Rational Functions

Front

What is a rational function?

Back

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 a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Front

How do you find the horizontal asymptote of a rational function?

Back

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 numerator / leading coefficient of denominator; if the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is a vertical asymptote?

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A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a certain value, typically where the function is undefined.

Front

How do you find the vertical asymptote of a rational function?

Back

To find the vertical asymptote, set the denominator equal to zero and solve for x.

Front

What does it mean if a rational function has no horizontal asymptote?

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It means that as x approaches positive or negative infinity, the function's values do not approach a finite number.

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What is the significance of the leading coefficient in determining horizontal asymptotes?

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The leading coefficient of the numerator and denominator helps determine the horizontal asymptote when the degrees of both polynomials are equal.

Front

What is the relationship between the degree of the numerator and the degree of the denominator in rational functions?

Back

The degree of the numerator and denominator determines the behavior of the function at infinity and the existence of horizontal asymptotes.

Front

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 happens to the graph of a rational function at its vertical asymptotes?

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The graph approaches the vertical asymptote but never touches or crosses it.