Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. Do you recall how we could test the functions for symmetry? If not, here are the tests.

1.

A graph has symmetry with respect to the y-axis if, whenever (x, y) is on the graph, so is the point (-x, y).

2.

A graph has symmetry with respect to the origin if, whenever (x, y) is on the graph, so is the point (-x, -y).

3.

A graph has symmetry with respect to the x-axis if, whenever (x, y) is on the graph, so is the point (x, -y).

The big question is how do we test for symmetry of an equation in polar coordinates? Let us look at the following diagrams to determine the answer to this question.

x-axis symmetry

y-axis symmetry

symmetry about the origin

 So here are the symmetry tests for polar graphs.

1.

Symmetry about the x-axis: If the point (r, q ) lies on the graph, then the point  (r, -q ) or (-r, p - q ) also lies on the graph.

2.

Symmetry about the y-axis: If the point (r, q ) lies on the graph, then the point (r, p - q ) or (-r, -q ) also lies on the graph.

3.

Symmetry about the origin: If the point (r, q ) lies on the graph, then the point (-r, q ) or (r, q + p ) also lies on the graph.

EXAMPLE 1:

Identify the symmetries of the curve r = 2 + 2 cos q and then sketch the graph.

SOLUTION:

(r, - q ) Þ r = 2 + 2 cos (-q ) Þ r = 2 + 2 cos q

(Remember that cosine is an even function.)

(-r, - q ) Þ -r = 2 + 2 cos (-q )Þ -r = 2 + 2 cos q Þ r = -2 - 2 cos q

(-r, q ) Þ -r = 2 + 2 cos q Þ r = -2 - 2 cos q

Now, let us compare our findings with the graph of this function.

Notice that the graph's only symmetry is with respect to the x-axis, and this is what we determine with our testing.

EXAMPLE 2:

Identify the symmetries of the curve r = 2 + sin q and then sketch the graph.

SOLUTION:

(r, - q ) Þ r = 2 + sin (-q ) Þ r = 2 -sin q

(Remember that sine is an odd function.)

(-r, - q ) Þ -r = 2 + sin (-q ) Þ -r = 2 - sin q Þ r = -2 + sin q

Is this a correct answer? No! Let us look at the graph of these two functions on the same coordinate axis.

We have the same graph, but they start in different places. Therefore, this function does have y-axis symmetry.

Sometimes it is best to look at the graph of the polar function instead of trusting algebraic manipulation.

EXAMPLE 3:

Identify the symmetries of the curve r ² = cos q and then sketch the graph.

SOLUTION:

(r, - q ) Þ r ² = cos (-q ) Þ r ² = cos q

(Remember that cosine is an even function.)

(-r, - q ) Þ (-r) ² = cos (-q ) Þ r ² = cos q

(-r, q ) Þ (-r) ² = cos q Þ r ² = cos q

Now, let us compare our findings with the graph of this function.

Yes this graph does fit the results that we received from algebraic manipulation.

Now let us look at how to determine the slope of a polar curve r = f (q ). Remember that the slope of any curve is given by dy/ dx not dr/ dq , so we will have to derive out the formula for dy/dx.

Let x = r cos q = f (q ) cos q and y = r sin q = f (q ) sin q . If f is a differentiable function of q, then so is x and y. When dx/ dq ¹ 0, we can find dy/ dx from the parametric formula.

EXAMPLE 4:

Find the slope of the curve r = -1 + sin q at q = p .

SOLUTION:

Now evaluate dy/dx at q = p .

EXAMPLE 5:

Find the slope of the curve r = cos 2 q at q = p / 2.

SOLUTION:

Now evaluate dy/ dx at q = p / 2.

There are two types of intersection points. They are (1) simultaneous, and (2) non-simultaneous. Here is how your find both types of points.

To find the simultaneous intersection points, set the two equations equal to each other and solve for q .

To find the non-simultaneous intersection points, graph both equations and determine where the graphs cross each other.

EXAMPLE 6:

Find the points of intersection (both types) of the pair of curves r = 1 + sin q and r = 1 - sin q .

SOLUTION:

1 + sin q = 1 - sin q ® 2sin q = 0 ® q = 0 and q = p

When q = 0, then r = 1 + sin 0 = 1 ® (1, 0).

When q = p , then r = 1 + sin p = 1 ® (1, p ).

Let us graph both equations on the same axis.

Notice that the graphs cross each other at the point (0, 0), so this is the non-simultaneous intersection point. This is the only one.

EXAMPLE 7:

Find the points of intersection (both types) of the pair of curves r = cos q and r = 1 - cos q .

SOLUTION:

The graphs cross each other at the origin, so the only non-simultaneous intersection point is (0, 0).

EXAMPLE 8:

Find the points of intersection (both types) of the pair of curves r ² = cos 2 q and r ² = sin 2 q .

SOLUTION:

The only non-simultaneous intersection point for these two graphs is the origin, (0, 0).