https://file.aiquickdraw.com/mathbot/file/1729712962514p4cok4aq.jpg

The given function is:

\[
y = \frac{1}{5}(x + 3)^2 - 2
\]

Let's analyze this function step by step:

### 1. Vertex Form
This is a quadratic function in vertex form:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex, and \(a\) determines the shape and direction of the parabola. For the given equation:
- \(h = -3\) (the sign is flipped inside the parentheses),
- \(k = -2\) (the vertical shift),
- \(a = \frac{1}{5}\), which means the parabola opens upwards and is relatively wide due to the small value of \(a\).

So, the vertex of the parabola is at \((-3, -2)\).

### 2. Parabola Characteristics
- **Direction:** Since \(a > 0\), the parabola opens **upward**.
- **Width:** The factor \(\frac{1}{5}\) means the parabola is wider than the standard parabola, which opens with a narrower curve.
- **Vertex:** The vertex is at \((-3, -2)\), so the parabola is shifted to the left by 3 units and down by 2 units.

### 3. Graph Matching
Now, let's compare this with the provided graphs:
- We need to look for a graph with its vertex at \((-3, -2)\) and opening upwards with a wide curve.

From the given graphs, the **bottom left graph** has the vertex at \((-3, -2)\) and opens upwards. This matches the equation provided.

### Answer: The graph that corresponds to the given equation is the **bottom left graph**.

---

Would you like more details or have any questions about this?

Here are some related questions to explore:
1. How do you find the axis of symmetry for a parabola in vertex form?
2. What role does the coefficient \(a\) play in determining the width of a parabola?
3. How does shifting the graph horizontally and vertically affect its equation?
4. What is the general form of a quadratic equation, and how does it differ from vertex form?
5. How can you convert a quadratic equation from standard form to vertex form?

**Tip:** When graphing parabolas, always check the vertex and the direction of opening first to make matching with the graph easier.

Match the function y = 1/5(x + 3)^2 - 2 with its graph.

Learn how to match the quadratic function y = 1/5(x + 3)^2 - 2 with its graph. This problem involves identifying the vertex, direction of the parabola, and analyzing how changes in 'a', 'h', and 'k' affect the graph. Ideal for Grades 9-10 students.

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