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Cogito

AP Precalculus · Chapter 3 · Lesson 2

Practice Worksheet

End Behaviour of Rational Functions

Comparing the degrees, top and bottom.

11 problems · about 22 minutes · AP Precalculus 1.12

NameDate

Figure — use these to answer the problems

A bottom-heavy rational function flattens towards y = 0 at both ends.
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y = 2/x + 0
a2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. (9x² + 2)/(3x² − 7). Horizontal asymptote y = ?

    Answer

  2. 5/(x + 2). Horizontal asymptote y = ?

    Answer

  3. (4x² − 1)/(x² + 3). Horizontal asymptote y = ?

    Answer

  4. (x³ + 1)/(x + 2). How many horizontal asymptotes?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Sort each function by its end behaviour.

    Write each item under the heading it belongs to: 1/x · 2x/(x + 1) · x²/(x + 1) · x/(x³ + 2)

    Approaches y = 0

    Approaches a non-zero y

    No horizontal asymptote

  2. (2x + 7)/(5x − 1). Horizontal asymptote y = ? Give it as a decimal.

    Answer

  3. Top degree exceeds bottom by exactly 1. Is there a slant asymptote? 1 for yes, 0 for no.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. (x² + 5)/(x² + 5). Horizontal asymptote y = ?

    Answer

  2. Can a graph cross its horizontal asymptote? 1 for yes, 0 for no.

    Answer

  3. Bottom Heavy

    For x / (x² + 1), what is the horizontal asymptote? Give the y value.

    Answer

  4. Equal Degrees

    For (6x + 1) / (3x − 2), what is the horizontal asymptote? Give the y value.

    Answer