Which graph represents the function of f(x) = the quantity of 4 x squared minus 16, all over 2 x minus 4?

To determine which graph represents the function \( f(x) = \frac{4x^2 - 16}{2x - 4} \), we need to first simplify the function.

### Step 1: Factor the numerator
The numerator \( 4x^2 - 16 \) can be factored as:
\[
4x^2 - 16 = 4(x^2 - 4) = 4(x - 2)(x + 2)
\]

### Step 2: Simplify the function
Now, we substitute the factored form into the function:
\[
f(x) = \frac{4(x - 2)(x + 2)}{2(x - 2)}
\]
We can cancel out the \( (x - 2) \) term, but with a note that \( x \neq 2 \) since the original denominator would be zero at \( x = 2 \). After canceling:
\[
f(x) = 2(x + 2) \quad \text{for } x \neq 2
\]

### Step 3: Analyze the simplified function
The function \( f(x) = 2x + 4 \) is a linear function with a slope of 2 and a y-intercept at \( y = 4 \). However, due to the cancellation, the function has a hole at \( x = 2 \).

### Graph Description
- **Linear Function**: The graph is a straight line with a slope of 2 and crosses the y-axis at \( y = 4 \).
- **Hole at \( x = 2 \)**: The graph will have a point missing at \( x = 2 \), where the y-value would have been \( f(2) = 2(2) + 4 = 8 \).

### Summary
The correct graph is a line with a slope of 2, a y-intercept at 4, and a hole at the point \( (2, 8) \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you determine the slope and y-intercept of a linear function?
2. What is the significance of a hole in the graph of a function?
3. How can you find the domain and range of the simplified function?
4. What are the differences between a function's graph with and without holes?
5. How do you solve for the x-intercepts of a function like \( f(x) \)?

**Tip:** Always check for values that might be excluded from the domain after canceling terms when simplifying rational functions.

Explore and understand the graph of the rational function f(x) = (4x^2 - 16) / (2x - 4). This problem involves factorization of polynomials, cancellation of terms in rational functions, and analysis of linear functions with holes. Suitable for Grades 10-12 students studying algebra and functions.

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