Transformations of Reciprocal (Rational) Functions

This applet investigates transformations of reciprocal (rational) functions, learning the meaning of the word asymptote and its importance in describing the graph.

Author and programmer: Ron Barrow

UK Years 12-13, KS5, Core Mathematics, AQA OCR Edexcel MEI C3 C4
US - Grades 11 - 12


Instructions below    See also:   Modulus Functions   PRINT

[Applet failed to run. No Java plug-in was found.]

How to Use this Applet

The applet on this page is designed to help you understand and visualize the behaviour of reciprocal functions and their graphs.
The general reciprocal function has the form: A/(Bx + C) + D.
Varying A, B, C and D changes the curve. By investigating the effects of varying these different parameters, you should develop a feel for the behaviour of these very important functions.
Sometimes you will see grey lines parallel to the x- and y- axes. These are asymptotes. The function gets closer and closer to these lines, but never crosses them.
If you tick the box marked 'Original Curve' then you will see a dull image of the basic curve y = 1/x to help you compare.
Two forms of the equation are given. The equation on the left is in the form given above. On the right, the expression has been rewritten as a single fraction, with coefficients rounded to 1 d.p.

The curves generated by this function are called rectangular hyperbolas. The rest of the instructions are easy to guess at, so I'll let you get on with it!
Challenge - To test your understanding, hitting the 'Random' button will generate a random curve. By varying the parameters, try to cover the random line with the other one, and hence find the equation of the random curve. There is usually more than one correct answer.
If you want an easier start, then clicking on '1' at the top of the page will set the program so that only one of A, B, C, D changes on the random curve - the others stay the same. It get more complicated as you click '2', '3' and '4'! Have a go.
If the 'Easier' box is ticked, then the value of B in the random curve is never negative, which makes things a little easier. Clear the tick if you want to allow negative values of B in the random curve - a trickier challenge.
Have some fun! Studying this function should also help understand functions and their transformations in general.