Rational numbers, also known as fractions, aren’t a whole lot of fun, but they are an important part of arithmetic.
Rational functions play in important role in algebra and higher mathematics.
This tool will help you understand some important features of rational functions.
One of the most obvious features of rational functions are the vertical asymptotes. A rational function may have several of these, and any one of them may occur multiple times. The number of times a single vertical asymptote occurs is called its multiplicity.
The multiplicity of a vertical asymptote is not apparent from the graph of a rational function, but can be seen as an exponent on a factor of the function algebraic representation. This function would have a vertical asymptote of multiplicity 7.
The locations of the vertical asymptotes may be entered by typing a numerical value into the field, or by using the slider. The multiplicity can take any integer value from 0 to 9. When the value is 0, the entry is ignored, so you can leave the x-value in place but not have it included in calculations while you explore.
Rational functions always have a non-vertical asymptote as well. It may be horizontal …
… or it may be oblique, a linear asymptote that is neither vertical nor horizontal.
The non-vertical asymptote may be specified by values for the slope and y-intercept of the line and selecting the appropriate checkboxes. When both boxes are unchecked, any values in the boxes are ignored and the non-vertical asymptote corresponds to the x-axis.
Checking only the y-intercept box will allow you to enter a horizontal asymptote different from the x-axis.
Checking the slope box will produce an asymptote that forms an angle with the x-axis.
You can also specify where your rational function crosses or touches its non-vertical asymptote by typing in the x-values or by using the slider. The crossing points correspond to x-intercepts when the x-axis is the horizontal asymptote.
The sum of the multiplicities for crossing points must always be less than the sum of the multiplicities for vertical asymptotes. For this reason, the first piece of information you enter must be a vertical asymptote.
If you want to see specific points on the graph of your rational function plotted, or if there are holes in the graph that need to be seen, type in their x-values, or use the slider. The y-values will be computed based on the other features of your function.
Given any combination of vertical asymptotes and crossing points, there are an infinite number of rational functions, but you may optionally specify the x- and y-values of a point known to lie on the graph of your rational function to determine a unique function, as long as the x-value hasn’t been used previously.
Be careful about using x-values for more than a single purpose. No x-value can be the location of a vertical asymptote and a point at the same time, nor can an x-value be the location of both a point and a crossing point.
As you enter information, the algebraic form of the function is built up and displayed in the area at bottom of the interface. The graph of the function is also displayed in the graphing area.
Experiment with the different features of the graph, and try as many combinations as you can. Above all, look for connections and always think mathematically!