Graphing Rational Functions Example

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Graphing Rational FunctionsExample 6
We want to graph this rational function showing
all relevant characteristics.
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Graphing Rational FunctionsExample 6
First we must factor both numerator and
denominator, but dont reduce the fraction
yet. Both factor into 2 binomials.
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Graphing Rational FunctionsExample 6
Note the domain restrictions, where the
denominator is 0.
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Graphing Rational FunctionsExample 6
Now reduce the fraction. In this case, it doesn't
reduce.
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Graphing Rational FunctionsExample 6
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 6
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. Thus, there are
no holes in this case.
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Graphing Rational FunctionsExample 6
Next look at the degrees of both the numerator
and the denominator. Because both the
denominator's and the numerator's degrees are the
same, 2, there will be a horizontal asymptote at
y(the ratio of the leading coefficients) and
there is no oblique asymptote.
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Graphing Rational FunctionsExample 6
Next we need to find where the graph of f(x)
would intersect the H.A. To do this we set the
reduced form equal to the number from the H.A.,
and solve for x.
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Graphing Rational FunctionsExample 6
Since the equation has a solution, the
intersection will be the point with x-coordinate
of the solution of the equation, and the
y-coordinate will be the number from the H.A.
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Graphing Rational FunctionsExample 6
We find the x-intercepts by solving when the
function is 0, which would be when the numerator
is 0. Thus, when x10.
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Graphing Rational FunctionsExample 6
Now find the y-intercept by plugging in 0 for x.
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Graphing Rational FunctionsExample 6
Plot any additional points needed. Here I only
plotted one more point at x2 since a point
hadn't been plotted to the right of that V.A. You
can always choose to plot more points than
required to help you find the graph.
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Graphing Rational FunctionsExample 6
Finally draw in the curve. For the part to the
right of the V.A., x1, we use that it can't
cross the H.A. and it has to approach the V.A.
and the H.A.
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Graphing Rational FunctionsExample 6
For the section between the V.A.'s, we use that
it can't cross the H.A., it has to approach both
V.A.'s and the multiplicity of the x-int. of -1
is 2, so the graph bounces of the x-axis.
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Graphing Rational FunctionsExample 6
For -3ltxlt-2, we use that the graph has to
approach the V.A. of x-2, and the graph can't
cross the x-axis in this interval.
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Graphing Rational FunctionsExample 6
For xlt-3, we have to use that the graph has to
approach the H.A., and to find out that the graph
crosses the H.A., we can use either that the
intersection w/ the H.A. has a multiplicity of 1,
or we could just plot another point at x-4.
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Graphing Rational FunctionsExample 6
This finishes the graph.
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