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The equation provided is \( y = \tan(6x) \). 

For the tangent function, vertical asymptotes occur where the function is undefined. Specifically, the tangent function is undefined where the cosine in the denominator of \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) is equal to zero. This happens at:

\[
6x = \frac{\pi}{2} + n\pi \quad \text{(for all integer values of \(n\))},
\]
since the general solution for vertical asymptotes of \( y = \tan(kx) \) occurs at \( kx = \frac{\pi}{2} + n\pi \).

Solving for \( x \), we divide both sides by 6:

\[
x = \frac{1}{6} \left( \frac{\pi}{2} + n\pi \right) = \frac{\pi}{12} + \frac{n\pi}{6}.
\]

Thus, the vertical asymptotes occur at:

\[
x = \frac{\pi}{12}, \frac{\pi}{12} + \frac{\pi}{6}, \frac{\pi}{12} + 2\left(\frac{\pi}{6}\right), \dots
\]

For the first three vertical asymptotes, set \( n = 0 \), \( n = 1 \), and \( n = 2 \):

1. When \( n = 0 \): \( x = \frac{\pi}{12} \)
2. When \( n = 1 \): \( x = \frac{\pi}{12} + \frac{\pi}{6} = \frac{\pi}{4} \)
3. When \( n = 2 \): \( x = \frac{\pi}{12} + 2\left(\frac{\pi}{6}\right) = \frac{5\pi}{12} \)

Thus, the equations of the three asymptotes are:

\[
x = \frac{\pi}{12}, \quad x = \frac{\pi}{4}, \quad x = \frac{5\pi}{12}.
\]

Would you like further explanation on asymptotes or details on this function?

Here are 5 follow-up questions you might find helpful:
1. How do vertical asymptotes affect the graph of a trigonometric function?
2. Can you explain why \( \tan(x) \) has asymptotes at \( x = \frac{\pi}{2} + n\pi \)?
3. What happens to the tangent graph between two vertical asymptotes?
4. How would the asymptotes change if the equation was \( y = \tan(3x) \)?
5. Can the tangent function have horizontal asymptotes?

**Tip**: For tangent functions, the period is \( \frac{\pi}{k} \), where \( k \) is the coefficient of \( x \) in \( y = \tan(kx) \).

Write the equations for three asymptotes in the graph of y = tan(6x).

This guide covers how to find the vertical asymptotes of the tangent function y = tan(6x). It explains how asymptotes occur where the cosine function in the tangent's denominator equals zero and provides a detailed step-by-step solution for finding the first three asymptotes.

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