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rational functions.docx

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University of Toronto Scarborough
Ken Butler

Reciprocal of a Linear Function  The reciprocal of a linear function has the form: f(x) = 1 / kx – c  The restriction on a domain of a reciprocal linear function can be determined by finding the value of x that makes the denominator equal to zero, that is x = c / k.  The Vertical Asymptote of a reciprocal linear function has an equation of the form x = k / c.  The horizontal asymptote of a reciprocal linear function has equation y = 0.  If k > 0, the left branch of a reciprocal linear function has a negative, decreasing slope, and the right branch has a negative, increasing slope.  Basically occupies Q3 and Q1.  If k < 0, the left branch of a reciprocal linear function has a positive, increasing slow, and the right branch has a positive, decreasing slope.  Basically occupies Q2 and Q4. Reciprocal of a Quadratic Function  Rational functions can be analyzed using key features: asymptotes,intercepts, slope (positive or negative, increasing or decreasing),domain, range, and positive and negative intervals.  Reciprocal of quadratic functions with two zeros have three parts, with the middle one reaching a maximum or minimum points. This point is equidistant from the two vertical asymptotes.  The behavior near asymptotes is similar to that of reciprocals of linear functions.  All of the behaviors listed above can be predicted by analyzing the roots of the quadratic relation to the denominator. Rational Functions of the form f(x) = (ax + b) / (cx + d)  A rational function of the form f(x) = (ax + b) / (cx + d) has the following key features:  The vertical asymptote can be found by setting the denominator equal to zero and solving for x, provided the numerator does not have the same zero.  The horizontal asymptot
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