Assignment 1

By: Olamide Alli

Exploration: Graph the following equations:

What do you expect for the graph of

or

The following images are graphs that I have used for the exploration in order to produce a hypothesis on what the graphs for the equations and will produce.

Expectations for the graph of

Based on the graphs of  and , the graph of  should resemble the shape of a quadrilateral, specifically a square. When using even exponents for the variables x and y, as the exponents increase, the constructed graph will increasingly resemble a square. I’ve noticed that there is still slight curvature at the “edges” of the square. I’m willing to make the assumption that regardless of the value of the exponent, the equation will never produce a perfect square. Using the equation , where a = b, and a and b are even exponents. The only way to construct a perfect square or a construction with 90° angles is to implement a piecewise function.
After constructing the  graph, I’ve concluded that my original assumption is correct. I have also graphed the equation  and  in order to further explore my assumption about the appearance of the graph as well to determine whether or not the graph of the quadrilateral could have perfect 90° angles.  The graph may appear to be perfect but if you decide to zoom in on the pictorial view of the graph, you will notice that there is still slight curvature at the four “edges”. This subsequently proves the initial assumption that regardless of the value of the exponent, the equation will never produce the graph of a perfect square.

Expectations for the graph of

Based on the graphs of  and , the graph of  will have precise angles and there will be a change in the direction of the slope at the points (1,-1) and (-1, 1). There will also be a horizontal line at y=1 from x=-1 to x=1. My assumption is deduced from the pattern found from the equation , when a is an even number. As the value of the exponent a increased, the preciseness of the graph improved as well. This means that the resemblance to constructed edges on the points (-1, 1) and (1,-1) were similar to precise angles with no curvature.

After constructing the graphs of , I’ve concluded that my original assumption is correct. I have also graphed the equation for  and  in order to further the exploration of my original assumption about the appearance of the graph having a distinct change in the direction of the slope at points (1,-1) and (-1, 1). The graph may appear to have a distinct change in the slope but there is still slight curvature around the change in slope at the points (1, -1) and (-1, 1). This subsequently proves shows that there can’t be an exact change in the slope. There equation must employ a piece wise function as well as parameters for x.