#### Answer

IV

#### Work Step by Step

To plot the point $(r, \theta )=(5, -\frac{\pi }{4})$, begin with the $-\frac{\pi }{4}$ angle. Because $-\frac{\pi }{4}$ is a negative angle, draw $\theta = -\frac{\pi }{4}$ clockwise from the polar axis. Now consider $r=5$. Because $r \gt 0$, plot the point by going out five units on the terminal side of $\theta$.
So, looking at the graph, we see that the point lies in the quadrant IV.