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The given question asks to graph the function \( f(x) = (x + 5)^2 - 4 \) by using transformations starting from the parent function \( y = x^2 \). Let's walk through the steps of how this transformation works:

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### **Transformations Breakdown:**
1. **Start with the parent function**:  
   \( y = x^2 \).  
   This is a standard parabola that opens upwards, with the vertex at \( (0, 0) \).

2. **Horizontal Shift**:  
   The function \( f(x) = (x + 5)^2 - 4 \) has a horizontal shift.  
   - \( x + 5 \) shifts the graph **5 units to the left** (because the shift is opposite to the sign inside the parentheses).

3. **Vertical Shift**:  
   The term \( -4 \) moves the graph **4 units down**.  
   - So, the new vertex will be at \( (-5, -4) \).

4. **No Reflection or Stretch/Compression**:  
   - The coefficient inside the squared term is 1, so the shape (width) of the parabola remains the same as the parent function. 
   - There is no reflection (as there is no negative sign).

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### **Summary of Transformations:**
- **Shift left by 5 units**: \( x + 5 \).
- **Shift down by 4 units**: \( -4 \).
- **No stretching/compression or reflection**.

Thus, the vertex of the transformed parabola will be at **(-5, -4)**, and it will maintain the same "U" shape as the original \( y = x^2 \). 

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### **Graphing Instructions:**
1. Plot the vertex at \( (-5, -4) \).
2. Since the graph is not stretched or compressed, follow the usual pattern:  
   - From the vertex, move 1 unit right/left and 1 unit up to get additional points.
   - Then move 2 units right/left and 4 units up (since \( 2^2 = 4 \)).
3. Draw a smooth curve passing through these points.

Would you like me to create an image of this graph, or do you need more clarifications?

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### Related Questions:
1. What happens if we change \( (x + 5)^2 - 4 \) to \( -(x + 5)^2 - 4 \)?
2. How do horizontal and vertical shifts affect the domain and range of a function?
3. What would happen if we introduced a stretching factor, like \( 2(x + 5)^2 - 4 \)?
4. How would the graph change if the equation was \( (x + 5)^2 + 4 \) instead?
5. What is the algebraic process to find the vertex of any quadratic function in vertex form?

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**Tip:** When dealing with transformations, always check the sign inside the parentheses for horizontal shifts—it behaves opposite to intuition.

Graph the function f(x) = (x + 5)^2 - 4 by starting with the graph of y = x^2 and using transformations (shifting, stretching/compressing, and/or reflecting).

Learn how to graph the quadratic function f(x) = (x + 5)^2 - 4 by using transformations. This problem involves a horizontal and vertical shift based on the parent function y = x^2. Suitable for students in Grades 8-10.

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