Cristina Aurrecoechea Fall 2005 |

We explore the following set of equations:

The following set of graphs are plotted for
**n = 1,2,3,4,5,6** and **n = 25** (in yellow)

For **n = 1** the equation represents a
straight line **x + y = 1**, with negative slope.

For **n = 2** the equation represents a
circle centered at (0,0) and radius = 1.

Observations:

- All these lines, generated for
**positive integer values of n**, intersect at the points (1,0) and (0,1). - For
**n = 2,4,6,...**(even values) the line is closed. As**n**increases it seems to approximate a square 2x2 centered at (0,0). - For
**n = 1,3,5,7....**(odd values) the line is open. As**n**increases it seems to approximate x+y=0 (the square diagonal) for x > 1 and x < -1.

This Graphing Calculator 3.5 movie
will show you the evolution from straight line, to circumference,
to a pseudo-square while varying **n** **from 1 to 15**.
There is a problem with the movie though. What is wrong? As **n**
varies from 1 to 15 the line only exists in the first quadrant.
But we KNOW the lines for **n = 1 to 15** exist at least in
3 of the quadrants (see Figure 1 above). Why then is only the
first quadrant plotted?

The problem is that, as **n** varies from
1 to 15, Graphing Calculator calculates the line for intermediate
NON-integer values of **n**. (Observation: maybe the line only
exists in the first quadrant for non=integer values of **n**?
We explore that below). To solve the problem we modify the equation
by introducing the ceiling() function --so **n** only hits
integer values. This improved movie
shows clearly the sequence of plotted lines for different integer
values of **n**.

Another possible exploration is varying **m**
in . We look at how
each of the lines in this family changes, as we modify **m**
from 1 to larger values. We observe that as **n** increases
the variation introduced by higher values of **m** is hardly
noticeable: while for **n = 1**, **x+y=1** is very different
from **x+y=10000, **for **n = 25** there is hardly any difference
between **m = 1** or **m = 1000**. Figure 2 shows the equation
for **n = 25** and **m = 1, 100, 10000**.

(Since **m** does not affect the shape of
the curve for a given value of **n**, in the following discussion
we will default **m** to 1.)

Lets explore now
for **n = 1 to 25**.

Figure 3 shows he lines that correspond to
odd values of **n** (1,3,5 up to 25 in yellow). For **negative
m, n cannot be an even value** since that would make the terms
and
always positive and therefore their sum would never be negative.

Now lets explore negative values of **n. (**Let's
assume **m**=1). For **n= -1** the equation becomes:

which is a rectangular hyperbola (we let you
check that out). Explore for other negative values of **n**
in the movie we have prepared.

Next, lets look at **non-integer values of
n**. Figure 5 shows , for
**m = 4** and for **n = 1, 1.5, 2** and **25**. (By now
you should know which one corresponds to **n = 1**, **n =
2**, and **n = 25**.)

We come back to the question: Do non-integer
values of **n** make the line to exist only in the first quadrant?
The answer is: it depends on the rational value. For **n = 1.5**
we would have . If
there is a square root, both **x** and **y** have to be
positive for the line to exist in the real plane. On the other
hand, for **n = 1/3**,
is a real number for negative values of x (e.g., for x=-8, the
result is -2).

Figure 6 plots the equation for **n = 4/3**
and **m = 1**.

This movie
will show you the graph as the exponent in
varies **between 1/7 and 7/2** (you can easily change those
values). (Please, do not get confused by the fact that in the
GC file we use the name "n" for the numerator in the
exponent; "n" varies between 1 and 7 while the denominator
"a" varies between 2 and 7).

It is interesting to see how the line changes
from **concave** to **convex** when the exponent increases
from <1 to >1. Why is the curve closed and it exists in
the four quadrants for **n = 4/3** and **n = 2/3**, while
for **n = 1/3** it is open and it only exists in 3 quadrants?

Finally the following gcf file will allow you
to visualize all this in 3D and play
with **n** and **m** values. Figure 7 is a snapshot for
**n = 25** and **m = 1** and **m = 100000**; it is the
3D version of Figure 2 above.

Return to Cristina's page with all the assignments.