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.