To determine the correct vertex form of the parabola, we can analyze the given options and the characteristics described.
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Vertex Form of a Parabola: The vertex form of a quadratic function is generally expressed as: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola.
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Concavity: Since the parabola is described as concave down, we know that the coefficient \(a\) will be negative.
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Determining the Vertex from the Options: We must choose the vertex that aligns with the characteristics of the graph observed in the coordinate plane described.
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Candidates Analysis: To find the vertex based on the provided options:
- Option 1: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
- Option 2: \(y = -\frac{1}{4}(x - 2)^2 + 4\)
- Option 3: \(y = -\frac{1}{4}(x - 4)^2 + 2\)
- Option 4: \(y = -\frac{1}{4}(x + 2)^2 + 4\)
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Identify the Vertex:
- For Option 1, vertex is at \((-4, 2)\).
- For Option 2, vertex is at \((2, 4)\).
- For Option 3, vertex is at \((4, 2)\).
- For Option 4, vertex is at \((-2, 4)\).
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Choose the Correct Option Based on Parabola's Location: Since the parabola is said to pass through 2nd, 3rd, and 4th quadrants, the vertex must be located between the quadrants effectively along the y-axis. Considering the y-coordinates and which x-coordinates still allow it to arch downwards while intersecting the defined quadrants, we find:
- Quadrants:
- 2nd quadrant: where \(x < 0\) and \(y > 0\).
- 3rd quadrant: where \(x < 0\) and \(y < 0\).
- 4th quadrant: where \(x > 0\) and \(y < 0\).
Considering the choices, Option 1 has a vertex positioned into quadrant 2 with a maximum height of 2 and opens downward, thereby intersecting the 3rd quadrant successfully.
Thus the correct answer is:
Option 1: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
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