4-21 Vertical and Horizontal Asymptotes from Equations

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 values that the function cannot take.

Front

How do you find Vertical Asymptotes from a rational function?

Back

Set the denominator equal to zero and solve for x. The solutions are the vertical asymptotes.

Front

What is a Horizontal Asymptote?

Back

A horizontal asymptote is a line y = b that a function approaches as x approaches infinity or negative infinity. It indicates the behavior of the function at extreme values.

Front

How do you find Horizontal Asymptotes for rational functions?

Back

1. If the degree of the numerator is less than the degree of the denominator, y = 0 is the horizontal asymptote. 2. If the degrees are equal, divide the leading coefficients. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What does it mean if a function has no Horizontal Asymptote?

Back

It means that as x approaches infinity or negative infinity, the function does not approach a specific value.

Front

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

Back

The degrees determine the behavior of the function at infinity, which helps in identifying the horizontal asymptote.

Front

If a rational function has a vertical asymptote at x = 3, what does this imply?

Back

It implies that the function is undefined at x = 3 and approaches infinity or negative infinity as x approaches 3.

Front

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

Back

Vertical asymptotes occur at the values of x that make the denominator zero, provided those values do not also make the numerator zero.

Front

Can a function have more than one Vertical Asymptote?

Back

Yes, a function can have multiple vertical asymptotes if the denominator has multiple factors that can be set to zero.

Front

What happens to the graph of a function at a Vertical Asymptote?

Back

The graph will approach the vertical asymptote line but will never touch or cross it.