Graphing Rational Functions Flashcardizz Review

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What is a horizontal asymptote?

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A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) approaches positive or negative infinity. It indicates the behavior of the function at extreme values.

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What is a vertical asymptote?

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A vertical asymptote is a vertical line that the graph of a function approaches as the input (x) approaches a specific value where the function is undefined. It indicates where the function tends to infinity.

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What is a slanted (oblique) asymptote?

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A slanted asymptote occurs when the degree of the numerator is one greater than the degree of the denominator in a rational function. It represents the linear behavior of the function as x approaches infinity.

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How do you find the horizontal asymptote of a rational function?

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To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the asymptote is y = 0. If they are equal, the asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

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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. The values of x that make the denominator zero are the vertical asymptotes.

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What is the significance of asymptotes in graphing rational functions?

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Asymptotes help to understand the behavior of rational functions, indicating where the function does not exist (vertical asymptotes) and how it behaves at extreme values (horizontal asymptotes).

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The vertical asymptote is at x = 1, where the denominator equals zero.

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The horizontal asymptote is y = 2, since the degrees of the numerator and denominator are equal and the leading coefficients are 2 and 1 respectively.

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What is the process to find a slanted asymptote?

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To find a slanted asymptote, perform polynomial long division on the rational function. The quotient (ignoring the remainder) will give the equation of the slanted asymptote.

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The vertical asymptotes are at x = 1 and x = -1, and there is a horizontal asymptote at y = 0.