Langara College - Department of Mathematics and Statistics

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Internet Resources for the Calculus Student - Topics in Precalculus

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Rational Functions

Ratios of polynomials are called rational functions. You can use
our Graph
Explorer (or any Computer Algebra System or Graphing Calculator) to
graph such functions and to see how their singularities and intercepts
are related to the zeros of the denominator and numerator (either by factoring,
or by graphing the numerator and denominator separately along with the
ratio).
Notice that if the degree of the numerator is larger than that of the
denominator, then the y-value on the graph "goes to infinity" as x goes
to +or- infinity (you may have to zoom out quite a bit to see this in some
cases), but if the denominator has larger degree, then the graph comes
down close to the x-axis (which we then say is an "asymptote" for the graph).
When the degrees are equal there is again a horizontal asymptote, but now
at a y-value equal to the ratio of leading coefficients rather than to
zero. If the degree of the numerator is just one bigger than that of the
denominator, then the graph will have a slant asymptote. You can guess
an equation for the asymptote and then adjust the coefficients to get a
good fit (play around with moving about and zooming out and in to check
that your line fits well for large |x|), and then compare with what you
get by long division. More detailed instructions for some of these activities
are included in our on-line lab on rational
functions.

A geometric
construction of the graph of 1/f(x) from that of f(x) is provided by
the IES group in Japan.

The University of Saskatchewan's Exercises
in Math Readiness (EMR) site has a section on

Graphing Rational
Functions .

The 'Partial Fractions Expansion'
expresses a rational function with complicated (high degree) denominator
as a sum of simpler cases (with linear and quadratic denominators)

See The
Method of Partial Fractions, from S.O.S.
MATHematics.

or The Method of Partial
Fractions, from the Review
of Algebra Techniques, part of Oregon State University's extensive
Web
Study Guide project.

or Partial Fraction Decompositions
in USask's EMR site. Included are an introduction and three sets of exercises
: Introductory,
Moderate,
and Advanced.

You might also check our 'raw list' (of links provided without comment)
to see if there are any

more examples
there that we haven't yet included here.

If you have come across any more good web-based illustrations of these
and related concepts,

please do let
us know and we will add them here.