10.4 Polar Coordinates: Graphs

Test the equation $r = 2 \sin \theta$ for symmetry.

Test the equation for symmetry: $r = - 2 \cos \theta .$

Using the equation in the example, find the zeros and maximum $\left|r\right|$ and, if necessary, the polar axis intercepts of $r = 2 \sin \theta .$

Without converting to Cartesian coordinates, test the given equation for symmetry and find the zeros and maximum values of $\left|r\right| :$ $r = 3 \cos \theta .$

Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.

What are the steps to follow when graphing polar equations?

What part of the equation determines the shape of the graph of a polar equation?

Test the equation for symmetry.

$r = 3 - 3 \cos \theta$

Test the equation for symmetry.

$r = 3 \sin 2 \theta$

Test the equation for symmetry.

$r = 2 \theta$

Test the equation for symmetry.

$r = \frac{2}{\theta}$

Test the equation for symmetry.

$r = \sqrt{5 \sin 2 \theta}$

Use a graphing utility to graph each pair of polar equations on a domain of $\left[0 , 4 \pi\right]$ and then explain the differences shown in the graphs.

$r = \theta , r = \theta + \sin \theta$

Use a graphing utility to graph each pair of polar equations on a domain of $\left[0 , 4 \pi\right]$ and then explain the differences shown in the graphs.

$r = 2 \sin \left(\frac{\theta}{2}\right) , r = \theta \sin \left(\frac{\theta}{2}\right)$

Use a graphing utility to graph each pair of polar equations on a domain of $\left[0 , 4 \pi\right]$ and then explain the differences shown in the graphs.

On a graphing utility, graph $r = \sin \left(\frac{16}{5} \theta\right)$ on $\left[\right. 0$, $4 \pi \left]\right.$, $\left[\right. 0$, $8 \pi \left]\right.$, $\left[\right. 0$, $12 \pi \left]\right.$, and $\left[\right. 0$, $16 \pi \left]\right. .$ Describe the effect of increasing the width of the domain.

Use a graphing utility to graph each pair of polar equations on a domain of $\left[0 , 4 \pi\right]$ and then explain the differences shown in the graphs.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

Use a graphing utility to graph each pair of polar equations on a domain of $\left[0 , 4 \pi\right]$ and then explain the differences shown in the graphs.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

Draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} = 6 - 4 \cos \theta , r_{2} = 4$

Draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} = 1 + \cos \theta , r_{2} = 3 \cos \theta$

Draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} = \sin^{2} \left(2 \theta\right)$, $r_{2} = 1 - \cos \left(4 \theta\right)$

Draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} ^{2} = \sin \theta , r_{2} ^{2} = \cos \theta$