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To map the function \( f(x) = 6(x-5)^2 - 1 \) back to \( f(x) = x^2 \), we need to reverse all the transformations applied to \( x^2 \) to obtain the given function. Let us analyze these transformations step by step:

### 1. Start with the base function: \( x^2 \)
   - This is the simplest quadratic function.

### 2. **Horizontal shift**:
   - The term \( (x - 5) \) inside the square indicates a horizontal shift **5 units to the right**.
   - To reverse this, shift **5 units to the left**.

### 3. **Vertical scaling**:
   - The coefficient \( 6 \) outside the squared term indicates a vertical stretch by a factor of 6.
   - To reverse this, divide the function's vertical values by 6 (apply a vertical compression by \( \frac{1}{6} \)).

### 4. **Vertical shift**:
   - The term \( -1 \) indicates a vertical shift **1 unit downward**.
   - To reverse this, shift **1 unit upward**.

### Final steps to map back to \( x^2 \):
- Reverse the vertical shift: Add 1 to the function.
- Reverse the vertical scaling: Divide by 6.
- Reverse the horizontal shift: Shift 5 units to the left.

### Writing these transformations mathematically:
1. Start with \( f(x) = 6(x - 5)^2 - 1 \).
2. Add 1: \( f(x) + 1 = 6(x - 5)^2 \).
3. Divide by 6: \( \frac{f(x) + 1}{6} = (x - 5)^2 \).
4. Shift horizontally 5 units to the left: \( \sqrt{\frac{f(x) + 1}{6}} + 5 = x^2 \).

These steps reverse all transformations to recover the original \( f(x) = x^2 \).

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Let me know if you want further details or clarification!

Here are 5 questions that build on this explanation:

1. How would the transformations change if the coefficient \( 6 \) were replaced by \( -6 \)?
2. What effect does the term \( (x - 5) \) have on the function's graph?
3. How would you describe the sequence of transformations in reverse order?
4. Can you represent the inverse of this transformation step-by-step as a general rule?
5. How do these transformations affect the vertex of the parabola?

**Tip:** Always perform transformations in the order of parentheses, scaling, and then shifts for clarity.

Identify the transformations that map the function f(x) = 6(x - 5)^2 - 1 back to f(x) = x^2.

Understand the transformations required to map the quadratic function f(x) = 6(x - 5)^2 - 1 back to the standard form f(x) = x^2. This step-by-step guide explains horizontal shifts, vertical scaling, and vertical shifts. Suitable for high school algebra students.

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