91Ó°ÊÓ

The given equation is

$r=3\cos \left(4\mathrm{θ}\right)$

Substitute $-\mathrm{θ}$for $\mathrm{θ}$in the equation

$$\begin{gathered} r=3\cos \left(4\left(-\mathrm{θ}\right)\right) \\ r=3\cos \left(4\mathrm{θ}\right) \end{gathered}$$

The equation matches with the original equation so the test is satisfied.

the graph is symmetric with respect to the polar axis

Substitute $\mathrm{π}-\mathrm{θ}$for $\mathrm{θ}$in the equation

role="math" localid="1646686393219" $$\begin{gathered} r=3\cos \left(4\left(\mathrm{π}-\mathrm{θ}\right)\right) \\ r=3\cos \left(4\mathrm{π}-4\mathrm{θ}\right) \\ r=3\cos \left(4\mathrm{θ}\right) \end{gathered}$$

The equation matches the original equation so the test is satisfied.

the graph is symmetric with respect to the line$\mathrm{θ}=\frac{\mathrm{π}}{2}$

The equation is symmetric with respect to the polar axis and line $\mathrm{θ}=\frac{\mathrm{π}}{2}$

so the equation is symmetric with respect to the pole

Consider different values for $\mathrm{θ}$in interval$\left(0,\frac{\mathrm{π}}{2}\right)$and determine coordinates of several points for graph

Locates points and use them to plot the graph of the equation

The graph state that the equation is a rose with eight petals

Plot the graph of the equation $r=3\cos \left(4\mathrm{θ}\right)$using a graphing utility

Graph matches so our graph is correct.