## Explorations with Graphs

### Sue Meredith

EMT 668
Assignment 1

1. Using the formula:

x(x2 -4) = y(y2-1)

Changing only the constant 4, the graph of the equation reverses after 1. As the values of this constan t decrease, the size of the loops gets smaller. When the 4 is replaced by 1 the graph becomes a line with ellipse. If the value continues to decrease, the graph then reverses.

In order to see the changes more clearly, graph only values of 4, 2 and 1 on a grid where the section of the graph near the origin is magnified.

The following graph starts with the original function , the 4 is replaced by 3, which gives the shadow. To get the flatter line across the bottom, 1 is subtracted from the x side of the function. The fourth line occured when the function is changed to :

x(x^2 -3/2) -1 = y(y^2 -1)

Another investigation involves the graphing of the following functions:

It becomes clear that the even powers of x and y will cause a closed figure and the odd powers of x and y will continue infinitely. The following graph illustrates the previous functions.

To extend this investigation, the functions were advanced to the 24th and 25th powers of x and y. The resulting graph fulfills the prediction.

x^24 + y^24 = 1
and
x^25 + y^25 = 1

Graphs of trigonometric functions produce very interesting curves. For the following let

f(x) = a sin (bx + c) and g(x) = a cos (bx + c)

where the values of a, b, and c are changed in a methodical pattern. In the following graph; a, b, and c are all 2.

When a new function is formed by the sum of the two trigonometric functions, the following function is graphed: