**Investigate the following equations with different values
of p:**

**for k > 1, k = 1, and k < 1.**

We will start by examining the graphs when **k** = 1 and
**p** is varied.

If we let **p** = 1 in the first equation, we get the following
equation and graph:

which is a parabola.

If we let **p** = 2 and graph this equation on top of the
first one, we get the following equation and graph:

By letting **p** = 2, we have essentially doubled the radius
at each point which has the effect of "spreading out"
our curve.

If we now let **p** = 0.5, we would expect to half the radius
at each point of our original graph. This would have the effect
of "pulling in" or shrinking our curve. The equation
and graph, laid over the original, are as follows:

We can see that we obtained the expected results.

We will now look at negative values of **p**. If we make
**p** negative, we are essentially multiplying the radius by
-1 at each point. This should have the effect of reversing the
direction of the radius from the origin. The length of the radius
should remain unchanged. We should see a reflection of the graph
through the origin.

We will look at our original equation again.

If we now let **p** = -1 and lay this graph over the original,
we get the following:

We can see that we obtained the expected results.

If we let **p** = 1 in the second equation, we get the following
equation and graph:

which is a parabola facing the opposite direction from the
first equation. Varying **p** will have the same effect on
this graph as it did on the first.

If we let **p** = 1 in the third equation, we get the following
equation and graph:

which, again, is a parabola. Varying **p** will have the
same effects as above.

And finally, if we let **p** = 1 in the fourth equation,
we get the following equation and graph:

which is a parabola facing the opposite direction from the
third equation. And again, varying **p** will produce the same
effects as above.

To examine the graphs when **k** < 1 and **p** is
varied, click **here**.

To examine the graphs when **k** > 1 and **p** is
varied, click **here**.