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.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find vertical asymptotes?

Back

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

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

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).

Tags

CCSS.HSF-IF.C.7D

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).

Tags

CCSS.HSF-IF.C.7D

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).

Tags

CCSS.HSF-IF.C.7D