Let's break down the given function $f(x) = \frac{4(x - 1)}{3(x - 1)^3}$ and determine each feature of the graph as requested:

1. Simplifying the function:

The function simplifies as follows: $f(x) = \frac{4(x - 1)}{3(x - 1)^3} = \frac{4}{3(x - 1)^2}$ Since the $(x - 1)$ in the numerator and denominator cancel out, we are left with the simplified function: $f(x) = \frac{4}{3(x - 1)^2}$

This will help us identify key features.

2. Vertical Asymptote:

A vertical asymptote occurs where the denominator equals zero and the function is undefined. Setting the denominator to zero: $3(x - 1)^2 = 0$ This gives $x = 1$. Therefore, there is a vertical asymptote at $x = 1$.

3. Horizontal Asymptote:

To determine the horizontal asymptote, observe the behavior of the function as $x \to \infty$ or $x \to -\infty$. The term $(x - 1)^2$ in the denominator grows large, making the whole fraction approach zero. Therefore, the function approaches: $\lim_{x \to \pm \infty} f(x) = 0$ Thus, there is a horizontal asymptote at $y = 0$.

4. Holes:

Holes occur when a factor cancels out from both the numerator and denominator, making the function undefined at that point. However, after simplifying the original function, we see that $x = 1$ does not result in a hole, but instead leads to a vertical asymptote. There are no holes in this graph.

5. X-intercept:

The x-intercept is found by setting $f(x) = 0$. However, in the simplified function, the numerator is a constant (4), meaning the function never equals zero. There is no x-intercept.

6. Y-intercept:

To find the y-intercept, we evaluate the function at $x = 0$: $f(0) = \frac{4}{3(0 - 1)^2} = \frac{4}{3}$ Therefore, the y-intercept is at $\left(0, \frac{4}{3} \right)$.

Summary of Features:

1. Vertical Asymptote: $x = 1$
2. Horizontal Asymptote: $y = 0$
3. Holes: None
4. X-intercept: None
5. Y-intercept: $\left(0, \frac{4}{3} \right)$

Would you like more details on any of these features?

Here are five related questions to expand this topic:

1. How would the function change if the numerator were $4(x - 2)$ instead of $4(x - 1)$?
2. What would happen to the vertical asymptote if the denominator were $(x - 2)^3$?
3. Can a function have multiple vertical asymptotes? Provide an example.
4. What is the significance of a hole in a function graph, and how does it differ from an asymptote?
5. How can we find oblique asymptotes for rational functions, and when do they occur?

Tip: Always start by simplifying a rational function to better understand its behavior and features like asymptotes and intercepts.