Horizontal and Vertical Asymptotes

Front

What is a horizontal asymptote?

Back

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

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the function's value approaches positive or negative infinity. It typically occurs at values of x that make the function undefined.

Front

How do you find horizontal asymptotes for rational functions?

Back

To find horizontal asymptotes of a rational function, compare the degrees of the numerator and denominator: If the degree of the numerator is less than the degree of the denominator, y=0 is the horizontal asymptote. If they are equal, divide the leading coefficients.

Front

How do you find vertical asymptotes for rational functions?

Back

Vertical asymptotes occur at values of x that make the denominator zero (and the numerator non-zero). Solve the equation of the denominator for x.

Front

What is the equation of the horizontal asymptote for the function f(x) = 3x^2 + 2 / 5x^2 + 1?

Back

The horizontal asymptote is y = 3/5, since the degrees of the numerator and denominator are equal.

Front

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

Back

The vertical asymptote is x = 2, where the denominator equals zero.

Front

Which of the following is NOT a vertical asymptote: x = 3, x = -1, y = 2?

Back

y = 2 is NOT a vertical asymptote; it is a horizontal line.

Front

What happens to the graph of a function as it approaches a horizontal asymptote?

Back

As the graph approaches a horizontal asymptote, the function values get closer to the y-value of the asymptote but do not necessarily touch or cross it.

Front

What is the significance of asymptotes in graphing functions?

Back

Asymptotes help identify the end behavior of functions and indicate where the function may be undefined or where it approaches certain values.

Front

Find the vertical asymptotes of the function f(x) = 1 / (x^2 - 4).

Back

The vertical asymptotes are x = 2 and x = -2, where the denominator equals zero.