Rational Function Graphs

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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 vertical asymptote?

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A vertical asymptote is a line x = a where the function f(x) approaches infinity or negative infinity as x approaches a. It occurs where the denominator Q(x) = 0 and P(a) ≠ 0.

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What is a horizontal asymptote?

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

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How do you find vertical asymptotes of a rational function?

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To find vertical asymptotes, set the denominator Q(x) = 0 and solve for x. The values of x that satisfy this equation are the vertical asymptotes.

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How do you find horizontal asymptotes of a rational function?

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

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What is the effect of adding a constant to a rational function?

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What is the effect of subtracting a constant from the denominator of a rational function?

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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 Q(x) = 0.

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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 horizontal asymptote value, if it exists.