To graph the function $g(x) = -3(x + 3)^2 - 5$ using transformations of the basic function $f(x) = x^2$, we can break down the equation into a sequence of transformations applied to the parent function $f(x) = x^2$.

Step-by-Step Transformation:

1. Start with the parent function $f(x) = x^2$.

2. Horizontal Shift: The term $(x + 3)$ inside the square indicates a horizontal shift. Specifically, the graph of $f(x)$ is shifted 3 units to the left.

3. Vertical Stretch and Reflection: The coefficient $-3$ outside the squared term affects both the stretching and the reflection:

   • The negative sign indicates that the graph is reflected about the x-axis.
   • The coefficient 3 indicates a vertical stretch by a factor of 3.

4. Vertical Shift: The term $-5$ at the end of the function indicates that the entire graph is shifted 5 units down.

Conclusion:

Based on these transformations, the correct options are:

• B: Stretch the graph vertically by a factor of 3.
• E: Shift the graph 3 units to the left.
• I: Reflect the graph about the x-axis.
• J: Shift the graph 5 units down.

Final Answer:

• B, E, I, J are the correct transformations needed to graph the function $g(x) = -3(x + 3)^2 - 5$.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. How does a horizontal shift differ from a vertical shift in a graph?
2. What effect does a vertical stretch have on the shape of a parabola?
3. How do you determine the direction of a reflection in a graph?
4. What happens to a graph when you apply multiple transformations in a different order?
5. How would the graph change if the equation was $g(x) = -3(x - 3)^2 - 5$?

Tip: Always apply horizontal shifts first, then vertical stretches/compressions, reflections, and finally vertical shifts when transforming graphs.