In the second part of the problem, you're asked to use a graphing tool to plot the function $g(x) = -2(x + 3)^2 - 2$ based on the transformations identified in the first part. Here’s how you would graph the function step-by-step:

Step-by-Step Guide to Graphing $g(x)$:

1. Start with the Base Graph:

   • Begin with the graph of $f(x) = x^2$, which is a parabola opening upwards with its vertex at the origin $(0, 0)$.

2. Apply Transformations:

   • Horizontal Shift (3 units to the left):

     • Move the entire graph 3 units to the left. The new vertex of the parabola is now at $(-3, 0)$.

   • Reflection about the x-axis:

     • Reflect the graph over the x-axis. The parabola, which initially opened upwards, will now open downwards. The vertex remains at $(-3, 0)$.

   • Vertical Stretch by a factor of 2:

     • Stretch the parabola vertically by a factor of 2, making it narrower. This affects the "width" of the parabola, making it steeper.

   • Vertical Shift (2 units down):

     • Finally, shift the entire graph 2 units down. The vertex moves to $(-3, -2)$.

Final Graph:

• The vertex of the transformed parabola is at $(-3, -2)$.
• The parabola opens downwards.
• It is narrower than the standard parabola due to the vertical stretch.

Using the Graphing Tool:

• Plot the vertex at $(-3, -2)$.
• Draw the parabola opening downwards with a narrower shape compared to the base graph of $f(x) = x^2$.
• Ensure the axis of symmetry is the line $x = -3$.

This should correctly represent the function $g(x) = -2(x + 3)^2 - 2$ based on the transformations.

Do you have any questions on how to use the graphing tool or need further clarification on the transformations?

Related Questions:

1. How would the graph change if the coefficient in front of $(x + 3)^2$ was $-1$ instead of $-2$?
2. What would be the vertex of the graph if the function was $g(x) = -2(x - 3)^2 + 4$?
3. How do you determine the axis of symmetry for a quadratic function?
4. How does a vertical stretch or shrink affect the shape of a parabola?
5. What are the effects of combining horizontal and vertical shifts on the vertex of the graph?

Tip: When using graphing tools, always mark the vertex and the direction the parabola opens first. This will help you sketch the graph more accurately.