Horizontal & Vertical Asymptotes, & Holes

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.

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, the asymptote is y=0. If they are equal, the asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

Front

What is a vertical asymptote?

Back

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 vertical asymptotes?

Back

Vertical asymptotes can be found by setting the denominator of a rational function equal to zero and solving for x.

Front

What is a hole in a function?

Back

A hole in a function occurs at a point where both the numerator and denominator of a rational function are zero, indicating that the function is undefined at that point.

Front

How do you find holes in a rational function?

Back

To find holes, factor both the numerator and denominator, and identify common factors. The x-values of these common factors are the locations of the holes.

Front

What is the significance of the degrees of the numerator and denominator in determining asymptotes?

Back

The degrees of the numerator and denominator help determine the behavior of the function as x approaches infinity, which in turn helps identify horizontal asymptotes.

Front

If a rational function has a degree of 2 in the numerator and 3 in the denominator, what is the horizontal asymptote?

Back

The horizontal asymptote is y = 0, since the degree of the numerator is less than the degree of the denominator.

Front

What happens to the graph of a function at a vertical asymptote?

Back

At a vertical asymptote, the graph of the function will approach infinity or negative infinity, indicating that the function is undefined at that x-value.

Front

Can a function have both vertical asymptotes and holes?

Back

Yes, a function can have both vertical asymptotes and holes. Holes occur at points where the function is undefined due to common factors, while vertical asymptotes occur where the function approaches infinity.