The problem shown in the image involves determining the equation of a quadratic function given its graph. Let's walk through each step based on what’s visible:

Step 1:

• Vertex identification:
  From the graph, the vertex is given as $(-3, -2)$. This is correct, and this point will be useful for finding the standard form of the quadratic equation.

Step 2:

• Quadratic function standard form:
  The standard form of a quadratic function is: $f(x) = a(x - h)^2 + k$ Where $(h, k)$ is the vertex of the parabola. Since the vertex is $(-3, -2)$, we substitute $h = -3$ and $k = -2$: $f(x) = a(x + 3)^2 - 2$

Step 3:

• Finding the value of $a$:
  To find $a$, we use the additional point $(0, 7)$ on the graph. Substituting $x = 0$ and $f(x) = 7$ into the equation: $7 = a(0 + 3)^2 - 2$ Simplifying: $7 = a(9) - 2$ Add 2 to both sides: $9 = 9a$ Solving for $a$: $a = 1$

Thus, the quadratic function in standard form is: $f(x) = (x + 3)^2 - 2$

Final Answer:

The quadratic function that describes the parabola is: $f(x) = (x + 3)^2 - 2$

Let me know if you'd like more details, or if you have any questions!

Here are some follow-up questions:

1. How can you verify that the quadratic function is correct?
2. What is the effect of the parameter $a$ on the shape of the parabola?
3. How would the equation change if the parabola opened downward?
4. Can you rewrite the quadratic function in expanded form?
5. What are the x-intercepts of the parabola?

Tip: The value of $a$ determines the "width" and "direction" of the parabola. If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.