Asymptotes

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line x = a where a function approaches infinity or negative infinity as the input approaches a. It indicates values that the function cannot take.

Front

How do you find vertical asymptotes of a rational function?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line y = b that a function approaches as x approaches infinity or negative infinity. It indicates the end behavior of the function.

Front

How do you determine horizontal asymptotes for rational functions?

Back

1. If the degree of the numerator is less than the degree of the denominator, y = 0 is the horizontal asymptote. 2. If the degrees are equal, y = leading coefficient of numerator / leading coefficient of denominator. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What does it mean if a function has no horizontal asymptote?

Back

If a function has no horizontal asymptote, it means that as x approaches infinity or negative infinity, the function does not approach a specific value.

Front

What is the significance of asymptotes in graphing functions?

Back

Asymptotes help identify the behavior of a function near certain values and guide the sketching of the graph, indicating where the function will not cross or touch.

Front

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

Back

The vertical asymptotes are found by setting the denominator equal to zero: x^2 - 4 = 0, which gives x = 2 and x = -2.

Front

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

Back

The horizontal asymptote is y = 3/2, since the degrees of the numerator and denominator are equal.

Front

What is the relationship between vertical asymptotes and the domain of a function?

Back

Vertical asymptotes indicate values that are excluded from the domain of the function, as the function approaches infinity at these points.

Front

Can a function have both vertical and horizontal asymptotes?

Back

Yes, a function can have both vertical and horizontal asymptotes, indicating different behaviors at different points.