Rational Functions, Asymptotes and Curve Sketching · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Taming quotients of polynomials

Mathematics · Algebra · ages 17-18
Name ______________________   Date ____________

  1. Let f(x) = (2x + 1) / (x^2 - 4). What is its horizontal asymptote?

    • y = 0
    • y = 2
    • y = 1/2
    • It has none

  2. Let f(x) = (x + 1) / (x - 2). Where is its vertical asymptote?

    • x = 2
    • x = -2
    • x = 1
    • x = -1

  3. Let f(x) = (x + 1) / (x - 2). Where is its vertical asymptote?

    • x = 2
    • x = -2
    • x = 1

  4. A cancelled shared factor leaves a vertical asymptote at that point.

    Circle one:   True   False

  5. Let f(x) = (4x + 7) / (2x - 1). Degrees match, so far out the graph settles near which y value?

    Answer: ______________

  6. Let f(x) = (x - 4) / (x + 2). What is its y-intercept?

    • -2
    • 2
    • 4
    • -4

  7. Let f(x) = (x - 3)(x + 1) / ((x - 3)(x - 2)). Where is the hole in its graph?

    • x = 3
    • x = 2
    • x = -1
    • x = -3

  8. Let f(x) = (x - 3)(x + 1) / ((x - 3)(x - 2)). Where is the hole in its graph?

    • x = 2
    • x = 3
    • x = -1

  9. A student sets the denominator to zero without factoring first and calls each zero an asymptote. What is the error?

    • Fractions never have any asymptotes
    • The zeros of the top were ignored
    • Skipped cancelling turned a hole into a fake asymptote

  10. Let f(x) = (x cubed + 1) / (x squared - 4). The top degree is heavier. What happens far out?

    • It grows without bound
    • It settles exactly at y = 1
    • It settles to y = 0
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Taming quotients of polynomials W1-mt_U-f9YTNb2o-s1

  1. y = 0 · The bottom degree is bigger, so the graph settles to zero far out.
  2. x = 2 · The denominator is zero at x = 2 and nothing cancels, so the graph breaks there.
  3. x = 2 · The denominator is zero at x = 2 and nothing cancels there.
  4. False · Cancelled factors leave a hole, a single missing point.
  5. 2 · Equal degrees settle at 4/2, which is 2.
  6. -2 · f(0) = -4 / 2 = -2, which plots below the origin as the sketch predicts.
  7. x = 3 · The factor (x - 3) cancels, leaving a single missing point at x = 3.
  8. x = 3 · The factor (x - 3) cancels, leaving a missing point at x = 3.
  9. Skipped cancelling turned a hole into a fake asymptote · Factor first and cancel, or holes get mislabeled as asymptotes.
  10. It grows without bound · A heavier top outgrows the bottom, so no horizontal settling exists.
Worksheet · LightMySky