The given quadratic function is:

$f(x) = -5(x + 3)^2$

To identify the graph of this function, let's break down its key characteristics:

1. Vertex form: The function is in vertex form, $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case:

   • $a = -5$
   • $h = -3$
   • $k = 0$

   So, the vertex is $(-3, 0)$.

2. Direction of the parabola: Since $a = -5$, which is negative, the parabola opens downward.

3. Stretch/Compression: The value of $|a| = 5$ means the parabola is vertically stretched by a factor of 5 compared to the basic parabola $y = x^2$. This makes the graph narrower.

4. Shift: The parabola is shifted 3 units to the left from the origin, as indicated by the $(x + 3)$ term.

Summary of the graph:

• Vertex: $(-3, 0)$
• Opens downward (due to $a = -5$)
• Narrower than the basic parabola
• No vertical shift (since $k = 0$)

Would you like more details or have any specific questions on this?

Here are 5 related questions to expand your understanding:

1. What is the effect of changing the value of $a$ on the graph of a parabola?
2. How would the graph change if the function were $f(x) = 5(x + 3)^2$?
3. What would be the equation if the vertex were shifted 2 units up?
4. How does the graph of $f(x) = -5(x + 3)^2$ compare to $f(x) = -x^2$?
5. How can you find the x-intercepts of the function $f(x) = -5(x + 3)^2$?

Tip: The vertex form of a quadratic function is particularly useful for quickly identifying the vertex and the direction in which the parabola opens.