8.4 Polar coordinates: graphs  (Page 2/16)

A General Note

Symmetry tests

A polar equation    describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry, as shown in [link] .

Graphing polar equations by plotting points

To graph in the rectangular coordinate system we construct a table of $x$ and $y$ values. To graph in the polar coordinate system we construct a table of $\theta$ and $r$ values. We enter values of $\theta$ into a polar equation    and calculate $r.$ However, using the properties of symmetry and finding key values of $\theta$ and $r$ means fewer calculations will be needed.

Finding zeros and maxima

To find the zeros of a polar equation, we solve for the values of $\theta$ that result in $r=0.$ Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for $x.$ We use the same process for polar equations. Set $r=0,$ and solve for $\theta .$

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of $\theta$ into the equation that result in the maximum value of the trigonometric functions. Consider $r=5\cos \theta ;$ the maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when $\theta =0,$ so our polar equation is $5\cos \theta ,$ and the value $\theta =0$ will yield the maximum $\left|r\right|.$

Similarly, the maximum value of the sine function is 1 when $\theta =\frac{\pi }{2},$ and if our polar equation is $r=5\sin \theta ,$ the value $\theta =\frac{\pi }{2}$ will yield the maximum $\left|r\right|.$ We may find additional information by calculating values of $r$ when $\theta =0.$ These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.