The images you see to the right were all created with IFS Function.* Click here or click the logo above for a test drive! IFS Function is free
software. Download it at Sourceforge! * The x and y values shown give the coordinates of the yellow highlighted point on the graph. 
mfunction.uifs 

IFS Linear is a program for creating affine fractal interpolation functions. Though you don't need to know it in order to use the program, the functions it produces are actually iterated function system (IFS) fractals. That's where it gets the "IFS" part of its name.
When the applet starts up, you should see a gridded design area. That grid is where you will design your function. This diagram shows the various parts of the design area.

Clicking the "Restart" button returns the default starting parabola to the design area.
Clicking "Snap to Grid" makes mouse dragging motions "snap" from one location to another. With "snap" on, the tools can only be placed in limited locations. This is useful, for example, if you want to place data points exactly on grid crossings or halfway points. Notice that when you click the button, it turns into a "Freehand" button. Clicking the "Freehand" button removes "snapping."
When you want a closer look at what you see in the preview area, you might want to click on the "Render Fractal" button. This will bring up a dialog box that allows you to create a larger, more colorful version of your interpolating function. This is more useful for some of the other IFS Tools, but you may find it interesting to try here as well.
When IFS Function starts up, the interpolating function you see is actually a parabola. Not just a good imitation, but a legitimate parabola. It is the parabola satisfying the equation
If you click and drag any of the data points vertically, the interpolating function remains a parabola. Different parabolas for different configurations of data points, of course, but parabolas nonetheless.
Each segment (interval between two consecutive data points) of the interpolating function is an exact replica, scaled and sheared, of the entire interpolating function. For a segment with a plus sign, the copy has the same lefttoright orientation as the whole funtion. For a segment with a minus sign, the copy has the opposite (righttoleft) orientation.
The .uifs files hold the coordinates of the data points, whether each segment should be replicated forward or backward, and the scaling factor for each segment. In other words, they store the mathematical description of the interpolating function.