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 and Q(x) ≠ 0.

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What is the general form of a rational function?

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The general form of a rational function is f(x) = \frac{P(x)}{Q(x)}, where P(x) and Q(x) are polynomials.

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What is an asymptote in the context of rational functions?

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An asymptote is a line that a graph approaches but never touches. For rational functions, there are vertical and horizontal asymptotes.

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How do you find vertical asymptotes of a rational function?

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Vertical asymptotes occur where the denominator Q(x) = 0, provided that the numerator P(x) is not also zero at those points.

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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 infinity or negative infinity.

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How do you determine 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 degree of P < degree of Q, y=0; if degree of P = degree of Q, y = leading coefficient of P / leading coefficient of Q; if degree of P > degree of Q, there is no horizontal asymptote.

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What does the transformation f(x) = \frac{1}{x - 2} represent?

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This transformation represents a horizontal shift of the basic rational function f(x) = \frac{1}{x} to the right by 2 units.

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What does the transformation f(x) = \frac{1}{x} + 2 represent?

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This transformation represents a vertical shift of the basic rational function f(x) = \frac{1}{x} upwards by 2 units.

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What is the effect of the transformation f(x) = \frac{1}{x - 4} + 1?

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This transformation shifts the graph of f(x) = \frac{1}{x} to the right by 4 units and up by 1 unit.

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What is the domain of the rational function f(x) = \frac{1}{x - 2}?

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The domain is all real numbers except x = 2, where the function is undefined.