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 :
Another investigation involves the graphing of the following 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.
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: