Rational Functions: Holes and Vertical Asymptotes

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not zero.

Front

What is a hole in a rational function?

Back

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

Front

How do you find holes in a rational function?

Back

To find holes, factor both the numerator and denominator, and identify common factors. The x-values that make these common factors zero are the locations of the holes.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a rational function approaches infinity or negative infinity as x approaches a.

Front

How do you find vertical asymptotes in a rational function?

Back

To find vertical asymptotes, set the denominator equal to zero and solve for x, excluding any x-values that correspond to holes.

Front

What is the significance of vertical asymptotes in graphing rational functions?

Back

Vertical asymptotes indicate where the function is undefined and help determine the behavior of the graph near those x-values.

Front

What happens to the function at a hole?

Back

At a hole, the function is not defined, but the limit of the function exists as x approaches the hole's x-value.

Front

Can a rational function have both holes and vertical asymptotes?

Back

Yes, a rational function can have both holes and vertical asymptotes, as they arise from different factors in the numerator and denominator.

Front

What is the relationship between the degree of the numerator and denominator in rational functions?

Back

The degree of the numerator and denominator determines the end behavior of the function and the presence of horizontal asymptotes.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b that the function approaches as x approaches infinity or negative infinity.