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Lecture

# rational functions.docx

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University of Toronto Scarborough

Mathematics

MATA30H3

Ken Butler

Winter

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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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