Graphing Rational Functions with a Hole

Front

What is a rational function?

Back

Front

What is a hole in a graph of a rational function?

Back

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

Front

How do you find the x-value of a hole in a rational function?

Back

To find the x-value of a hole, factor both the numerator and denominator, and identify the common factors. The x-value of the hole is the value that makes the common factor zero.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity.

Front

How do you determine the horizontal asymptote of a rational function?

Back

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the function's value approaches infinity or negative infinity, typically occurring at values of x that make the denominator zero.

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. The values of x that make the denominator zero are the locations of the vertical asymptotes.

Front

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

Back

The degree of the numerator and denominator helps determine the behavior of the function, including the existence of horizontal and vertical asymptotes.

Front

What is the relationship between holes and vertical asymptotes?

Back

Holes and vertical asymptotes are related in that both occur at values that make the denominator zero. However, a hole indicates a removable discontinuity, while a vertical asymptote indicates a non-removable discontinuity.

Front

How do you graph a rational function with a hole?

Back

To graph a rational function with a hole, first identify the hole's location, plot the vertical and horizontal asymptotes, and then sketch the graph, leaving a gap at the hole.