Asymptotes in Rational Functions Review

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What is a vertical asymptote?

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A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a. It occurs when the denominator of a rational function equals zero and the numerator does not.

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How do you find vertical asymptotes in a rational function?

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To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

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What is a horizontal asymptote?

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A horizontal asymptote is a line y = b that a function approaches as x approaches infinity or negative infinity. It indicates the behavior of the function at extreme values.

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How do you determine horizontal asymptotes in rational functions?

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To determine horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less, y = 0; if equal, y = leading coefficient of numerator/leading coefficient of denominator; if greater, there is no horizontal asymptote.

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What is a hole in a rational function?

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A hole occurs in a rational function at a point where both the numerator and denominator are zero, indicating that the function is undefined at that point but can be simplified.

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How do you identify holes in a rational function?

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To identify holes, factor both the numerator and denominator, and find common factors. The x-values of these common factors indicate the location of the holes.

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What is the significance of vertical asymptotes?

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Vertical asymptotes indicate values of x where the function is undefined and help in understanding the behavior of the graph near those points.

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What is the significance of horizontal asymptotes?

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Horizontal asymptotes indicate the end behavior of a function as x approaches infinity or negative infinity, helping to predict the function's long-term behavior.

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There is a hole at x = 1 because both the numerator and denominator equal zero at this point.

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The vertical asymptotes are at x = 2 and x = 3, found by setting the denominator equal to zero.