How do you graph rational functions?
See explanation...
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To graph rational functions, follow these steps:
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Determine the domain of the function by identifying any values that would make the denominator zero. Exclude these values from the domain.
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Find the vertical asymptotes by setting the denominator equal to zero and solving for x. These are the vertical lines that the graph approaches but never touches.
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Determine the horizontal asymptotes by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.
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Find any holes in the graph by canceling out common factors between the numerator and denominator.
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Determine the x-intercepts by setting the numerator equal to zero and solving for x.
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Determine the y-intercept by evaluating the function at x = 0.
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Plot the vertical asymptotes, horizontal asymptotes, holes, x-intercepts, and y-intercept on the graph.
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Use additional points or symmetry to sketch the graph between the asymptotes and intercepts.
Remember to label the axes and provide a title for the graph.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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