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Internet Resources for the Calculus Student - Topics in Precalculus

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.

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