From Jim
Wilson’s website: Examine:

.

**See the graph.**What happens if
the 4 is
replaced by other numbers (not necessarily integers)? Try 5, 3, 2, 1,
1.1, 0.9,
-3. Any unusual event?
Interpret.

What
equation would give the following graph:

What
happens if a constant is added to one side of the equation? Try several
graphs
in some systematic way. Click **here** for one set of graphs.

Try
graphing

By exploring the graph of for various values of *n*, we
discover that there are three
classes of graphs: n<0,
0<
*n* < 1, *n*=1,
and *n*>1 .

For each of these classes, the *y*-intercepts will remain the same. The
y-intercepts occur when x = 0, so the
y-intercepts are solutions to . By factoring, we have . Every graph of the
form will have
will have
y-intercepts 0, 1, and -1.

When n > 1, we have a graph such
as
below. (Here with n = 4)

The most apparent characteristics of
this
graph are it’s intercepts.
Note that x-intercepts occur when y = 0, so
the x-intercepts are solutions to . By recognizing the
difference of two squares, we can solve this equation easily by
transforming it
to When
*n* > 1, we
have three x-intercepts:
.

When n = 1, we have:

This graph appears to be the union of
an
ellipse and the line y=x. Algebraically,
we have: . So
. By distributing and
combining like terms, we have
. By
factoring, we arrive at . The zero product
property states that if the product of two numbers is zero, then one of
the two
numbers must be zero. So we have or . In the first case, means and we the line y = x
appears graphically as expected. In the
second case, we have ,
which is an ellipse and confirms the second dominant shape
in the graphical representation.

When n < 1, we have a graph such
as
below. (Here with n = -4)

There is only one x-intercept in the
case
when n < 1. In factored form we have . For integers less
than 1, this means *n* is negative, or *n*
is zero. If *n*
is zero, the only solution to is x = 0. If *n*
is negative, one solution remains zero, while the remaining two
solutions are
complex.

When *n*
is between zero and 1, a case similar to when *n* > 1
occurs. This is the
result of three real solutions to the equation . The graph below
shows when *n* = .5

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