Problem 1 is an exploration of the function below for different values of n.

In the first set of graphs n is replaced by 5,3,2,1,1.1,0.9,and-3.This is the resulting graph.

In first three graphs
**n=5,n=3, and n=2**, the graphs have the same shape. The shape
is a curved one that has two curves that go up and down then goes
on indefinitely on each end. The graph where **n=5** is **red**
and is the largest one. The other two graphs are **blue**
and **green**, respectively, and each one has the same shape
as the first. Each graph is smaller than the one before
it.

When you get to
the fourth graph where **n=1**, this is the **light blue**
one, the graph changes shape. The graph is made up of an elliptical
shape that has a linear path that goes right through the center
of the ellipse. The next graph, where **n=1.1**, is **yellow**.
This graph seems to trace the previous graph, but does not have
the exact same elliptical shape. This graph also continues
on in a linear path that continues on indefinitely. The
next graph, where **n=0.9**, is **pink**. This graph
also traces the graph where n=1. This graph traces almost
on the inside of the graph where n=1, where the previous graph
traced mostly on the outside of the graph.

The last graph,
where **n=-3**, changes shape once again. This is the
**gray** one. This graph goes along the y-axis, where
the others seemed to go along the y-axis. This shape is
also more flattened out than the others. It also has two
curves in the graph like the first three and goes on indefinitely
after the curves in opposite directions.

**The next part of problem 1 asks for an equation for a given
graph**.

The equation that I found to give the closest graph to the given one is

In this set of graphs, I added constants to each side of the equation
to test what would happen. First, I added **5, 4, 3**,
and **-1** on the left.

**This is the resulting graph of the equations.**

These graphs are also curved, like the original set of graphs,
but they have different shapes. The first graph is the **purple
**one where the constant **5 **is added. This graph
has only one curve and then goes on forever in both directions.
The next graph, where the constant **4** is added, is the **red
**graph. This graph has the same shape as the purple one,
but it is smaller. The third graph, where **3** is added,
is the **blue** graph. This graph has a more pointed
curve and had also added a circular shape onto it. The last
graph, where **ñ1** is added, is the **green **one.
This graph has two curves on it and then goes on forever in both
directions.

In the next set of graphs, I added the constants **5,4,3**,
and **ñ1** onto the right side of the equation.

**These were the graphs resulting from adding
the constants on the right.**

These graphs are basically the same as the graphs in the previous example, but they have been flipped onto the opposite side of the axis. Each graph has the same color as the one in the previous example for the same constant. So, you can tell by looking at both graphs that they are the exact same shape.

**Therefore, I
suggest that adding the same constant onto the left or the right
side of the equation will result in the same shape but on opposite
sides of the axis. **

The last graph that I made on this set of problems has both sets of equations on it. This graph includes the graphs where the constants are added on the left and the right.

You can definitely tell from this graph how the shapes are the same but on opposite sides of the axis. The ones on the top left are the ones where the constant is added on the left.The ones on the bottom right are the ones where the constant is added on the right.

The resulting graph is a **three dimensional graph**.
This graph is also curved like all of the other graphs of the
function.