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Lauren K Williams
Associate Professor, Mathematics
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Graphing Rational Functions




What is the \(y\) intercept of this function? Enter an ordered pair \((x,y)\) or type 'none' if the function has no \(y\)-intercept.
    Submit Hint

Finding the y-intercept:
Find the y-intercept by evaluating the function at 0.

If the function is not defined at x=0, it has no y-intercept.
For what values of \(A\) does the function have a horizontal asymptote \(y = A\)? Enter a single number, a list of numbers separated by a comma, or 'none' if the function has no horizontal asymptotes.
    Submit Hint

Finding horizontal asymptotes:
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is 0.

If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
For what values of \(A\) does the function have a vertical asymptote \(x = A\)? Enter a single number, a list of numbers separated by a comma, or 'none' if the function has no horizontal asymptotes.
    Submit Hint

Finding vertical asymptotes:
After completely simplifying the function, find all values of x (if any) where the denominator is equal to 0.
Click the graph to add a few additional points of the function. Your points will be accepted if they are reasonably close to an actual point on the graph. Once you find at least two points on either side of a vertical asymptote, the function will be plotted for you. Hint

Finding points on the graph:
Choose at least two values of x on each side of the vertical asymptotes and evaluate the function at that value to find y. Then, click on the graph near the point (x,y).

About Rational Functions

A rational function is one of the form \(\frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials. A sketch of these functions can be obtained by finding certain properties, such as intercepts and asymptotes.

Using the Applet

This applet can be used to practice sketching rational functions, without tools used in calculus (such as the first and second derivative tests). By following the prompts, you'll collect sufficient information about the function to draw an approximation of its graph.

Enter the required information in the prompts on the right. Note that the \(y\)-intercept must be entered as a point, such as (0, 2) or (0,2). The parentheses and comma should be included, and the order of the numbers in the pair will also be tested. As each correct answer is given, you'll be asked to find the next piece of information about the function. In the last step, you'll click on the plot itself near a point on the graph (some flexibility is provided here - as long as your point is within 0.5 units of the actual graph, your point will be accepted). When enough points are gathered, the actual graph of the function will be drawn.

About this Applet

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.

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