Amberly Roberts

**Assignment 1: An Investigation of **

**where n = 1, 2, 3, 4, 5, ...**

By observing the graphs of the six relations, do you notice any interesting patterns?

I think that it's important to note that in all the relations with **even** exponents, the relation is satisfied by points in all four quadrants.

In all of the relations with **odd** exponents, the relation is satisfied in only quadrants 1, 2, and 4. Why?

In observing this pattern, I will reorganize the graphs. I will group the "even" ones together and the "odd" ones together.

What do you think will happen if we make the exponent 24? what about 100?

As the odd exponent increases, the curve seems to sharpen. The hill in the first graph becomes a steep mountain in the third graph.

What do you think will happen if we make the exponent 25? what about 101?

In both cases, the odd and even, the graphs seem to move from smooth to rigid.

Now, let's bring everything together and look at an animation of everything we've observed. We will animate

where 0 < n < 10

Do you notice anything strange? Why is the graph flashing the images we previously observed?

Well...here's the reason.

Our study has been limited to integer exponents. When expanding the exponents to include real numbers, we get several cases in which the relation appears only in the first quadrant. Why?

What happens when we try to find x and y values that satisfy the equation when 1.5 is the exponent?

Why do only pairs of positive x and y values work?