1. **State the problem:** We are given the function $y = \tan 5x$ and asked to analyze or solve it. 2. **Recall the formula and properties:** The tangent function is defined as $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and has vertical asymptotes where $\cos \theta = 0$. 3. **Period of the function:** The period of $\tan x$ is $\pi$. For $y = \tan 5x$, the period is compressed by a factor of 5, so the period is $\frac{\pi}{5}$. 4. **Find zeros:** The zeros of $\tan 5x$ occur where $5x = n\pi$, for integers $n$. Thus, zeros are at $x = \frac{n\pi}{5}$. 5. **Find vertical asymptotes:** Vertical asymptotes occur where $\cos 5x = 0$, i.e., where $5x = \frac{\pi}{2} + n\pi$, so $$x = \frac{\pi}{10} + \frac{n\pi}{5}$$ for integers $n$. 6. **Summary:** The function $y = \tan 5x$ has zeros at $x = \frac{n\pi}{5}$ and vertical asymptotes at $x = \frac{\pi}{10} + \frac{n\pi}{5}$, with period $\frac{\pi}{5}$. This completes the analysis of the function $y = \tan 5x$.