To determine the correct vertex form of the parabola equation based on the provided information, we need to understand a few key aspects of the vertex form of a parabola, which is generally written as:
\[ y = a(x - h)^2 + k \]
Here, \((h, k)\) represents the vertex of the parabola, and \(a\) determines the direction and "width" of the parabola.
From your description:
- The parabola is concave down, indicating that \(a\) should be negative.
- The given coordinate plane suggests that the vertex is likely positioned within the value ranges described for both axes. Since the parabola is drawn through the 2nd, 3rd, and 4th quadrants, it indicates that the vertex must have a positive y-value.
Considering the options provided, we examine their vertex components:
-
Option 1: \(y = -\frac{1}{4}(x + 4)^2 + 2\)
- Vertex: \((-4, 2)\) (not in the desired y-range).
-
Option 2: \(y = -\frac{1}{4}(x + 2)^2 + 4\)
- Vertex: \((-2, 4)\) (a valid option as the y-value is positive).
-
Option 3: \(y = -\frac{1}{4}(x - 4)^2 + 2\)
- Vertex: \((4, 2)\) (also a valid option but located in quadrant 1).
-
Option 4: \(y = -\frac{1}{4}(x - 2)^2 + 4\)
- Vertex: \((2, 4)\) (also in quadrant 1, not fitting the description).
Out of these options, the second option, \(y = -\frac{1}{4}(x + 2)^2 + 4\) appears to be the best fit considering the direction, vertex location and that it resides in the necessary quadrants.
Thus, the correct answer is:
y = -\frac{1}{4}(x + 2)^2 + 4.