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AP Calculus AB: 8.5.1 Vertical Asymptotes

Mathematics7 CardsCreated 8 months ago

This content explains how to identify vertical asymptotes in rational functions by factoring, canceling common factors, and setting the denominator equal to zero. It also distinguishes vertical asymptotes from holes, which occur when factors cancel in both numerator and denominator.

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Vertical Asymptotes

  • Asymptotes are lines that the graph of a function approaches. A vertical asymptote to the graph of a function f is a line whose equation is x = a

  • Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero.

  • If a given point makes both the numerator and denominator of a function equal zero, then there might be a hole in the graph of the function at that point

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Key Terms

Term
Definition

Vertical Asymptotes

  • Asymptotes are lines that the graph of a function approaches. A vertical asymptote to the graph of a function f is a line whose equation is...

note

  • The graphs of some rational functions have
    vertical asymptotes.

  • To determine vertical asymptotes you must follow three st...

Find the vertical asymptotes of the curve:

y = x−2 / x^2+4x+3.

x = −1 and x = −3

Find the vertical asymptotes.

y = x^2+2x / x^3−4x

x = 2

Find the vertical asymptotes of the curve:

y = x^2−x−6 / x^2+x−2.

x = 1

Find the vertical asymptotes.

y = x / x−2

x = 2

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AP Calculus AB: 8.5.1 Vertical Asymptotes
TermDefinition

Vertical Asymptotes

  • Asymptotes are lines that the graph of a function approaches. A vertical asymptote to the graph of a function f is a line whose equation is x = a

  • Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero.

  • If a given point makes both the numerator and denominator of a function equal zero, then there might be a hole in the graph of the function at that point

note

  • The graphs of some rational functions have
    vertical asymptotes.

  • To determine vertical asymptotes you must follow three steps. First, factor the numerator and denominator. Second, cancel any factors they have in common. Third, set the resulting denominator equal to zero and solve for x.

  • For the reciprocal function, no factoring or canceling is possible. When you set the denominator equal to zero, the result is x = 0. This is the equation of the vertical asymptote.

  • If a function has a vertical asymptote, its graph will get extremely close to the asymptote, but it will never cross it.

  • For the rational function on the far left, no factoring or canceling is possible.

  • When you set the denominator equal to zero, you get an equation that you can solve to produce x = 3. This is the equation of the vertical asymptote for the function.

  • The function on the near side is more complicated. It can be factored, and the numerator and denominator have a common factor. Canceling produces the same function as on the far left, except that it is not defined for x = 3. This produces a hole in the graph, not a vertical asymptote.

Find the vertical asymptotes of the curve:

y = x−2 / x^2+4x+3.

x = −1 and x = −3

Find the vertical asymptotes.

y = x^2+2x / x^3−4x

x = 2

Find the vertical asymptotes of the curve:

y = x^2−x−6 / x^2+x−2.

x = 1

Find the vertical asymptotes.

y = x / x−2

x = 2

Find the vertical asymptotes of f(x).

f(x) = x^2−x−2 / x^2+x−2

x = 1, x = −2