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Let's examine the function \( g(x) = 3(x - 6)^2 - 4 \) and determine the correct transformation from the basic function \( f(x) = x^2 \).

1. **Vertical Stretch**: The coefficient 3 in front of \( (x - 6)^2 \) indicates a vertical stretch by a factor of 3. This stretches the graph of \( f(x) = x^2 \) by making it narrower.
   
2. **Horizontal Translation**: The term \( (x - 6) \) indicates a shift to the right by 6 units, as \( x - h \) inside the square represents a horizontal shift to the right when \( h > 0 \).

3. **Vertical Translation**: The \( -4 \) at the end of the equation represents a downward shift of 4 units, as subtracting a constant outside the squared term moves the graph down.

Based on these transformations:
- The graph of \( f(x) = x^2 \) is vertically stretched, shifted 6 units to the right, and 4 units down.

The correct answer is:
**C. It is the graph of \( f(x) = x^2 \) vertically stretched, and then translated 6 units down and 4 units to the right.**

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How do different values of vertical stretches or compressions affect the graph?
2. What changes would occur if the equation had a negative coefficient instead of positive?
3. How would the transformation differ if we added a constant inside the squared term?
4. What are the effects of horizontal shifts on parabolas?
5. How can we identify transformations just by looking at the equation?

**Tip:** Always remember that adding or subtracting inside the function (next to \( x \)) affects horizontal shifts, while changes outside the function affect vertical shifts.

Each statement describes a transformation of the graph of f(x) = x^2. Which statement correctly describes the graph of the function shown below?

g(x) = 3(x - 6)^2 - 4

Learn how to identify transformations of the quadratic function f(x) = x^2 in the form g(x) = a(x - h)^2 + k. Understand vertical stretches, horizontal translations, and vertical translations with this detailed explanation. Suitable for high school students in Grades 9-12.

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