rational functions and their asymptotes

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What are Vertical Asymptotes?

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Vertical asymptotes are lines x = a where a rational function approaches infinity or negative infinity as the input approaches a. They occur at values of x that make the denominator zero, provided the numerator is not also zero at that point.

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How do you find Vertical Asymptotes?

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To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

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What is an Oblique Asymptote?

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An oblique (or slant) asymptote is a diagonal line that a function approaches as x approaches infinity or negative infinity. It occurs when the degree of the numerator is one greater than the degree of the denominator.

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How do you find an Oblique Asymptote?

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

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What are Horizontal Asymptotes?

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Horizontal asymptotes are lines y = b where a rational function approaches a constant value b as x approaches infinity or negative infinity.

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How do you determine Horizontal Asymptotes?

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To determine horizontal asymptotes, compare the degrees of the numerator and denominator: If the degree of the numerator is less than the denominator, y = 0; if they are equal, y = leading coefficient of numerator / leading coefficient of denominator; if the numerator's degree is greater, there is no horizontal asymptote.

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What is the significance of the degree of the numerator and denominator?

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The degree of the numerator and denominator helps determine the behavior of the rational function at infinity and the existence of horizontal or oblique asymptotes.

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What are the characteristics of a rational function?

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A rational function is a function of the form f(x) = P(x)/Q(x), where P and Q are polynomials. It can have vertical, horizontal, and oblique asymptotes.

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What happens at a Vertical Asymptote?

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At a vertical asymptote, the function approaches infinity or negative infinity, indicating that the function is undefined at that x-value.

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Can a rational function have more than one Vertical Asymptote?

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Yes, a rational function can have multiple vertical asymptotes, depending on the number of factors in the denominator that can be set to zero.