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Describing Transformations of Rational Functions

Describing Transformations of Rational Functions 1.3 - Solution

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Let's start by recalling possible transformations of the parent function

Function Transformation of the Graph of
Horizontal translation by units.
If the translation is to the right.
If the translation is to the left.
Vertical translation by units.
If the translation is up.
If the translation is down.
Horizontal stretch or compression by a factor of
If it is a horizontal stretch.
If it is a horizontal compression.
Reflection across the axis.

Now, let's consider the given function. We can see that and Also, there is a sign in the denominator. From here, we can determine the transformations.

  1. A horizontal stretch by a factor of
  2. A reflection across the axis.
  3. A horizontal translation units to the right.
  4. A vertical translation unit up.

Using these transformations, we can find the asymptotes and the reference points of the graph of Note that the horizontal stretch, the reflection across the axis, and the horizontal translation affect only the coordinates, while the vertical translation affects only the coordinates.

Feature
Vertical asymptote

Horizontal asymptote

Reference point

Reference point

Next, we will use the table above to graph and