Graphing Rational Functions Example

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Graphing Rational FunctionsExample 3
We want to graph this rational function showing
all relevant characteristics.
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Graphing Rational FunctionsExample 3
First we must factor both numerator and
denominator, but dont reduce the fraction
yet. Numerator Factor out a GCF of
x2. Denominator Factor as a perfect square
trinomial.
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Graphing Rational FunctionsExample 3
Note the domain restrictions, where the
denominator is 0.
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Graphing Rational FunctionsExample 3
Now reduce the fraction. In this case, there
isn't a common factor. Thus, it doesn't reduce.
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Graphing Rational FunctionsExample 3
Any places where the reduced form is undefined,
the denominator is 0, forms a vertical asymptote.
Remember to give the V. A. as the full equation
of the line and to graph it as a dashed line.
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Graphing Rational FunctionsExample 3
Any values of x that are not in the domain of the
function but are not a V.A. form holes in the
graph. In other words, any factor that reduced
completely out of the denominator would create a
hole in the graph where it is 0. Since this
example didn't reduce, it has no holes.
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Graphing Rational FunctionsExample 3
Next look at the degrees of both the numerator
and the denominator. Because the denominator's
degree,2, is exactly 1 less than the numerator's
degree,3, there will be an oblique asymptote, but
no horizontal asymptote.
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Graphing Rational FunctionsExample 3
To find the O.A. we must divide out the rational
expression. In this case, since the fraction
didn't reduce we will use the original form.
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Graphing Rational FunctionsExample 3
The O.A. will be y(what is on top of the
division).
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Graphing Rational FunctionsExample 3
Now we need to find the intersections between the
graph of f(x) and the O.A. Usually the easiest
way to do this is to set the remainder from the
division equal to 0 and solve for x.
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Graphing Rational FunctionsExample 3
Next we need to find the y coordinate of the
intersection by plugging the x we just found into
the equation from the O.A.
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Graphing Rational FunctionsExample 3
We find the x-intercepts by solving when the
function is 0, which would be when the numerator
is 0. Thus, when x20 and x40.
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Graphing Rational FunctionsExample 3
Now find the y-intercept by plugging in 0 for x.
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Graphing Rational FunctionsExample 3
Plot any additional points needed. In this case
we don't need any other points to determine the
graph. Though, you can always plot more points if
you want to.
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Graphing Rational FunctionsExample 3
Finally draw in the curve. Let's start on the
interval for xlt-1, the graph has to pass through
the point (-4,0) and approach both asymptotes.
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Graphing Rational FunctionsExample 3
For -1ltxlt0, the graph to go through the points
(0,0) and (-2/5,8/5) and approach the V.A. The
graph has to approach the V.A. going up since it
can cross the x-axis between x-1 and x-2/5.
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Graphing Rational FunctionsExample 3
For xgt0, the multiplicity of the x-intercept of 0
is 2. Which since it is even the graph must
bounce off the x-axis. Then the graph must
approach the O.A.
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Graphing Rational FunctionsExample 3
This finishes the graph.
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