Graphical Maths Project

The first task to be carried out was to find several perfect tangents of the graph y=x2. This presented a challenge, as there is no easy way to discover lines that are perfect tangents of a curve. In my first attempt, I created a line in autograph and gradually changed the y intercept parameters in an attempt to “move” the line into position as a tangent of the graph y=x2. This proved time consuming and ineffective however, as the following screenshots show;

Here the blue line appears to cross the red line at y=0.25, x=0.45, but this is unclear because of the distance from the point.

Further zooming reinforces the possibility that the line could be an exact tangent.

Further zooming still makes the possibility of a potential discovery significantly less likely; however, when even more zooming takes place, the red and blue lines appear to separate. Depending on the exact y intercept parameters, the lines either do not touch at all, or continue to touch at more than 1 point despite how much zooming is done.

My second attempt at trying to find perfect tangents of the graph y=x2 involved using a somewhat more technical and advanced method. In autograph, there is a feature that can be applied to any 2D line called “evolute” – this brings up a menu on which if one chooses “tangents” and changes the “x-step” value to 1, a set of lines is created, all of which are tangents of the originally selected line.

Because the set of tangents are part of one entity in autograph, one does not have the option to see the gradients of the lines by any simple means, instead, one must simply “count squares” in order to find the necessary information.

Some Tangents found using this method include:

This shows a clear pattern; for every increase of 1 to the x value at which a tangent crosses y=x2, the gradient is increased by 2. In addition, the y-intercept value of each graph is equal to the “crosses at x” value, squared. The relationship between the “crosses at x” and “crosses at y” values is of course natural, and explained logically by the name of the graph.