# Examining Polar Equations

### Assignment 11

### Erin Horst

Investigate with different values of p

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

For notes on the derivations of these formulas, please see
Dr
Jim Wilson's page.

Let us begin by examining

for values of k > 1.

Let us view the graph in steps for k = 2, k = 4, k = 5, k =
7, and k = 10.

**k = 2**

**k = 4**

**k = 5**

**k = 7**

**k = 10**

To view a movie of k moving from k = 2 to k = 10, please download the QuickTime file.

Notice that when k > 1, the graphs of r are hyperbolas,
whose respective asymptotes are also illustrated.

Next, let us view when k = 1 for

**k = 1**

Notice that when k = 1 the graphs of r are parabolas.

Lastly, let us consider when k < 1 for

Let's view the graph in steps for k = -2, k = -4, k = -5, k
= -7, and k = -10.

**k = -2**

**k = -4**

**k = -5**

**k = -7**

**k = -10**

To view a movie of k moving from k = -2 to k = -10, please
download the QuickTime file.

Notice that when k < 1, the graphs of r are hyperbolas,
whose respective asymptotes are also illustrated.

Do you observe anything else when k < 1, k = 1, and k >1?
Do you see that there is symmetry presented in graphs in a multitude
of ways? Notice for example

represented by red and purple, respectively, that these functions
of r are reflected across the vertical axis. This is similar for

represented by blue and green, respectively, although they
are reflected across the horizontal axis. Are there other forms
of symmetry represented?

(In further exploration, it is also interesting to note that
when 0 < k <1 (similarly for -1 < k < 0) that the
graphs of r transition from parabolas to ellipsis to circles as
k decreases. You can view this movement by downloading
the QuickTime file.)

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