https://file.aiquickdraw.com/mathbot/file/172611386337236iihzix.jpg

To determine which equation best matches the graph shown, let's analyze the characteristics of the parabola.

### Key Features of the Parabola in the Graph:
1. **Vertex**: The vertex of the parabola is at \((-6, -5)\). This can be observed as the lowest point of the U-shaped curve.
2. **Direction**: The parabola opens **upwards** because it has a "U" shape rather than an "n" shape.
3. **Stretch/Compression**: The parabola is narrower than the standard \(y = x^2\), indicating a vertical stretch. If the parabola opens upwards, the coefficient of the squared term should be positive.

### Analyzing the Given Equations:
The given options are in **vertex form**:  
\[ y = a(x - h)^2 + k \]  
Where \((h, k)\) is the vertex of the parabola.

Given equations:
1. \( y = -4(x + 6)^2 - 5 \)
2. \( y = -4(x - 6)^2 - 5 \)
3. \( y = 4(x + 6)^2 - 5 \)
4. \( y = 4(x - 6)^2 - 5 \)

### Steps to Find the Correct Equation:
1. **Vertex Matching**:  
   The vertex is \((-6, -5)\), so \(h = -6\) and \(k = -5\). This means the expression inside the parentheses should be \((x + 6)\).

2. **Coefficient Sign**:  
   Since the parabola opens upwards, \(a > 0\). So, we need a positive coefficient. The correct form should have a coefficient of \(4\) (not \(-4\)).

3. **Select the Correct Option**:  
   The correct equation must be \(y = 4(x + 6)^2 - 5\) as it satisfies both the vertex condition and the coefficient sign.

### Conclusion:
The correct answer is:

\[
\boxed{y = 4(x + 6)^2 - 5}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you convert a parabola from standard form to vertex form?
2. What does the coefficient \(a\) in the vertex form tell us about the parabola?
3. How do we determine the direction in which a parabola opens?
4. What effect does the value of \(k\) have on the parabola?
5. How would the equation change if the parabola were reflected over the x-axis?

### Tip:
Remember, the vertex form of a parabola makes it easy to identify the vertex and direction of the parabola's opening.

Which equation best matches the graph shown below?

Learn how to match a quadratic equation in vertex form to a parabola graph. This problem involves identifying the vertex, direction of opening, and stretch factor of a quadratic function in Grades 9-11.

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