Asymptotes of Rational Functions

Front

What is a horizontal asymptote?

Back

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

Front

What is a vertical asymptote?

Back

A vertical line that the graph of a function approaches as x approaches a specific value where the function is undefined.

Front

What is a hole in a rational function?

Back

A point on the graph where the function is not defined due to a factor in the numerator and denominator canceling out.

Front

How do you find vertical asymptotes?

Back

Set the denominator of the rational function equal to zero and solve for x.

Front

How do you find horizontal asymptotes for rational functions?

Back

Compare the degrees of the numerator and denominator: If degree of numerator < degree of denominator, y=0; if equal, y = leading coefficient of numerator/leading coefficient of denominator; if greater, no horizontal asymptote.

Front

What happens to the function at a hole?

Back

The function is not defined at that point, but the graph will approach that point from either side.

Front

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

Back

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

Front

Can a rational function have both vertical asymptotes and holes?

Back

Yes, a rational function can have vertical asymptotes where the function approaches infinity and holes where the function is not defined.

Front

What is the relationship between the factors of the numerator and denominator in finding holes?

Back

A hole occurs when a factor in the numerator cancels with a factor in the denominator.

Front

If a rational function has a hole at x = -3, what does this imply?

Back

It implies that the function is not defined at x = -3, but the graph will approach this value.