Now that you’ve identified the correct transformations needed to graph the function $g(x) = -3(x + 3)^2 - 5$, the next step is to actually plot the graph using these transformations.

Steps to Graph $g(x) = -3(x + 3)^2 - 5$:

1. Start with the graph of $f(x) = x^2$:

   • This is a standard parabola that opens upwards with its vertex at the origin (0, 0).

2. Apply the Horizontal Shift:

   • Shift the graph 3 units to the left. The vertex of the parabola moves from (0, 0) to (-3, 0).

3. Apply the Vertical Stretch and Reflection:

   • Reflect the graph about the x-axis (it now opens downwards).
   • Stretch it vertically by a factor of 3, making the parabola narrower and steeper.

4. Apply the Vertical Shift:

   • Shift the graph 5 units down. The vertex moves from (-3, 0) to (-3, -5).

Graph Description:

• The vertex of the transformed parabola will be at (-3, -5).
• The parabola opens downward because of the reflection across the x-axis.
• It’s narrower than the standard parabola due to the vertical stretch by a factor of 3.

How to Use the Graphing Tool:

• Start by plotting the basic parabola $f(x) = x^2$.
• Move the vertex 3 units to the left to (-3, 0).
• Reflect the parabola across the x-axis.
• Stretch the parabola vertically so it becomes narrower.
• Shift the entire graph 5 units down to place the vertex at (-3, -5).

After applying these transformations, you should see a parabola opening downward, with its vertex at (-3, -5) and stretched vertically.

Would you like more details or have any questions?

Here are some related questions to further explore the topic:

1. What happens to the graph of $f(x) = x^2$ if you apply a vertical shrink instead of a stretch?
2. How do you determine the direction of the parabola opening based on the coefficient of $x^2$?
3. How would the graph change if the equation was $g(x) = -3(x - 2)^2 + 4$?
4. What effect does a horizontal stretch have on the width of the parabola?
5. How would you graph the function $g(x) = 2(x + 3)^2 + 5$?

Tip: When graphing functions using transformations, always pay attention to the order in which transformations are applied—start with shifts, then stretches/reflections, and finish with vertical shifts.