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Use the given graph of f(x) and sketch in the same grid above, -f(x+4)-3

To sketch the transformation of the function \(-f(x+4) - 3\) on the same grid, we need to apply the following steps:

1. **Horizontal Shift:** The term \(f(x+4)\) represents a **left shift** of the function by 4 units. This means every point on the original function moves 4 units to the left.
   
2. **Reflection in the x-axis:** The negative sign before the function \(-f(x+4)\) reflects the function across the x-axis. This flips all the y-values of the points on the graph. So, if a point was at \((x, y)\), after reflection, it will be at \((x, -y)\).

3. **Vertical Shift:** The \(-3\) represents a **downward shift** by 3 units. After reflecting the function, every point is shifted 3 units down.

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### Instructions to sketch:

1. Start by shifting the entire graph of \(f(x)\) **4 units to the left**.
2. Then **reflect** the new graph across the x-axis, flipping it upside down.
3. Finally, **shift the reflected graph downward by 3 units**.

Let me know if you'd like me to guide you through each step further. Would you like to see an example of one of the points transformed?

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Here are 5 relative questions for further exploration:

1. How does horizontal shifting affect the shape of the graph?
2. What happens when a function is reflected over the x-axis?
3. Can you identify the transformations in terms of function composition?
4. What effect does adding or subtracting from \(x\) have on the function’s behavior?
5. How can you describe the domain and range of the transformed function?

**Tip:** Always apply transformations step by step to avoid confusion, especially when dealing with multiple changes like shifts and reflections.

Learn how to sketch the transformation of the function -f(x+4)-3 on the same grid. This problem involves understanding reflections over the x-axis, horizontal shifts to the left, and vertical translations downward. Suitable for students in Grades 10-12.

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