Asymptotes

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or slant.

Front

What is a horizontal asymptote (HA)?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Front

How do you find the horizontal asymptote of a rational function?

Back

For rational functions, compare the degrees of the numerator and denominator. If the degree of the numerator is less, HA is y=0. If equal, HA is y=(leading coefficient of numerator)/(leading coefficient of denominator). If greater, there is no HA.

Front

What is a vertical asymptote (VA)?

Back

A vertical asymptote is a vertical line that the graph approaches as the function approaches a certain x-value, typically where the function is undefined.

Front

How do you find vertical asymptotes?

Back

Set the denominator of a rational function equal to zero and solve for x.

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is one greater than the degree of the denominator.

Front

How do you find the slant asymptote of a rational function?

Back

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

Front

What is the horizontal asymptote of the function f(x) = (3x^2 + 2)/(6x^2 + 5)?

Back

y = 1/2 (since the degrees are equal, HA is the ratio of leading coefficients).

Front

What is the vertical asymptote of the function f(x) = 1/(x^2 - 4)?

Back

x = 2 and x = -2 (set the denominator x^2 - 4 = 0 and solve for x).

Front

What is the horizontal asymptote of the function f(x) = 5x/(2x + 1)?

Back

y = 2.5 (since the degree of the numerator is equal to the degree of the denominator, HA is the ratio of leading coefficients).