Key Features of Rational Functions

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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, where the denominator is not zero.

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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) = 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 the graph of a function approaches but never touches or crosses.

Front

What are the types of asymptotes associated with rational functions?

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There are three types of asymptotes: vertical asymptotes, horizontal asymptotes, and slant (oblique) asymptotes.

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How do you find vertical asymptotes of a rational function?

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Vertical asymptotes occur at the values of x that make the denominator Q(x) equal to zero, provided that these values do not also make the numerator P(x) equal to zero.

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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 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 the degree of the numerator is less than the degree of the denominator, y=0; if they are equal, y = leading coefficient of P / leading coefficient of Q; if the numerator's degree is greater, there is no horizontal asymptote.

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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 greater than the degree of the denominator.

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How do you find the x-intercept of a rational function?

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To find the x-intercept, set the numerator P(x) equal to zero and solve for x.

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What is the significance of the y-intercept in a rational function?

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The y-intercept is the point where the graph crosses the y-axis, found by evaluating f(0) if the function is defined at x=0.