Graphing Rational Functions

Front

What is a horizontal asymptote?

Back

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

Front

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

Back

To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the asymptote is y=0. If they are equal, the asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a certain value, typically where the function is undefined.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes can be found by setting the denominator of the rational function equal to zero and solving for x.

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 point where both the numerator and denominator are zero, indicating that the function is undefined at that point.

Front

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

Back

To find the coordinates of a hole, factor the numerator and denominator, cancel the common factors, and then substitute the x-value of the canceled factor into the simplified function.

Front

What is the significance of the leading coefficient in determining horizontal asymptotes?

Back

The leading coefficient of the numerator and denominator helps determine the horizontal asymptote when the degrees of both are equal.

Front

If a rational function has a degree of 2 in the numerator and 3 in the denominator, what is the horizontal asymptote?

Back

The horizontal asymptote is y=0 because the degree of the numerator is less than the degree of the denominator.

Front

What happens to the graph of a rational function near a vertical asymptote?

Back

As the graph approaches a vertical asymptote, the function values increase or decrease without bound, leading to a steep rise or fall.

Front

Can a rational function have more than one vertical asymptote?

Back

Yes, a rational function can have multiple vertical asymptotes, depending on the number of distinct factors in the denominator.