This graph is a circle with radius 1 and center (0,0).
This graph starts as a line with a negative slope and looks as if it will hit the origin, but then at y=1 the line loops up as if it is trying to go around a circle. It comes back down at x=1 and turns back into a line. It is as if the circle opened up where it should have connected in the 3rd quadrant and extended to infinity in the 2nd and 4th quadrant.
This graph is similar to the first graph, but instead of a circle, 4 corners are beginning to assemble.
This graph looks more like a square than a circle but the edges of the "square" are rounded out
This graph is similar to the second graph but the loop that emerges at y=1 is not quite as rounded. This loop is more pointed but still pretty rounded off. It looks as if it is trying to go around a square.MYPREDICTIONS:
My predictions held to be true, except the corners of the "square" in are not as pointed and exact as I had thought they would be. So, I examined the graph of x^100 +y^100 = 1 and was satisfied that the corners of this square are more exact points.
x^even number + y^even number = 1 is a graph that goes from a circle to a square as the exponent increases in size. The domain is -1 to 1.
x^odd number + y^odd number = 1 is a grpah of a line with negative slope that loops out and becomes more pointed towards (1,1) as the exponent increases in size. The domain is all values of x.
I have always known the equation for a circle. ; where the squareroot of r is the radius, and (h,k) is the center. However, I did not know that if you increase the exponents of this equation you begin to turn it into a square. When I thought of an equation of a square I think about four seperate linear equations being pieced togethor at their intersections. This assignment has created a connection between circles and squares for me.
What will happen if I change not only the exponents but the center?
It is a square- like shape with center at (5,2) .
Also, what will happen if I change the r^2.?
If I used the equation but then replace the 1 with 25, I would assume that like the equation of a circle this would make the square have a "radius" of 5, or side length of 10.
However, this does not occur. The intercepts are not exactly 1 and -1 anymore. They expanded just a little bit, but not what I was expecting. I was expecting the intercepts to be at 5 and -5. This equation then, does not behave just like the equation of a circle.
Another thing to examine is graphs where the exponent is a negative integer.
And look at
Now look at 2 positve and negative exponent graphs togethor: