The following is a discussion and representation
of equations and their graphs which begin with the circle and the cubic
function. The exploration will involve what happens to the graphs of the
equations as the value of the exponents increase and given that x and y
always maintain the same value. The focus of your attention should involve
particular consideration of the equations with odd powers versus the equations
with even powers and their corresponding graphs.
Let's begin by looking at the "parent"
equation of the circle, . The graph
is shown below in purple and as stated makes a complete, closed circle.
The equation is the "parent"
for a cubic function and its graph is given below.
Compare the two graphs. Given the powers of 2 for
x and y, the graph is circular and completely closed. When 3 is the exponent
for x and y, we are given a curved graph, but it is not closed.
Let's look at the graph of below. How does
this compare with the graph of and ? The graph given below
is once again a closed figure, but does not seem to be totally circular.
Which equation is it most similar to?
The graph of is given below. How does
this compare to and ? This is not closed , yet it is not as rounded
on the curved part of the graph. Which equation is it most similar to?
Now let's look at . Is it what you expected?
What seems to be happening to the curved parts of the graph?
How about ? Look at the graph below. Is it
what you expected this time? Are you beginning to have ideas about what
is happening to the graphs as the exponents' value increases?
Make a conclusion about the graphs of the types
of equations we have looked at, given odd powers for x and y and given even
powers for x and y and as those values increase.
What would you expect for the graph of
and to look like?
Scroll down to see if your conclusions are correct.
In conclusion, given the parent equation for a
circle and a cubic function, as the values of the exponents increase the
following will result:
Given even powers for x and y, the graphs will be closed
and maintain a similar shape graph of a circle. However, only when the exponents
2 for x and y, do we have a circle. As the value
of the exponents increase, the circle becomes more rigid, reshaping into
more of a curved, square
Given odd powers of x and y, the graphs will be open and
as the value of the exponents increase, the curved part of the graph also
rigid and less curved.