Graphing Rational Functions

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 are the key features to identify when graphing a rational function?

Back

Key features include intercepts, asymptotes (vertical and horizontal), and the behavior of the function as it approaches these asymptotes.

Front

How do you find the vertical asymptotes of a rational function?

Back

Vertical asymptotes occur where the denominator of the rational function is equal to zero, provided the numerator is not also zero at those points.

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

The horizontal asymptote can be determined by comparing the degrees of the numerator and denominator: if the degree of the numerator is less, the asymptote is y=0; if they are equal, y is the ratio of leading coefficients; if the numerator's degree is greater, there is no horizontal asymptote.

Front

What is the significance of intercepts in graphing rational functions?

Back

Intercepts (x-intercepts and y-intercepts) indicate where the graph crosses the axes, providing key points for sketching the graph.

Front

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

Back

X-intercepts are found by setting the numerator of the rational function equal to zero and solving for x.

Front

How do you find the y-intercept of a rational function?

Back

The y-intercept is found by evaluating the function at x=0, provided the function is defined at that point.

Front

What is the behavior of a rational function near its vertical asymptotes?

Back

As the function approaches a vertical asymptote, the function values will tend to positive or negative infinity, depending on the direction of approach.

Front

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

Back

A hole occurs at a point where both the numerator and denominator are zero, indicating that the function is not defined at that point, but the graph approaches it.