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Simple Rational Functions - Asymptotes

A simple rational function is any functionof the formwhereandare constants.

We can perform long division on this to obtain

This is a transformation of the graphThe transformation is in three steps:

  1. A translation to the left, the negative– direction, by

  2. A scaling in the- direction byIf this expression is negative, then the graph is reflected in the– axis and scaled.

  3. A translation in the– direction by

  4. The asymptotes are also transformed. The– asymptotebecomes the– asymptoteTheasymptotebecomes the– asymptote

Suppose for example that

Proceeding as above,

  1. 1. A translation to the left, the negative– direction, by

  2. A reflection in the x – axis and scaling in the- direction by

  3. A translation in the– direction byTheandasymptotes becomeandrespectively.

The transformation is shown below.