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Identify the correct graph for the following function:


To identify the correct graph for the function \( f(x) = \frac{x^3 - x^2 - 2x}{4x^2 - 20x + 24} \), we need to analyze the function for the following characteristics:

1. **Factorization**: Simplify the function by factoring the numerator and the denominator.
2. **Discontinuities (Vertical Asymptotes)**: Find the values of \( x \) that make the denominator zero (where the function is undefined).
3. **Zeros of the Function (X-intercepts)**: Find the values of \( x \) that make the numerator zero.
4. **Horizontal or Oblique Asymptotes**: Determine the behavior of the function as \( x \) approaches infinity.

### Step 1: Factor the Numerator and Denominator

1. **Numerator**: \( x^3 - x^2 - 2x \)
   - Factor out \( x \):
   \[
   x(x^2 - x - 2)
   \]
   - Factor \( x^2 - x - 2 \):
   \[
   x(x - 2)(x + 1)
   \]
   So, the numerator factors to:
   \[
   x(x - 2)(x + 1)
   \]

2. **Denominator**: \( 4x^2 - 20x + 24 \)
   - Factor out \( 4 \):
   \[
   4(x^2 - 5x + 6)
   \]
   - Factor \( x^2 - 5x + 6 \):
   \[
   4(x - 2)(x - 3)
   \]
   So, the denominator factors to:
   \[
   4(x - 2)(x - 3)
   \]

### Step 2: Simplify the Function
After factoring, the function becomes:
\[
f(x) = \frac{x(x - 2)(x + 1)}{4(x - 2)(x - 3)}
\]
Notice that \( (x - 2) \) cancels out, leaving:
\[
f(x) = \frac{x(x + 1)}{4(x - 3)}, \quad \text{for } x \neq 2
\]

### Step 3: Discontinuities (Vertical Asymptotes)
The denominator after cancellation is \( 4(x - 3) \), so the vertical asymptote occurs at:
\[
x = 3
\]
However, \( x = 2 \) is a removable discontinuity (hole) because it was canceled out.

### Step 4: X-intercepts
The zeros of the numerator are:
\[
x(x + 1) = 0 \quad \Rightarrow \quad x = 0, -1
\]

### Step 5: Horizontal Asymptote
Since the degree of the numerator (2) is equal to the degree of the denominator (1), there is no horizontal asymptote. The function has an oblique asymptote, which can be determined by dividing the numerator by the denominator.

### Conclusion:
- **Discontinuities**: \( x = 3 \) (vertical asymptote), and \( x = 2 \) (hole).
- **X-intercepts**: \( x = 0, -1 \).

Now, let's match this analysis with the options provided in the image:

- **Option C** is the correct graph. It shows a vertical asymptote at \( x = 3 \), a removable discontinuity (hole) at \( x = 2 \), and x-intercepts at \( x = 0 \) and \( x = -1 \).

Would you like further clarification or details?

Here are five related questions you might find useful:
1. How do you determine the oblique asymptote for rational functions?
2. What happens to the function when there is a hole at a certain point?
3. How do you interpret the behavior of a function near its vertical asymptote?
4. Can you explain how to factor higher-degree polynomials?
5. How does the degree of the numerator and denominator affect the end behavior of rational functions?

**Tip:** When analyzing a rational function, always simplify it first to identify any removable discontinuities and make it easier to find the asymptotes and intercepts.

Learn how to identify the correct graph for the rational function (x^3 - x^2 - 2x) / (4x^2 - 20x + 24) with a step-by-step solution. This problem covers key concepts such as rational functions, vertical asymptotes, removable discontinuities, x-intercepts, and oblique asymptotes. Suitable for Grades 11-12 students.

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