Question

Identify the type(s) of symmetry for the graph below.Select all that apply. aSymmetry with respect to the line \small \theta=\frac{\pi}{2} bSymmetry with respect to the polar axis cSymmetry with respect to the pole

Answer 1

The line θ=π/2 is the vertical line in the polar grid, the polar axis is the horizontal line and the pole is the center of coordinates. Now let's analyze the symmetries:

If the grpah is symmetric with respect to θ=π/2 then the graph at the left of this line has to be the mirrored image of the graph at the right side. This is the case of this graph so it does have symmetry with respect to θ=π/2.

For the polar axis is the same, the graph above the axis has to be the mirrored image of that below the axis. However in this case we have two "petals" above the polar axis and one below so the upper part is not the mirrored version of the lower part so it has no symmetry with respect to this axis.

For the pole we must rotate the graph 180°. If the graph remains unchanged then it is symmetric with respect to it. In this case if we rotate the graph 180° the lower petal ends up in the opposite direction so the graph changes after a 180° rotation and it has no symmetry with respect to the pole.

Then the only type of symmetry is with respect to the line θ=π/2 and the answer is option a.