We will look at the relationship of x and y in the equations that follow the pattern of
Here’s the first case where n = 1. So the equation is x + y = 1.
Here is the simplest case. N is 1 for the equation and we can see that the line is straight. It crosses the x-axis at 1 and the y-axis at 1 also. We can also see a perfect negative correlation between the two variables. As one goes up the other goes down the same value.
Now let’s look at the next value of n. .
As the graph shows we have produced a very famous circle. The Unit Circle has many significant qualities in geometry and trigonometry. But for our explorations we can see that the circle has a radius of 1 and the center is the origin. We note here that when n was 1 it was a straight line that never connected itself. Here, with n = 2 we have a connection in a perfect circle. Let’s pursue the next value of n.
Here is .
The graph of n = 3 has an interesting quality. We see some sort of combination of the two previous graphs. We have a straight line, but we also have an anomaly between y = 1,-1. We still have a negative correlation with x and y everywhere else outside of y = 1, -1. But inside the section we have a partial square with a rounded edge. So to summarize when n equaled 1 and 3 we had some sort of line that had no end, and when n = 2 we had a line that connected the beginning to the end in a circular shape.
Next we see the graph when n is 4.
This graph when n = 4 gives rise to a very interesting observation. So far when n is odd we have seen a line of some kind extending to infinity in both directions. And when n is even, such as this, we have a symmetric shape on both the x and the y axis, and the end of the line connects with the beginning. We’ll continue to increase n.
Here’s the graph of .
Once again we see that n is odd and it produced a curvy line that extends forever on each side of itself. We also note that the bend in the line seems to becoming more and more like a partial square. Our instinct tells us that if we continue to increase n then the bend in the curve will become more ‘sharp’ at the bend when n is odd. And the shape created when n is even will actually become a square the higher n becomes. Let’s just try a few more examples before we conclude.
Just to show similarities I graphed on the same graph as in blue and red respectively. We can see that the graphs follow the same pattern of a square in the first quadrant. Let’s do one more with a very large number for n and see how close we can get to a perfect square. We’ll try .
This looks pretty close to a square to me. The edges are almost parallel to each other and the corners seem to be perfect right angles, but we know that they are not from our earlier graphs. The only thing we can say it that this is much more like a square than any other graph we have produced so far.
Write Up by David Drew