Lines Approaching Infinity graphing rational functions asintoto [asymptote] by flickr user zinov
May 18, 2015
Lines Approaching
Infinitygraphing rational functions
asintoto [asymptote] by flickr user zinov
Graphing Rational Functions
where a(x) and b(x) are polynomial functions.
Examples
Functions of the form
Graphing Rational Functions
Appearance
Where n is even, the graph looks like this:
Where n is odd, the graph looks like this:
Graphing Rational Functions
Sketching (7 steps)
Step 1: Find the y-intercept (let x = 0)
Step 4: Find the vertical asymptotes by finding the roots of the denominator b(x).
Step 3: Find the roots of the function by finding the roots of the numerator a(x).
Step 2: Factor everything. (Use rational roots theorem if necessary.)
Graphing Rational FunctionsStep 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.
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.
ii When [degree of numerator = degree of denominator] the H.A. is the ratio leading coefficient of a(x)
leading coefficient of b(x)
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.
Graphing Rational Functions
Sketching: Example 1 of 4
Step 1:
Step 2:
Step 3:
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Step 7:
Step 5:
Sketch the graph of
Graphing Rational Functions
Sketching: Example 2 of 4
Step 1:
Step 2:
Step 3:
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Step 5:
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Step 5: