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**Problem 5:**

- The maximum point on the graph of the equation \( y = f(x) \) is \( (2, -3) \).
- The question asks: What is the range of the function represented by the equation \( y = 2f(x - 4) \)?

### Step-by-Step Solution:

#### Step 1: Understand the transformation in \( y = f(x) \)
The maximum point of the original function \( y = f(x) \) is at \( (2, -3) \). This tells us that the highest point on the graph for the original function is when \( x = 2 \) and the corresponding \( y \)-value is \( -3 \).

#### Step 2: Analyze the transformation \( y = f(x - 4) \)
The term \( f(x - 4) \) represents a **horizontal shift** of the graph of \( f(x) \). Specifically, the graph is shifted 4 units to the **right**. So, the new maximum point will be moved 4 units to the right of \( x = 2 \). 

- Old maximum point: \( (2, -3) \)
- New maximum point: \( (2 + 4, -3) = (6, -3) \)

The \( y \)-value remains unchanged at \( -3 \) because a horizontal shift does not affect the vertical values.

#### Step 3: Analyze the transformation \( y = 2f(x - 4) \)
The factor of 2 outside the function represents a **vertical stretch**. This means that all \( y \)-values of the function are multiplied by 2. Therefore, the maximum \( y \)-value, which was originally \( -3 \), is now stretched vertically by a factor of 2.

- Original maximum \( y \)-value: \( -3 \)
- New maximum \( y \)-value: \( 2 \times (-3) = -6 \)

Thus, the new maximum point after both the horizontal shift and vertical stretch is \( (6, -6) \).

#### Step 4: Determine the range of the new function
The range of a function is the set of possible \( y \)-values. Since \( f(x) \) has a maximum point at \( (2, -3) \), it means \( f(x) \) reaches \( -3 \) as its highest value and extends downward. After applying the vertical stretch of 2, the maximum value becomes \( -6 \), and the function will still extend downward to negative infinity.

Therefore, the range of \( y = 2f(x - 4) \) is:
\[
(-\infty, -6]
\]

#### Step 5: Final Answer
The range of the function represented by \( y = 2f(x - 4) \) is \( (-\infty, -6] \).

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Would you like further details or clarifications on any of the steps? Here are five related questions you might explore:

1. How do vertical stretches or compressions affect the graph of a function?
2. What is the difference between horizontal and vertical shifts in a function?
3. How would the range change if the stretch factor was \( \frac{1}{2} \) instead of 2?
4. How can you find the domain of a function after applying transformations?
5. What is the impact of negative values in transformations like \( y = -f(x) \)?

**Tip:** Always visualize transformations step by step, especially when dealing with multiple transformations like shifts and stretches.

The maximum point on the graph of the equation y = f(x) is (2, -3). What is the range of the function represented by the equation y = 2f(x - 4)?

Explore how to determine the range of a function after applying a horizontal shift and a vertical stretch. In this example, the function starts with y = f(x) having a maximum point at (2, -3). The transformed function y = 2f(x - 4) is analyzed step by step, with the final range calculated as (-∞, -6]. Suitable for students in Grades 9-12.

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