The graph shows a parabola that opens upwards, and we are looking for the equation that represents this parabola. The general form of a parabola in vertex form is:

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

where $(h, k)$ is the vertex of the parabola.

Step 1: Identify the vertex

From the graph, we see that the vertex of the parabola is at $(-5, -3)$.

Step 2: Match the vertex with the given equations

• In option C, the equation is $y = (x + 5)^2 - 3$. This equation has the vertex $(-5, -3)$, which matches the graph.
• The other options have different vertices:
  • Option A: Vertex at $(-4, 2)$
  • Option B: Vertex at $(2, -4)$
  • Option D: Vertex at $(3, 5)$

Thus, the correct equation is Option C: $y = (x + 5)^2 - 3$.

Would you like more details or have any other questions?

Here are five related questions to deepen your understanding:

1. How do you determine the direction in which a parabola opens from its equation?
2. What role does the "a" value play in the equation of a parabola?
3. How can you find the x-intercepts of a parabola from its equation?
4. What is the significance of the axis of symmetry in a parabola's graph?
5. How would the graph change if the constant term in the equation were increased?

Tip: The vertex form $y = a(x - h)^2 + k$ is very useful for quickly identifying the vertex of a parabola and its direction of opening.