Assignment #1 Problem #6
A discussion of the equation:
by: Kelli Nipper
What do you expect to find for the graph of:
In this exploration, I began looking at the behavior of the graphs as
"a" increases. It was obvious right away that there were two sets
of behaviors. One, when "a" was even; and the other when it was
odd. Using the algebra expresser program, it was easy to form two hypotheses.
Hypothesis 1: For even integer values of "a".
The graph began as a circle, and started to flatten as "a" increased.
This is seen in the change between the graphs of:
So you would expect the graph of "a"=24 to follow the pattern
by approaching a square as "a" increases:
Hypothesis 2: For odd integer values of "a".
The graph began as a curve that was tangent to the preceding even power
graph at two points:
I predicted that the curve would continue to be tangent to the shape
as "a" increased. However, I was surprised to find that the curve
began to approach the assymptote with a trace around the shape formed by
the preceding even power relation.
At this point, I concluded that as "a" increased (as and integer),
the relation would approach a square (for even values of "a")
and a straight line at y=-x with a trace around the preceding even power
graph between the values of -1 < x < 1 .
Graphing the equations of consecutive odd exponents on the same axis
leads to the observation that as "a" increases, the y value must
decrease for all odd values of "a".
To continue this investigation, I wanted
1) Are the behaviors consistent for values of "a" between 5 and
2) How does the function perform for values of "a" > 25
3) How does the function perform for values of "a" that are not
Using algebra Xpresser makes these easy to answer.
1) The behaviors were consistent for values of "a"
between 5 and 24.
The graphs of "a"=14 and 15 supports this conclusion:
2) When "a">25, it performs as I concluded above. The
following is a graph of "a" = 50 and 51.
3) For non-whole number values, the graphs begin to enter the circle
created by the relation with
"a" = 2. Other patterns begin to form. It is important to mention
- for "a"= 0 there are no points that satisfy. This makes
sense because when any number is raised to the zero power, it is one. No
points will satisfy the values of 1+1=0.
- for positive and negative (non-whole number) values of "a"
the pattern begins a whole different discussion.
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