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Published byJesse Underwood Modified over 3 years ago

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Rational Functions

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Definition: A Rational Function is a function in the form: f(x) = where p(x) and q(x) are polynomial functions and q(x) 0. In this section, p and q will have degree 1 or 0. For example:

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Very Important definitions: Vertical asymptote occurs at values of x for which the function is undefined (exception: unless there is a hole... well talk about that later). Horizontal asymptote occurs if the function approaches a specific value when x approaches infinity or negative infinity. Think of the warmup: what happens to y when x gets REALLY big or REALLY small? To graph rational functions, ALWAYS figure out the asymptotes FIRST. Then you can plot specific points!!!!

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Consider: Vertical asymptote will occur at x = 0. D f = (, 0), (0, ) Horizontal asymptote will occur at y = 0. Think what happens when you divide 1 by a VERY large number!!!!! Show your asymptotes!! Pick some xs on each side of the vertical asymptote to see the graph!!!

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xyxY 3 3 2 2 1 1.5 -.3333 -.5 -2.3333.5 1 2 R f = (, 0), (0, ) In most cases, the range will be closely related to the horizontal asymptote... be sure to check the graph. Note: the graph represents a hyperbola centered at (0, 0)

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Another type of rational function: The vertical asymptote is still x = h. Based on our observations, the hortizontal asymptote is y = k. D f = (, h), (h, ) So, this will be a hyperbola centered at (h, k)!!

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Show your asymptotes!! Pick some xs on each side of the vertical asymptote to see the graph!!! Vertical asymptote: x = –3 Horizontal asymptote: y = 2 xyxY 4 –2 5 –1 0 1 4 3 D f = (, 3), ( 3, ) R f = (, 2), (2, ) Use more points if you want...

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Last example: Vertical asymptote will correspond to the value that makes the denominator 0. Horizontal asymptote: y = Asymptotes:x = –2y = 3 xyxY 3 –1 4 0.5 –2 8 5.5 D f = (, 2), ( 2, ) R f = (, 3), (3, )

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x = 0; y = 0x = 3; y = 1

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x = 4; y = 1 Remember: Find the asymptotes FIRST. Show them on the graph!!! Pick x values to the right and to the left of the vertical asymptote(s). Use the points along with the asymptotes to sketch the graph!!!

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