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

Shifted  ------   by ------   Number   units

Shifted  ------   by ------   Number   units

Vertically stretched by a factor of   ----- Number

represents the graph of this function, considering the location (quadrant) of the vertex and the direction that the graph opens

To obtain the graph of \( f(x) = 2|x+7| + 8 \) from the graph of \( g(x) = |x| \), the following transformations occur:

1. **Shifted horizontally** by **7 units** to the **left**:
   - This corresponds to the \( +7 \) inside the absolute value. Shifting left by 7 units moves the vertex from \( (0, 0) \) to \( (-7, 0) \).

2. **Shifted vertically** by **8 units** **upwards**:
   - The \( +8 \) outside the absolute value causes the graph to shift 8 units up, moving the vertex from \( (-7, 0) \) to \( (-7, 8) \).

3. **Vertically stretched by a factor of 2**:
   - The coefficient 2 in front of the absolute value causes a vertical stretch by a factor of 2, making the graph steeper compared to the original \( g(x) = |x| \).

### Location and Direction of the Graph:

- **Location of the Vertex**: The vertex of the transformed graph is at \( (-7, 8) \).
- **Quadrant**: Since the vertex is at \( (-7, 8) \), the vertex is located in the **second quadrant** of the coordinate plane.
- **Direction**: The graph opens **upwards** since the coefficient in front of the absolute value is positive.

### Summary:

- **Shifted horizontally** by **7 units** to the **left**.
- **Shifted vertically** by **8 units** **upwards**.
- **Vertically stretched by a factor of 2**.

Would you like more details or have any questions?

### Follow-Up Questions:
1. How does changing the coefficient from 2 to -2 affect the graph of \( f(x) \)?
2. What is the impact on the graph if the vertical shift is changed to a downward shift by 8 units?
3. How would the graph change if the horizontal shift was to the right instead of to the left?
4. What would be the new vertex if the function was \( f(x) = 3|x-4| + 6 \)?
5. How does the transformation affect the width of the graph compared to \( g(x) = |x| \)?

### Tip:
When analyzing graph transformations, remember that horizontal shifts inside the function affect the x-axis, and vertical transformations outside the function affect the y-axis.

Explore the transformation of the function f(x)=2|x+7|+8 from the original absolute value function g(x)=|x|. Learn about horizontal and vertical shifts, vertical stretching, and analyze the location of the vertex and the direction of the graph. Suitable for high school students studying functions and transformations.

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