While playing around with your calculator one day, you create a polar plot that looks like this

Your teacher tells you that this is a polar plot with an equation

### Transforming Polar Graphs

Just as in graphing on a rectangular grid, you can also graph polar equations on a polar grid.

These equations may be simple or complex. To begin, you should try something simple like

#### Graphing Equations

1. On a polar plane, graph the equation

The solution is all ordered pairs of

2. On a polar plane, graph the equation

For this problem, the

To begin graphing more complicated polar equations, we will make a table of values for

3. Graph the following polar equations on the same polar grid and compare the graphs.

The cardioid is symmetrical about the positive

Changing the value of

It is also possible to create a sinusoidal curve called a limaçon. It has

As we've seen with cardioids, it is possible to create transformations of graphs of limaçons by changing values of constants in the equation of the shape.

### Examples

#### Example 1

Earlier, you were asked to graph a polar equation.

As you've seen in this section, transformations to the graph of a cardioid can be accomplished by 2 different ways. In this case, you want to rotate the graph so that it is around the "x" axis instead of the "y" axis. To accomplish this, you change the function from a sine function to a cosine function:

#### Example 2

Graph the curve

#### Example 3

Graph the curve

#### Example 4

Graph the curve

### Review

Graph each equation.

r=4 θ=60∘ r=2 θ=110∘

Graph each function using your calculator and sketch on your paper.

r=3+3sin(θ) r=2+4sin(θ) r=1−5sin(θ) r=2−2sin(θ) r=3+6sin(θ) r=−3+6sin(θ) - Analyze the connections between the equations and their graphs above. Make a hypothesis about how to graph \begin{align*}r=a+b\sin(\theta )\end{align*} for positive or negative values of a and b where \begin{align*}b\geq a\end{align*}.

Graph each function using your calculator and sketch on your paper.

- \begin{align*}r=3+3\cos(\theta )\end{align*}
- \begin{align*}r=2+4\cos(\theta )\end{align*}
- \begin{align*}r=1-5\cos(\theta )\end{align*}
- \begin{align*}r=2-2\cos(\theta )\end{align*}
- \begin{align*}r=3+6\cos(\theta )\end{align*}
- \begin{align*}r=-3+6\cos(\theta )\end{align*}
- Analyze the connections between the equations and their graphs above. Make a hypothesis about how to graph \begin{align*}r=a+b\cos(\theta )\end{align*} for positive or negative values of a and b where \begin{align*}b\geq a\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.3.