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**9.3 Rational Functions and Their Graphs**

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**If the graph is not continuous at x = a then the function has a point of discontinuity at x = a.**

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**Ex 1 Find any points of discontinuity.**

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**Ex 2 Find any points of discontinuity.**

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Vertical Asymptotes There is a point of discontinuity for each real zero of Q(x). If P(x) and Q(x) have no common real zeros, then the graph has a VA at each real zero of Q(x). If P(x) and Q(x) have a common real zero, a, then there is a hole in the graph or a VA at x = a.

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Ex 3 Find the VA or holes.

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Ex 4 Find the VA or holes.

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Ex 5 Find the VA or holes.

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**Horizontal Asymptotes (There is at most 1 HA per graph.)**

If the degree of the denominator is > the degree of the numerator then there is a HA at y = 0. If the degree of the numerator is > the degree of the denominator then there is NO HA. If the degree of the numerator = the degree of the denominator then the HA is y = a/b where a is the leading coefficient of the numerator & b is the LC of the denominator.

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**Ex 6 Find the VA, HA, and holes.**

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**Ex 7 Sketch the graph and identify the VA, HA, and holes.**

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**The zero of the numerator is the x-intercept!!**

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