Vertical and Horizontal Asymptotes

Front

What are vertical asymptotes?

Back

Vertical asymptotes are lines x = a where a function approaches infinity or negative infinity as the input approaches a. They occur where the denominator of a rational function is zero.

Front

What are horizontal asymptotes?

Back

Horizontal asymptotes are lines y = b that a function approaches as the input approaches positive or negative infinity. They indicate the behavior of the function at extreme values.

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

How do you find horizontal asymptotes in a rational function?

Back

To find horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less, y = 0; if equal, y = leading coefficient of numerator/leading coefficient of denominator; if greater, no horizontal asymptote.

Front

What is the vertical asymptote of the function f(x) = 1/(x + 5)?

Back

The vertical asymptote is x = -5.

Front

What is the horizontal asymptote of the function f(x) = 2x^2/(3x^2 + 1)?

Back

The horizontal asymptote is y = 2/3.

Front

What happens to a function as it approaches a vertical asymptote?

Back

As a function approaches a vertical asymptote, its values increase or decrease without bound, approaching positive or negative infinity.

Front

What is the significance of horizontal asymptotes?

Back

Horizontal asymptotes indicate the end behavior of a function, showing what value the function approaches as x goes to positive or negative infinity.

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 equal zero.

Front

Can a function have a horizontal asymptote and still cross it?

Back

Yes, a function can cross its horizontal asymptote; the asymptote describes the behavior as x approaches infinity, not necessarily at all points.