Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 5 cos 5θ

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. 

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