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To identify the correct equation for the given graph, let’s analyze the visual characteristics:

1. **Vertex of the Parabola**: The graph's vertex appears to be at the point \((-2, -1)\).
2. **Orientation of the Parabola**: The parabola opens upward, indicating a positive leading coefficient in the equation.
3. **Standard Form of Parabola**: Since the vertex is at \((-2, -1)\), the equation should resemble the vertex form of a parabola:
   \[
   f(x) = a(x + h)^2 + k
   \]
   where \((h, k)\) is the vertex of the parabola.

Given this, the equation that matches the vertex at \((-2, -1)\) is:

**Option (c)**: \( f(x) = (x + 2)^2 - 1 \)

This choice correctly places the vertex at \((-2, -1)\) and maintains an upward opening due to the positive leading term.

Would you like further details on the solution, or do you have any questions?

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Here are some additional questions to deepen understanding:
1. How do you determine the vertex of a parabola from its equation?
2. What effect does a positive or negative coefficient have on the orientation of a parabola?
3. How does shifting the parabola horizontally and vertically affect its equation?
4. What is the importance of the vertex form of a parabola in identifying its key features?
5. How would the equation change if the parabola opened downward?

**Tip:** When identifying a parabola's equation from a graph, locate its vertex and determine the direction it opens to match it with the correct form.

Write an equation for the given parabola on the graph.

Learn how to determine the equation of a parabola with a vertex at (-2, -1) by analyzing a graph. Understand the vertex form of quadratic functions and how vertex and orientation help in writing equations. This problem is suitable for students in Grades 8-10.

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