Rational Functions-Vertical Asymptotes and Holes

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a. It indicates that the function is undefined at that point.

Front

How do you find vertical asymptotes in a rational function?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

Front

What is a hole in a rational function?

Back

A hole occurs in a rational function when a factor in the numerator cancels with a factor in the denominator, indicating that the function is undefined at that point.

Front

How do you identify holes in a rational function?

Back

To identify holes, factor both the numerator and denominator, then find the values of x that make both the numerator and denominator zero.

Front

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

Back

As the graph approaches a vertical asymptote, the function values increase or decrease without bound, indicating that the function does not cross the asymptote.

Front

What is the significance of the degree of the numerator and denominator in rational functions?

Back

The degree of the numerator and denominator helps determine the behavior of the function, including the presence of vertical asymptotes and holes.

Front

If a rational function has a degree of the numerator greater than the degree of the denominator, what can be inferred?

Back

If the degree of the numerator is greater than the degree of the denominator, the function will have a slant (oblique) asymptote.

Front

What is the relationship between vertical asymptotes and the roots of the denominator?

Back

Vertical asymptotes occur at the roots of the denominator where the function is undefined.

Front

Can a rational function have both vertical asymptotes and holes?

Back

Yes, a rational function can have both vertical asymptotes and holes, depending on the factors in the numerator and denominator.

Front

What is the vertical asymptote of the function f(x) = 1/(x-3)?

Back

The vertical asymptote is x = 3.