We are given the function:

$y = 4 \tan\left(x + \frac{\pi}{6}\right)$

The vertical asymptotes of the tangent function occur at values where the argument of the tangent function is an odd multiple of $\frac{\pi}{2}$, i.e., where:

$x + \frac{\pi}{6} = \frac{\pi}{2} + n\pi$ for integer $n$.

Step 1: Solve for $x$

To find the vertical asymptotes, solve for $x$ by subtracting $\frac{\pi}{6}$ from both sides:

$x = \frac{\pi}{2} + n\pi - \frac{\pi}{6}$

We need to combine the fractions:

$x = \frac{3\pi}{6} - \frac{\pi}{6} + n\pi = \frac{2\pi}{6} + n\pi = \frac{\pi}{3} + n\pi$

Step 2: Identify the principal cycle

The period of the tangent function is $\pi$, meaning the principal cycle (the interval between two vertical asymptotes) is $\pi$ units wide. We typically look at one cycle, so we focus on the values of $x$ for $n = 0$ and $n = 1$.

• For $n = 0$, we have: $x = \frac{\pi}{3}$

• For $n = 1$, we have: $x = \frac{\pi}{3} + \pi = \frac{4\pi}{3}$

Step 3: Conclusion

The vertical asymptotes of the principal cycle are at $x = \frac{\pi}{3}$ (left) and $x = \frac{4\pi}{3}$ (right).

Thus, the equations of the vertical asymptotes are:

• Left vertical asymptote: $x = \frac{\pi}{3}$
• Right vertical asymptote: $x = \frac{4\pi}{3}$

Let me know if you need further details or clarification!

Related Questions:

1. How do you find the period of a tangent function?
2. What happens to the vertical asymptotes if the phase shift is different?
3. What is the effect of the coefficient of $4$ on the graph of the tangent function?
4. How would you graph the function $y = 4 \tan\left(x + \frac{\pi}{6}\right)$?
5. How can you find the phase shift for any tangent function?

Tip: Remember, the general form of the tangent function is $y = A \tan(Bx + C) + D$, where the period is $\frac{\pi}{|B|}$, and the phase shift is $-\frac{C}{B}$.