Asymptotes

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 is determined by the values that make the denominator of a rational function equal to zero.

Front

Which asymptote is determined by looking at the denominator of a rational function?

Back

Vertical asymptote.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b that a function approaches as the input approaches positive 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. Compare the degrees of the numerator and denominator. 2. If the degree of the numerator is less than the denominator, y = 0 is the horizontal asymptote. 3. If the degrees are equal, divide the leading coefficients. 4. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is the horizontal asymptote of the function f(x) = 2x^2 + 3 / 4x^2 + 5?

Back

y = 1/2 (the leading coefficients of the numerator and denominator are 2 and 4 respectively).

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, meaning it approaches positive or negative infinity.

Front

Can a function cross its horizontal asymptote?

Back

Yes, a function can cross its horizontal asymptote; the horizontal asymptote describes the end behavior of the function, not its values in between.

Front

What is the significance of identifying asymptotes in graphing functions?

Back

Identifying asymptotes helps in understanding the behavior of the function, predicting its graph, and determining limits.

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. It can be found using polynomial long division.

Front

How do you find a slant asymptote for the function f(x) = (2x^3 + 3) / (x^2 + 1)?

Back

Perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.