Based on the description, the graph has vertical asymptotes at $x = \pm n\pi$ and crosses the x-axis at $\left(\frac{\pi}{4}, 0\right)$ and $\left(-\frac{3\pi}{4}, 0\right)$, which is characteristic of a tangent function. Tangent functions exhibit vertical asymptotes at integer multiples of $\pi$ and oscillate between positive and negative infinity near those points.

Moreover, the graph's shift to cross the x-axis at $\frac{\pi}{4}$ and $-\frac{3\pi}{4}$ suggests a horizontal shift. The equation of the tangent function that matches this behavior is:

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

Thus, the correct answer is:

$\boxed{y = \tan\left(x - \frac{\pi}{4}\right)}$

This equation reflects the tangent function shifted to the right by $\frac{\pi}{4}$, fitting the described properties of the graph.

Would you like more details on how transformations affect trigonometric functions or any specific part of the explanation?

Related Questions:

1. How does the graph of $y = \tan(x)$ differ from $y = \cot(x)$?
2. What are the general characteristics of the tangent function’s graph?
3. How do horizontal shifts affect the graph of trigonometric functions?
4. What is the period of the tangent function, and how does it affect the graph?
5. How do asymptotes relate to the period of the tangent and cotangent functions?

Tip: Tangent functions have vertical asymptotes at odd multiples of $\frac{\pi}{2}$ when unshifted, which helps to identify transformations.