Graphing Rational Functions

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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 indicates values that the function cannot take.

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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 the input approaches positive or negative infinity. It indicates the behavior of the function at extreme values.

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How do you find the Vertical Asymptote of a rational function?

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

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How do you find the Horizontal Asymptote of a rational function?

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To find the horizontal asymptote, 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 the domain of a rational function?

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The domain of a rational function is all real numbers except where the denominator equals zero.

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What is the range of a rational function?

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The range of a rational function is all real numbers except for the value(s) that the function cannot take, often related to horizontal asymptotes.

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Example of finding a Vertical Asymptote: For f(x) = 1/(x-6), what is the vertical asymptote?

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The vertical asymptote is x = 6.

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Example of finding a Horizontal Asymptote: For f(x) = 2x/(3x+1), what is the horizontal asymptote?

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The horizontal asymptote is y = 2/3.

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What happens to the graph of a rational function at a Vertical Asymptote?

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The graph approaches the vertical asymptote but never touches or crosses it.

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What happens to the graph of a rational function at a Horizontal Asymptote?

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The graph approaches the horizontal asymptote as x approaches positive or negative infinity.