rational functions and their asymptotes

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What is a rational function?

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A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

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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 x approaches positive or negative 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 P / leading coefficient of Q.

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What is a vertical asymptote?

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A vertical asymptote is a vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

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How do you find vertical asymptotes of a rational function?

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Vertical asymptotes are found by setting the denominator Q(x) = 0 and solving for x.

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What is a slant (oblique) asymptote?

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A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

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

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To find the slant asymptote, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.

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What are the characteristics of the graph of a rational function near vertical asymptotes?

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Near vertical asymptotes, the graph of a rational function will approach infinity or negative infinity, indicating a discontinuity.

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What happens to the graph of a rational function at horizontal asymptotes?

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At horizontal asymptotes, the graph of the rational function approaches the line but may not touch or cross it.

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What is the significance of the leading coefficients in determining horizontal asymptotes?

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The leading coefficients of the numerator and denominator determine the value of the horizontal asymptote when the degrees of the polynomials are equal.