal and slant
asymptotes at the same time.
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