## Explorations with Graphs

#### by

### 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:

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