Graphing Rational Functions

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

Front

What is the domain of a rational function?

Back

The domain of a rational function includes all real numbers except where the denominator is zero.

Front

How do you find vertical asymptotes of a rational function?

Back

Vertical asymptotes occur at the values of x that make the denominator equal to zero, provided that these values do not also make the numerator 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 positive or negative infinity.

Front

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

Back

To find the horizontal asymptote, 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.

Front

What is the significance of holes in the graph of a rational function?

Back

Holes occur in the graph of a rational function at x-values that make both the numerator and denominator zero, indicating that the function is undefined at those points.

Front

What is the process of simplifying a rational function?

Back

To simplify a rational function, factor both the numerator and denominator and cancel out any common factors.

Front

What is the relationship between the degree of the numerator and the degree of the denominator in determining asymptotes?

Back

The degree of the numerator and denominator helps determine the type of asymptotes: vertical asymptotes are found from the denominator, while horizontal asymptotes depend on their degrees.

Front

What does it mean for a function to be undefined?

Back

A function is undefined at points where the denominator equals zero, as division by zero is not possible.

Front

How do you find the x-intercepts of a rational function?

Back

The x-intercepts of a rational function occur where the numerator equals zero, provided the denominator is not also zero at those points.