Graphing Rational Functions

Questions and Answers

What are the x-intercepts?

(X, 0); they are on the x-axis where y is 0.

What are the y-intercepts?

(X, 0); they are on the x-axis where y is 0.

What are vertical asymptotes?

Where the function is undefined; happens where the denominator equals zero.

What is a horizontal asymptote?

Determined by the relationship between the highest degree of the numerator and denominator. Signup and view all the answers

What is Case 1 for horizontal asymptotes?

If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y=0. Signup and view all the answers

What is the first step to graphing a linear over quadratic function?

First find the x-intercept (numerator=0) and then find the y-intercept by plugging in 0 for x. Signup and view all the answers

If the x-intercept is (0,0), what is the y-intercept?

(0,0); it is also on both the x and y-axis. Signup and view all the answers

How can you find the y-intercept?

By looking for the constant/constant value in the function. Signup and view all the answers

What should be done when there is no horizontal asymptote?

The m over n value guides the end behavior and the graph should not cross the x-axis without a point. Signup and view all the answers

What happens when the vertical asymptote comes out to an imaginary number?

There is no vertical asymptote because you cannot graph imaginary numbers. Signup and view all the answers

What happens when the y-intercept comes out to #/0?

It is undefined because division by zero is not possible. Signup and view all the answers

What occurs when there is no y-intercept and no horizontal asymptote with an end behavior of y=x^2?

The graph follows the end behavior and hugs the asymptotes. Signup and view all the answers

What is an example of a graphing scenario where a quadratic crosses the horizontal asymptote?

Graphing a quadratic over cubic function that passes through intercepts and hugs the asymptotes. Signup and view all the answers

How do you determine where the graph goes during a sign evaluation?

Evaluate the point for x; if positive, the graph is positive at that x; if negative, the graph is negative. Signup and view all the answers

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.