The given graph represents a parabola, and the equation is required in the form:

$y = a(x - h)^2 + k$

Step 1: Identify the vertex

From the graph, the vertex of the parabola is at the point $(-2, -4)$. This gives us the values of $h$ and $k$, where:

• $h = -2$
• $k = -4$

So, the equation so far is:

$y = a(x + 2)^2 - 4$

Step 2: Find the value of $a$

To determine $a$, we need another point on the graph. From the graph, one clear point is $(0, 0)$.

Substitute $x = 0$ and $y = 0$ into the equation:

$0 = a(0 + 2)^2 - 4$

Simplifying:

$0 = a(4) - 4$ $4a = 4$ $a = 1$

Final Equation:

The equation of the parabola is:

$y = (x + 2)^2 - 4$

Let me know if you would like more details or have any questions.

Here are some related questions:

1. How does the vertex form of a parabola help in identifying the graph’s transformation?
2. What role does the value of $a$ play in the shape of the parabola?
3. Can you explain how to complete the square to convert a standard quadratic equation to vertex form?
4. What are other forms of a parabola's equation and when are they useful?
5. How can you determine if a parabola opens upwards or downwards from its equation?

Tip: The coefficient $a$ in the vertex form controls the direction and width of the parabola: positive $a$ makes it open upwards, and larger values of $|a|$ make it narrower.