Today we will be exploring the graph of x +y= 1 as x and y are increasing in exponents and we will be able to see the changes in the graphs and see if we can find a pattern.

Our first graph is:

Lets look at

The Next graph will give us the following:

So it looks like the even numbers start out as a circle and the odd numbers almost form half a square,but the graph of should looks more like a square than a circle. So up to this point we see that both are getting more square around the edges as we increase our exponents.

this should go around the origin forming half a square. Is it getting closer to two sides of a square, but near the bottom of the graph, it is still curving as it tails off.

So lets look at the next. As we are getting larger even numbers it looks like we are getting closer to a square. Lets see what this gives us.

So lets now look at what it would be like to the 25th. We would think that the edges will be less rounded than before on the turns. Lets see what gives us:

This gives us closer to two sides of a square with the line starting maybe near x = -y then taking a turn across the y-axis then almost making a ninety degree turn at (1,1), then crossing the x-axis and almost making another ninety degree turn downward along the x = -y line. So we get almost half a square with this graph.

So here is a combination of all the graphs as x and y increase and one can be able to see that as the odd powers go higher we draw closer to the line x = -y. In doing so, we get approximately half of the square formed by the even exponents. Here is a picture of all the above graphs in one:

is the red circle.

is the green line that goes around the outside of the square.

is the blue line that is in between the circle and the square.

is the sky blue line inbetween the green and purple lines near the x = -y line.

is the black box that almost forms a square with round barely on the corners.

is the purple line that looks like it traces the black square.

The following graph below we can see that with the x = -y green line drawn in each increasing odd exponent it touches the line for a longer time.

So as we can see the odd exponents draw closer to forming half of a squre while starting out on the x = -y line and then making turns sharper turns as it increases. The even exponents go from forming a circle to forming a squre.