Graphing 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.

Front

What is an asymptote in the context of graphing functions?

Back

An asymptote is a line that a graph approaches but never touches or crosses.

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 polynomials. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If they are equal, the horizontal asymptote is 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 the significance of x-intercepts in a rational function?

Back

X-intercepts are the points where the graph of the function crosses the x-axis, which occur when the numerator of the rational function is equal to zero.

Front

How do you determine the x-intercepts of a rational function?

Back

To find the x-intercepts, set the numerator of the rational function equal to zero and solve for x.

Front

What is a vertical asymptote?

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A vertical asymptote is a vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

Front

How do you find vertical asymptotes in a rational function?

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Vertical asymptotes occur at the values of x that make the denominator of the rational function equal to zero, provided that these values do not also make the numerator zero.

Front

What is the difference between horizontal and vertical asymptotes?

Back

Horizontal asymptotes describe the behavior of a function as x approaches infinity, while vertical asymptotes describe the behavior of a function as x approaches a specific finite value.

Front

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

Back

If a rational function has no horizontal asymptote, it means that the degree of the numerator is greater than the degree of the denominator, and the function will increase or decrease without bound as x approaches infinity.

Front

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

Back

The leading coefficient of the numerator and denominator polynomials determines the value of the horizontal asymptote when the degrees of the polynomials are equal.