**Explorations of Functions and Relations**

**By Mary
Negley**

For this exploration, I will look at the equation for various *a*
values and use those graphs to predict higher *a* values.

First letŐs look at .

Notice
how it is a circle of radius 1 centered at the origin.

Next letŐs look at in relation to
the previous graph.

Notice
how the function is tangent to the previous one at and, the function decreases in the second quadrant and fourth
quadrant, and it curves around the previous graph in the first quadrant.

Now letŐs add to our graph.

In
this case, the new function is centered at the origin just like the first
function and it is tangent to the first function at,. Instead of
being a circle like the first equation, the corners are slightly rounded. Notice how for each, the y-value for the third equation is greater than the y-value
for the second equation.

Finally letŐs add to our graph.

This
new function is similar to the second function. Like the second function, it decreases in the second and
fourth quadrants and it curves around the first function in the first
quadrant. Moreover for each, the y-value for the third equation is greater than the
y-values for the other equations.

Now I must predict what the graphs of and look like.

Above I noticed that the graph of is tangent to
the graph of at four points,
but it resembles a square with rounded corners. Based on the graphs of and, I would predict that would resemble a
square even more than , but it would still be tangent to at the same four
points as . Below is the
graph of .

Above I noticed that and were tangent to at two points.
They both also decreased in the second and fourth quadrants and curved
around in the first quadrant.
Notice that when and , the y-value for is less than the y-value for ,
but when , the y-value for is greater than
the y-value for. I would predict that when and , the y-value for would be less than the y-value for, but when , the y-value for would be greater
than the y-value for. Now letŐs add to our
graphs.

In both cases, my predictions were true.