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Cogito

AP Precalculus · Chapter 3 · Lesson 1

Practice Worksheet

Vertical Asymptotes and Holes

Two different things a denominator of zero can mean.

11 problems · about 22 minutes · AP Precalculus 1.10, 1.11

NameDate

Figure — use these to answer the problems

The reciprocal function has a vertical asymptote where its denominator vanishes.
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y = 1/x + 0
a1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. (x + 2)/((x + 2)(x − 7)). At what x is the hole?

    Answer

  2. 1 / (x + 6). Vertical asymptote at which x?

    Answer

  3. (x − 1)/((x − 1)(x + 4)). Hole at which x?

    Answer

  4. Same function. Asymptote at which x?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Match each feature to what causes it.

    Draw a line from each item on the left to its match on the right.

    • Vertical asymptote
    • Removable hole
    • x-intercept
    • A denominator factor that does not cancel
    • A factor shared by numerator and denominator
    • A numerator factor that does not cancel

  2. Is the function defined at a hole? 1 for yes, 0 for no.

    Answer

  3. x/(x² − 9). How many vertical asymptotes?

    Answer

Stretch Yourself

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

4 problems

  1. Same function. What is the x-intercept?

    Answer

  2. 1/(x² + 1). How many vertical asymptotes?

    Answer

  3. The Denominator

    f(x) = 1 / (x − 4). At what x is there a vertical asymptote?

    Answer

  4. The Cancelling Factor

    f(x) = (x − 3) / ((x − 3)(x + 1)). At what x is there a hole?

    Answer