To solve this problem, we need to determine the transformations that are applied to the base function $f(x) = x^2$ to obtain the new function $g(x) = -2(x + 3)^2 - 2$.

Step-by-Step Analysis:

1. Starting Function:

   • The base function is $f(x) = x^2$, which is a standard parabola opening upwards with its vertex at the origin $(0, 0)$.

2. Transformation to $g(x) = -2(x + 3)^2 - 2$:

   • Horizontal Shift:

     • The expression inside the square, $(x + 3)$, indicates a shift.
     • $x + 3$ implies a shift 3 units to the left (opposite to the sign inside the parentheses).
     • Corresponding Transformation: $A$ (Shift the graph 3 units to the left).

   • Reflection:

     • The negative sign in front of the expression $-2(x + 3)^2$ indicates a reflection about the x-axis.
     • Corresponding Transformation: $B$ (Reflect the graph about the x-axis).

   • Vertical Stretch:

     • The factor of $-2$ outside the square implies a vertical stretch by a factor of 2 (the negative sign is for reflection, and the absolute value 2 is the stretch factor).
     • Corresponding Transformation: $E$ (Stretch the graph vertically by a factor of 2).

   • Vertical Shift:

     • The $-2$ at the end of the function $-2(x + 3)^2 - 2$ indicates a shift 2 units down.
     • Corresponding Transformation: $C$ (Shift the graph 2 units down).

Conclusion:

The transformations needed to graph $g(x) = -2(x + 3)^2 - 2$ from $f(x) = x^2$ are:

• A: Shift the graph 3 units to the left.
• B: Reflect the graph about the x-axis.
• C: Shift the graph 2 units down.
• E: Stretch the graph vertically by a factor of 2.

These are the correct transformations.

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Would you like more details on how these transformations affect the graph?

Related Questions:

1. What would happen if the function was $g(x) = 2(x + 3)^2 + 2$?
2. How would the graph change if the function was $g(x) = -\frac{1}{2}(x - 3)^2 - 2$?
3. What is the vertex of the function $g(x) = -2(x + 3)^2 - 2$?
4. How do horizontal and vertical stretches affect the graph of a quadratic function?
5. What is the difference between reflecting a graph about the x-axis and the y-axis?

Tip: When analyzing transformations, always apply the horizontal shifts and reflections first, followed by vertical transformations.