Vertical Asymptotes, Horizontal Asymptotes, & Holes

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as x approaches a. It occurs when the denominator of a rational function equals zero and the numerator does not equal zero at that point.

Front

What is a horizontal asymptote?

Back

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

Front

How do you find vertical asymptotes in a rational function?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x. The values of x that make the denominator zero are the vertical asymptotes.

Front

How do you find horizontal asymptotes in a rational function?

Back

To find horizontal asymptotes, compare the degrees of the numerator and denominator: 1) If the degree of the numerator is less than the degree of the denominator, y = 0 is the horizontal asymptote. 2) If they are equal, divide the leading coefficients. 3) If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is a hole in a function?

Back

A hole occurs in a function at a point where both the numerator and denominator equal zero. It indicates that the function is not defined at that point, but the limit exists.

Front

Given the function f(x) = (x^2 - 1)/(x - 1), identify the hole.

Back

The hole is at x = 1, because both the numerator and denominator equal zero at that point.

Front

What is the significance of identifying asymptotes in graphing functions?

Back

Identifying asymptotes helps in understanding the behavior of the function as it approaches certain values, guiding the sketching of the graph and indicating limits.

Front

Find 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 and the leading coefficients are 3 and 2.

Front

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

Back

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

Front

How do you determine if a function has a hole?

Back

A function has a hole if there is a common factor in the numerator and denominator that can be canceled out, resulting in a point where the function is not defined.