Holes and Vertical Asymptotes

Front

What is a hole in a rational function?

Back

A hole occurs when a factor in the denominator cancels with a factor in the numerator, resulting in an undefined point in the graph.

Front

How do you find the coordinates of a hole?

Back

To find the coordinates of a hole, set the canceled factor equal to zero and solve for x. Then, substitute this x-value into the simplified function to find the corresponding y-value.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line that a graph approaches but never touches or crosses, occurring where the denominator of a rational function is zero and not canceled by the numerator.

Front

How do you determine the vertical asymptotes of a rational function?

Back

To find vertical asymptotes, set the denominator equal to zero and solve for x, ensuring that these values are not canceled by the numerator.

Front

What does it mean if a rational function has no holes?

Back

If a rational function has no holes, it means that there are no factors in the denominator that cancel with factors in the numerator.

Front

What is the significance of the coordinates of a hole?

Back

The coordinates of a hole indicate a point on the graph where the function is undefined, representing a removable discontinuity.

Front

Can a rational function have multiple holes?

Back

Yes, a rational function can have multiple holes if there are multiple factors in the numerator and denominator that cancel.

Front

What is the relationship between holes and vertical asymptotes?

Back

Holes are removable discontinuities where the function is undefined, while vertical asymptotes are non-removable discontinuities where the function approaches infinity.

Front

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

Back

It implies that the function is undefined at x = 3, but it can be simplified to a form that is defined at that point.

Front

How do you express the vertical asymptote in mathematical notation?

Back

Vertical asymptotes are expressed in the form x = a, where 'a' is the value that makes the denominator zero.