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3.7 Graphs of Rational Functions

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A rational function is a quotient of two polynomial functions. A rational function is a quotient of two polynomial functions.

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Parent function: has branches in 1 st and 3 rd quadrants. No x or y-intercepts. Branches approach asymptotes.

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Vertical asymptote – the line x = a is a VA for f(x) if f(x) approaches infinity or f(x) approaches negative infinity as x approaches a from either the left or the right. The VA is where the function is undefined or the value(s) that make the denominator = 0. Whenever the numerator and denominator have a common linear factor, a point discontinuity may appear. If, after dividing the common linear factors, the same factor remains in the denominator, a VA exists. Otherwise the graph will have point discontinuity. Whenever the numerator and denominator have a common linear factor, a point discontinuity may appear. If, after dividing the common linear factors, the same factor remains in the denominator, a VA exists. Otherwise the graph will have point discontinuity.

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Ex 1 find any VA or holes

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Horizontal Asymptote – the line y = b is a HA for f(x) if f(x) approaches b as x approaches infinity or as x approaches negative infinity. Can have 0 or 1 HA. Can have 0 or 1 HA. May cross the HA but it levels off and approaches it as x approaches infinity. May cross the HA but it levels off and approaches it as x approaches infinity.

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Ex 2 Determine the asymptotes

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Ex 3 find the asymptotes

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Ex 4 Find asymptotes

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Shortcut for HAs 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 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 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. 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 5 find asymptotes

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Slant asymptote There is an oblique or slant asymptote if the degree of P(x) is EXACTLY one degree higher than Q(x). There is an oblique or slant asymptote if the degree of P(x) is EXACTLY one degree higher than Q(x). oblique or slant asymptoteoblique or slant asymptote If this is the case the oblique asymptote is the quotient part of the division. If this is the case the oblique asymptote is the quotient part of the division. Can have 0 or 1 slant asymptote. Can have 0 or 1 slant asymptote. Can have a VA and slant, a HA and VA, but NOT a HA and slant. Can have a VA and slant, a HA and VA, but NOT a HA and slant.

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Ex 6 find the slant asymptote

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Ex 7 Graph and find everything!!

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Day 2

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x 2 – 3xy – 13x + 12y + 39 = 0

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