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We are tasked with finding the x-intercepts, vertical asymptotes, and holes for the given rational function:

\[
f(x) = \frac{3x^2 - 48}{x^2 - 11x + 28}
\]

### Step 1: Factor the numerator and denominator

1. **Numerator**: \( 3x^2 - 48 \)
   \[
   3x^2 - 48 = 3(x^2 - 16) = 3(x - 4)(x + 4)
   \]

2. **Denominator**: \( x^2 - 11x + 28 \)
   To factor this, find two numbers that multiply to 28 and add to -11:
   \[
   x^2 - 11x + 28 = (x - 4)(x - 7)
   \]

So, the factored form of the function is:
\[
f(x) = \frac{3(x - 4)(x + 4)}{(x - 4)(x - 7)}
\]

### Step 2: Find x-intercepts
X-intercepts occur when the numerator is zero. From the factored numerator \( 3(x - 4)(x + 4) \), set it equal to zero:
\[
3(x - 4)(x + 4) = 0 \quad \Rightarrow \quad x = 4 \quad \text{or} \quad x = -4
\]
But since \( x = 4 \) will create a hole (as explained below), the only valid x-intercept is:
\[
x = -4
\]

### Step 3: Find vertical asymptotes
Vertical asymptotes occur when the denominator is zero and there is no cancellation. From the factored denominator \( (x - 4)(x - 7) \):
\[
x = 4 \quad \text{or} \quad x = 7
\]
However, \( x = 4 \) is a removable discontinuity (hole) due to the common factor of \( (x - 4) \) in both the numerator and denominator. Thus, the only vertical asymptote is:
\[
x = 7
\]

### Step 4: Find holes
A hole occurs where the numerator and denominator share a common factor, and that factor cancels out. Since both the numerator and denominator have \( (x - 4) \), there is a hole at:
\[
x = 4
\]
To find the y-coordinate of the hole, substitute \( x = 4 \) into the simplified function (after canceling \( x - 4 \)):

Simplified function after canceling \( (x - 4) \):
\[
f(x) = \frac{3(x + 4)}{x - 7}
\]
Substitute \( x = 4 \):
\[
f(4) = \frac{3(4 + 4)}{4 - 7} = \frac{3(8)}{-3} = -8
\]
So, the hole is at \( (4, -8) \).

### Final Answer:
- **x-intercept**: \( x = -4 \)
- **Vertical asymptote**: \( x = 7 \)
- **Hole**: \( (4, -8) \)

Would you like any more details or clarifications on this?

Here are some related questions to further explore the topic:
1. How do you find horizontal asymptotes for rational functions?
2. What is the behavior of the function near the vertical asymptote?
3. How do you determine if a function has a slant asymptote?
4. What changes occur in the graph of a function when a hole is present?
5. How can limits help in understanding asymptotic behavior?

**Tip**: Always cancel common factors before analyzing the function to avoid errors with asymptotes and holes.

Find any x-intercepts, vertical asymptotes, and holes for the graph of the following function: f(x) = (3x^2 - 48) / (x^2 - 11x + 28).

Learn how to find x-intercepts, vertical asymptotes, and holes for the rational function f(x) = (3x^2 - 48) / (x^2 - 11x + 28). This detailed solution involves factoring quadratic expressions, identifying points of discontinuity, and analyzing vertical asymptotes.

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