The image you provided contains a graph of a parabola, and the task is to find the equation of the parabola based on the graph.

To analyze:

• The graph shows a parabola opening upwards.

• The vertex of the parabola appears to be at the point $(0, -4)$, which suggests that the equation has a vertex form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Here, $(h, k) = (0, -4)$, so the equation becomes: $y = a(x - 0)^2 - 4 \quad \text{or} \quad y = ax^2 - 4$

• To find the value of $a$, we can use another point from the graph. From the graph, one point is $(1, -3)$. Substituting $x = 1$ and $y = -3$ into the equation $y = ax^2 - 4$:

  $-3 = a(1)^2 - 4$ $-3 = a - 4$ $a = 1$

Thus, the equation of the parabola is: $y = x^2 - 4$

The multiple-choice answers should confirm this equation.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do we derive the vertex form of a quadratic equation?
2. What happens to the graph when the value of $a$ changes in the equation $y = ax^2 + bx + c$?
3. Can the equation $y = x^2 - 4$ be written in standard form?
4. How do you find the vertex of a parabola from the general form of a quadratic equation?
5. What are the effects of shifting the graph horizontally or vertically?

Tip: Always check for key points on the graph, like the vertex and intercepts, to derive an accurate equation.