To graph the function \( y = 2g(x + 4) \), follow these steps:

### 1. **Understand the Transformations:**
   The function \( y = 2g(x + 4) \) represents a transformation of the base function \( y = g(x) \).

   - **Horizontal Shift**: The term \( x + 4 \) shifts the graph of \( y = g(x) \) horizontally to the left by 4 units.
   - **Vertical Stretch**: The coefficient 2 vertically stretches the graph, making it twice as tall as the original graph of \( y = g(x) \).

### 2. **Start with the Base Graph:**
   Begin by sketching the graph of the base function \( y = g(x) \). Since \( g(x) \) is not specified, treat it as a generic function.

### 3. **Apply the Transformations:**
   - **Horizontal Shift**: Shift the graph of \( y = g(x) \) 4 units to the left. This means every point \((x, y)\) on the graph of \( y = g(x) \) moves to \((x - 4, y)\).
   - **Vertical Stretch**: Multiply the \( y \)-values of the shifted graph by 2. This means if a point \((x, y)\) was on the graph after the horizontal shift, it now moves to \((x, 2y)\).

### 4. **Sketch the Final Graph:**
   Combine the transformations to produce the graph of \( y = 2g(x + 4) \).

### Example:
If \( y = g(x) \) is \( y = \sqrt{x} \) (a basic square root function):

- **Base Graph**: The graph of \( y = \sqrt{x} \) starts at \( (0, 0) \) and increases slowly, curving upwards.
- **Horizontal Shift**: Shifting it left by 4 units, the graph starts at \( (-4, 0) \).
- **Vertical Stretch**: Multiply all \( y \)-values by 2, so the graph becomes steeper, starting at \( (-4, 0) \) but increasing more rapidly.

Would you like to see the graph for a specific base function \( g(x) \) or have any questions?

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Here are some follow-up questions:

1. What function would you like to use for \( g(x) \)?
2. How does a vertical stretch affect the graph of a function?
3. Can you explain the difference between a horizontal shift and a vertical shift?
4. How would the graph change if the function were \( y = \frac{1}{2}g(x + 4) \)?
5. What if we had \( y = 2g(x - 4) \) instead?
6. How do you apply multiple transformations in sequence?
7. What does the graph look like if \( g(x) \) is a quadratic function?
8. What impact does a negative coefficient (e.g., \( -2g(x + 4) \)) have on the graph?

**Tip**: When multiple transformations are applied to a graph, it's important to perform horizontal shifts before applying vertical stretches or compressions.

Learn how to graph the function y = 2g(x + 4) with detailed steps and examples. Explore horizontal shifts and vertical stretches in function transformations. Suitable for students studying advanced algebra and function graphing.

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