Stephani Eckelkamp

Looking at Exponential Value Differences

The following graphs were produced using Graphing Calculator 3.5.

Question:

Using the above information what do you expect the graphs of x24 + y24 = 1 and x25 + y25 = 1 will produce?

First of all let us look at the graphs above.  The graph of x + y = 1 produces the unit circle (circle with center at (0,0) and radius of 1).  If we look at the next graph with even exponents we see a graph similar to the unit circle but more square (or perhaps a squared shape with rounded edges).

The grey line graphed above is the line y = -x.  As we can see these graphs are asymptotic to the line y = -x.

From the graphs of the equations with even exponents we can deduce that the graph of x24 + y24 = 1 will have a more squared off curve that approaches the point (1,1) and is still asymptotic to the line y = -x.

The dark grey line, x + y = 1, creates a line with an x intercept of 1 and a y intercept of 1.  We can also write this line as y = -x +1.  This line shows were all of the above graphs cross the x and y axis.  As well as all of these equations passing through the same two points, this is the most basic form of all of the variations of the above listed equations.  All we have done to alter them is to remove the exponent, in which we see the line x + y = 1 produced.

LetÕs investigate furtherÉ

Suppose we look at the same graphs as above, but with negative x and y values.

All four of these graphs cross the x and y axis at -1.

-x3 – y3 = 1

-x5 – y5 = 1

-x – y = 1

Now let us look at the graphs of the equations with odd exponents, both positive and negative, on the same graph.

As we can see from this graph, all of these equations are asymptotic to the line y = -x.  The more that we increase the exponent (and keep it odd) the closer our graph will come to the point (1,1) in quadrant I, and (-1, -1) in the quadrant III.

If we change one variable to a negative the equations produce this graph.

Now all of the odd exponential equations on the same graph.

Suppose we plug in a variable for the exponents.  Click on this link to see an  animated picture.