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 is determined by the values that make the denominator of a rational function equal to zero.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find vertical asymptotes in a rational function?

Back

Set the denominator equal to zero and solve for x.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find horizontal asymptotes for rational functions?

Back

Compare the degrees of the numerator and denominator: If the degree of the numerator is less than the degree of the denominator, y = 0; if they are equal, y = leading coefficient of numerator / leading coefficient of denominator; if the numerator's degree is greater, there is no horizontal asymptote.

Tags

CCSS.HSF-IF.C.7D

Front

What is a hole in a function?

Back

A hole occurs in a function at a point where both the numerator and denominator are zero, indicating that the function is undefined at that point but can be simplified.

Tags

CCSS.HSF-IF.C.7D

Front

How do you identify a hole in a rational function?

Back

Factor both the numerator and denominator, and find common factors. The x-value that makes these common factors zero indicates the location of the hole.

Tags

CCSS.HSF-IF.C.7D

Front

What is the significance of the degree of a polynomial in finding asymptotes?

Back

The degree of the polynomial helps determine the behavior of the function at infinity, which is crucial for identifying horizontal asymptotes.

Tags

CCSS.HSF-IF.C.7D

Front

If a function has a vertical asymptote at x = 3, what does this imply?

Back

As x approaches 3, the function's value approaches infinity or negative infinity, indicating that the function is undefined at x = 3.

Tags

CCSS.HSF-IF.C.7D

Front

Back

The function approaches infinity, indicating a vertical asymptote at x = 1.

Tags

CCSS.HSF-IF.C.7D

Front

Back

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

Tags

CCSS.HSF-IF.C.7D