Understanding Asymptotes and Holes

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or slant.

Front

What is a hole in a graph?

Back

A hole occurs in a graph at a point where a function is not defined due to a factor that cancels out in the function's expression.

Front

How do you find the slant asymptote of a rational function?

Back

To find the slant asymptote, perform polynomial long division on the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.

Front

What is the horizontal asymptote of a rational function?

Back

The horizontal asymptote is determined by the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If they are equal, it is y=ratio of leading coefficients.

Front

How do you determine x-intercepts of a function?

Back

To find x-intercepts, set the numerator of the rational function equal to zero and solve for x.

Front

What is the domain of a rational function?

Back

The domain consists of all real numbers except where the denominator equals zero.

Front

What is the range of a rational function?

Back

The range includes all real numbers except for any horizontal asymptotes or holes in the graph.

Front

How do you find the coordinates of a hole in a rational function?

Back

To find the coordinates of a hole, identify the value of x that makes both the numerator and denominator zero, then substitute this x-value into the simplified function.

Front

What is the significance of vertical asymptotes?

Back

Vertical asymptotes indicate values of x where the function approaches infinity or negative infinity, typically where the denominator is zero.

Front

What is the difference between a hole and a vertical asymptote?

Back

A hole indicates a removable discontinuity where the function is not defined, while a vertical asymptote indicates a non-removable discontinuity where the function approaches infinity.