Scenario: Players find themselves in an enchanted forest, where the trees' growth has been disrupted by a powerful spell. The task is to restore harmony by translating the distorted graphs of functions back to their original positions. Players will navigate through the forest, solving puzzles and interacting with magical creatures to unlock the secrets of graph translation.

Task:   Analyze the graph of the function  f(x)=2|x+7|+8
compared to the graph of the absolute value function  g(x)=|x|
.

To obtain the graph of  f(x)=2|x+7|+8
, the graph of  g(x)=|x|
has been

To analyze how the graph of the function \( f(x) = 2|x+7| + 8 \) is obtained from the graph of the absolute value function \( g(x) = |x| \), we can break down the transformation step by step:

1. **Horizontal Shift**:
   - The function \( f(x) = 2|x+7| + 8 \) involves \( |x+7| \). 
   - To obtain \( |x+7| \) from \( |x| \), we perform a horizontal shift to the left by 7 units. 
   - This means the entire graph of \( g(x) = |x| \) moves 7 units to the left, so the vertex of \( g(x) \), which is originally at \( (0, 0) \), will now be at \( (-7, 0) \).

2. **Vertical Stretch**:
   - The coefficient 2 in front of \( |x+7| \) indicates a vertical stretch by a factor of 2.
   - This means that every point on the graph of \( |x+7| \) is stretched vertically away from the x-axis by a factor of 2.
   - For example, the point \( (1, |1+7|) = (1, 8) \) on the graph of \( |x+7| \) becomes \( (1, 2 \times 8) = (1, 16) \) on the graph of \( f(x) \).

3. **Vertical Shift**:
   - The +8 outside of the absolute value indicates a vertical shift upwards by 8 units.
   - This means that after stretching the graph vertically, the entire graph is shifted up by 8 units.

### Summary of Transformations:

- **Start**: The graph of \( g(x) = |x| \).
- **Step 1**: Shift left by 7 units to get \( |x+7| \).
- **Step 2**: Stretch vertically by a factor of 2 to get \( 2|x+7| \).
- **Step 3**: Shift upwards by 8 units to get the final function \( f(x) = 2|x+7| + 8 \).

### Final Position of the Vertex:
- The vertex of the original graph \( g(x) = |x| \) is at \( (0, 0) \).
- After the horizontal shift left by 7 units, the vertex moves to \( (-7, 0) \).
- The vertical stretch does not change the x-coordinate, so the vertex remains at \( (-7, 0) \).
- After the vertical shift upwards by 8 units, the vertex moves to \( (-7, 8) \).

So, the graph of \( f(x) = 2|x+7| + 8 \) is a vertically stretched and shifted version of the graph of \( g(x) = |x| \), with the vertex located at \( (-7, 8) \).

Would you like more details or have any questions?

### Follow-Up Questions:
1. How would the graph change if the function was \( f(x) = |x+7| + 8 \) instead of \( 2|x+7| + 8 \)?
2. What happens if we subtract 8 instead of adding 8 in the function \( f(x) = 2|x+7| - 8 \)?
3. How does the graph of \( f(x) = -2|x+7| + 8 \) differ from the original function?
4. Can you describe the effect of shifting the graph horizontally by changing the inside of the absolute value function, e.g., \( f(x) = 2|x-3| + 8 \)?
5. How would the graph look if the function was \( f(x) = 2|x+7| - 5 \)?

### Tip:
When analyzing transformations of functions, always look at the order of operations: horizontal shifts and stretches/compressions are applied before vertical transformations.

Explore how the graph of f(x) = 2|x+7| + 8 is transformed from the graph of g(x) = |x|. Understand horizontal shifts, vertical stretches, and vertical shifts in function transformations. Suitable for advanced high school students studying functions and transformations.

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