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AP Calculus AB: 8.5.4 Functions with Asymptotes and Holes

Mathematics5 CardsCreated 3 months ago

This content explains how to analyze rational functions that have vertical and horizontal asymptotes, as well as holes (point discontinuities). It focuses on factoring, canceling common terms, and understanding how a removable discontinuity (hole) occurs when both the numerator and denominator share a factor that becomes undefined in the original function.

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Functions with Asymptotes and Holes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where
, or .
• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y = a, where , or .
• A hole (or point discontinuity) occurs in the graph of a function f at a point c if not equal to exists and f(c) is undefined or .

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Key Terms

Term
Definition

Functions with Asymptotes and Holes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where th...

note

  • When graphing a rational function, first look for
    vertical asymptotes.

  • This function can be factored. The numerator and

Which of the following curves is the graph of the equation

f(x) = x+3 / x^2+4x+3?

The correct graph is a rational function with vertical asymptotes at x=−1x = -1x=−1, a hole at x=−3x = -3x=−3, and approaches 0 as x→±∞x \to \pm\in...

Which of the following curves is the graph of the equation

f(x) = x^2−x−6 / x^2+x−2?

The correct graph has a vertical asymptote at x=1x = 1x=1, a hole at x=−2x = -2x=−2, and a horizontal asymptote at y=1y = 1y=1.

Which of the following curves is the graph of the equation

f(x) = −x^2+x+6 / x^2+x−2?

The correct graph has vertical asymptotes at x=1x = 1x=1 and x=−2x = -2x=−2, and a horizontal asymptote at y=−1y = -1y=−1.

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AP Calculus AB: 8.5.4 Functions with Asymptotes and Holes
TermDefinition

Functions with Asymptotes and Holes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where
, or .
• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y = a, where , or .
• A hole (or point discontinuity) occurs in the graph of a function f at a point c if not equal to exists and f(c) is undefined or .

note

  • When graphing a rational function, first look for
    vertical asymptotes.

  • This function can be factored. The numerator and
    denominator have a common factor of (x + 3), so cancel it. Make sure to promise not to evaluate the function at x = –3, because that would make the original expression undefined.

  • Now only x = 3 makes the denominator equal to zero, so that gives the location of the vertical asymptote.

  • Notice that the simplified expression for the function
    resembles a function you have already graphed. It has a horizontal asymptote at y = 1, but no critical point and no points of inflection.

  • The function is decreasing both to the left and right of x = 3. On the left it is concave down, and on the right it is concave up.

  • Notice that there is a hole at x = –3. Since the function cannot be evaluated at this point, the graph skips over it, as indicated by the open circle. Since the function is not continuous at that point, it is also called a point discontinuity.

Which of the following curves is the graph of the equation

f(x) = x+3 / x^2+4x+3?

The correct graph is a rational function with vertical asymptotes at x=−1x = -1x=−1, a hole at x=−3x = -3x=−3, and approaches 0 as x→±∞x \to \pm\inftyx→±∞.

Which of the following curves is the graph of the equation

f(x) = x^2−x−6 / x^2+x−2?

The correct graph has a vertical asymptote at x=1x = 1x=1, a hole at x=−2x = -2x=−2, and a horizontal asymptote at y=1y = 1y=1.

Which of the following curves is the graph of the equation

f(x) = −x^2+x+6 / x^2+x−2?

The correct graph has vertical asymptotes at x=1x = 1x=1 and x=−2x = -2x=−2, and a horizontal asymptote at y=−1y = -1y=−1.