Asymptotes of Rational Functions

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What is a vertical asymptote?

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A vertical asymptote is a line x = a where the function approaches infinity or negative infinity as x approaches a. It occurs when the denominator of a rational function equals zero and the numerator does not equal zero at that point.

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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 of the rational function equal to zero and solve for x. The values of x that make the denominator zero are the vertical asymptotes.

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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 a function approaches as x approaches infinity or negative infinity. It indicates the end behavior of the function.

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How do you determine horizontal asymptotes for rational functions?

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1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. 2. If the degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). 3. If the degree of the numerator 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 more than the degree of the denominator. It can be found using polynomial long division.

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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 is a removable discontinuity?

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A removable discontinuity occurs at a point where both the numerator and denominator of a rational function equal zero. It can be 'removed' by factoring and simplifying the function.

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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 value of the function when x = 0. It can be found by substituting x = 0 into the function.

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

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The x-intercept is the value of x where the function equals zero. It can be found by setting the numerator equal to zero and solving for x.

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What is end behavior in the context of rational functions?

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End behavior describes how the values of a function behave as x approaches positive or negative infinity. It is often represented by horizontal asymptotes.