Rational Functions

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What is a rational function?

Back

Front

What determines the vertical asymptotes of a rational function?

Back

Vertical asymptotes are determined by 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.

Front

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. It indicates the end behavior of the function.

Front

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

Back

To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator: If the degree of P is less than Q, the asymptote is y=0; if they are equal, y=\frac{a}{b} where a and b are the leading coefficients.

Front

What is a hole in the graph of a rational function?

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A hole occurs in the graph of a rational function at a value of x that makes both the numerator and denominator equal to zero. It indicates a removable discontinuity.

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How do you simplify a rational function?

Back

To simplify a rational function, factor both the numerator and denominator and cancel out any common factors.

Front

What is the significance of the end behavior of a rational function?

Back

The end behavior describes how the function behaves as x approaches positive or negative infinity, which is important for understanding the overall shape of the graph.

Front

What is the relationship between the degree of the numerator and the degree of the denominator in determining asymptotes?

Back

The degree of the numerator compared to the degree of the denominator determines the type of asymptotes: if the numerator's degree is greater, there is no horizontal asymptote; if they are equal, there is a horizontal asymptote.

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Back

The vertical asymptote is at x=3, where the denominator equals zero.

Front

How do you identify vertical asymptotes from a rational function?

Back

Set the denominator equal to zero and solve for x. The solutions are the vertical asymptotes.