Question

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.

r = 5 cos 5θ

Answer 1

x-axis

Explanation:

r=4 cos⁡5θ

∙Subsitute (r,-θ)in the given function

4 cos⁡〖5θ=4 cos⁡5(-θ) 〗,then r(θ)=r(π-θ)

∴Thus,it is symmetric about x-axis.

Since

4 cos⁡〖5θ≠-4 cos⁡5(π-θ) 〗

∴Thus,it is not symmetric about y-axis.

∙Since

4 cos⁡〖5θ≠-4 cos⁡5(π+θ) 〗

∴Thus,it is not symmetric about the origin

Answer 2

Short answer. X axis. This is a teaching lesson intended to show you what the odd and even graphs look like.
Remark
You should always try to draw these kinds of questions using a graphing calculator or some sort of program on the internet that imitates a graphing calculator. Begin by looking at the diagram on the left. Are there any clues which one you should pick?

There are. Notice the "leaf" that's on the plus x axis. It looks the same on both sides of the x axis.

There are 4 other leaves. Do they confirm your suspicion? They should. The next leaf counter clockwise above the one on the x axis is the mirror image of the leaf that is next clockwise to the one symmetrical about the x axis. Those two are mirror images.

Comment
What would happen if you were give an even number of leaves? Try 4.
r = 5*cos(4 theta). Notice that 2n leaves are produced (2*4 = 8 in this case). Notice also that you have a symmetry for the origin.

Finally
If the topic interests you, go to desmos and try r = 5*sin(4*theta) and r = sin(5*theta). Are there any differences?

Remember to be working in polar coordinates in desmos.