Graphing Rational Functions

Study Notes

Key Features of Graphing Rational Functions

X-Intercepts: Points where the graph intersects the x-axis; determined by setting the numerator equal to zero (resulting in points of the form (x, 0)).
Y-Intercepts: Points where the graph intersects the y-axis; found by evaluating the function when x=0 (resulting in points of the form (0, y)).

Asymptotes

Vertical Asymptotes: Occur where the denominator equals zero, indicating where the function is undefined; the graph cannot touch or cross these lines.
Horizontal Asymptotes: Determined by the degrees of the numerator and denominator; indicates the end behavior of the function and may or may not be crossed by the graph.

Cases for Horizontal Asymptotes

If the degree of the numerator (m) is less than the degree of the denominator (n), the horizontal asymptote is y=0.
If m > n, there is no horizontal asymptote; the end behavior aligns with y=ax^m/bx^n.

Steps to Graph Rational Functions

Find x-intercepts by setting the numerator equal to zero.
Determine y-intercepts by substituting x=0 into the function.
Identify vertical asymptotes by solving for when the denominator equals zero.
Establish horizontal asymptotes based on the degrees of the numerator and denominator.
Graph the x-intercepts and asymptotes.
Perform a sign check by testing points to the right of the asymptotes to determine graph behavior.
Use easy-to-calculate numbers near known points to verify graph continuity and direction.

Special Cases

When the x-intercept is (0, 0), it serves as both the x-intercept and y-intercept.
Y-intercept can also be found by evaluating the function at x=0, considering constant values.
In cases where no horizontal asymptote exists, the end behavior must still influence the graph, which will not cross the x-axis when no intercepts are present.
If vertical asymptotes yield imaginary numbers, they cannot be graphed.
A y-intercept represented as a constant divided by zero is undefined and cannot be plotted.

Graph Behavior without Y-Intercepts

In scenarios lacking both a y-intercept and horizontal asymptote, the graph will still align with its end behavior, following and hugging the identified asymptotes.

Example Scenarios

A quadratic over cubic function can demonstrate behavior where the graph crosses the horizontal asymptote while adhering to intercepts and asymptotic behavior.
When carrying out a sign evaluation, the positivity or negativity of substituted x-values dictates the direction in which the graph progresses.

Description

This quiz covers the key features of graphing rational functions, including x-intercepts, y-intercepts, and asymptotes. Understand how to identify vertical and horizontal asymptotes based on the degrees of the numerator and denominator. Test your ability to graph these functions accurately and evaluate their behavior.