To determine the correct vertex form of the parabola described, we need to identify the vertex of the parabola as indicated in the coordinate plane.
A parabola in vertex form is given by the equation: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex. Given that the parabola is concave down, the value of \(a\) will be negative.
From the given choices, we can analyze each one for the vertex's coordinates:
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\(y = -\frac{1}{4}(x - 4)^2 + 2\)
- Vertex is at (4, 2).
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\(y = -\frac{1}{4}(x + 2)^2 + 4\)
- Vertex is at (-2, 4).
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\(y = -\frac{1}{4}(x - 2)^2 + 4\)
- Vertex is at (2, 4).
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\(y = -\frac{1}{4}(x + 4)^2 + 2\)
- Vertex is at (-4, 2).
Considering the x-axis ranges from -10 to 2 and the y-axis ranges from -4 to 4, the vertex should likely fall within these ranges while also being consistent with the shape of a concave down parabola.
However, without specific coordinates for the vertex from the image, a confident selection can still be made based on the general trends of the functions.
Since the parabola is stated to pass through the 2nd, 3rd, and 4th quadrants, it suggests that the vertex likely needs to be in the 2nd quadrant (x negative and y positive).
The function \(y = -\frac{1}{4}(x + 2)^2 + 4\) has its vertex at (-2, 4), which falls in the 2nd quadrant and could maintain the criteria described. This makes it a probable candidate:
Thus, the most fitting response based on the clues given would be: y = −\(\frac{1}{4}\)(x + 2)² + 4.
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