To determine the correct vertex form of the equation for the concave down parabola that passes through the 2nd, 3rd, and 4th quadrants, we first need to identify the vertex of the parabola based on the provided information.
Since the parabola is concave down and passes through the 2nd, 3rd, and 4th quadrants, we can infer that its vertex must be located in the 2nd quadrant (where x is negative and y is positive). The vertex form of a quadratic function is usually represented as:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola.
Based on the provided options and the fact that the vertex should be in the negative x-domain and positive y-domain, the possibility of the vertex’s x-coordinate seems to be at points like -2 or -4, with a higher (and positive) y-coordinate.
Now let's analyze the responses given:
-
\( y = -\frac{1}{4}(x - 4)^2 + 2 \)
- The vertex would be at (4, 2) — In the 1st quadrant.
-
\( y = -\frac{1}{4}(x - 2)^2 + 4 \)
- The vertex would be at (2, 4) — In the 1st quadrant.
-
\( y = -\frac{1}{4}(x + 4)^2 + 2 \)
- The vertex would be at (-4, 2) — In the 2nd quadrant.
-
\( y = -\frac{1}{4}(x + 2)^2 + 4 \)
- The vertex would be at (-2, 4) — In the 2nd quadrant.
Given that the vertex must be in the 2nd quadrant and also considering the nature of the parabolas:
- The correct answers would likely be those that have their vertex coordinates fitting the description (either (-4, 2) or (-2, 4)) because both represent locations above the x-axis (positive y-coordinates) and to the left of the y-axis (negative x-coordinates).
Thus, both:
- \( y = -\frac{1}{4}(x + 4)^2 + 2 \)
- \( y = -\frac{1}{4}(x + 2)^2 + 4 \)
are plausible, but since we are trying to find only one of them based on the context provided, I will select:
- \( y = -\frac{1}{4}(x + 4)^2 + 2 \)
as the vertex form for this context. Therefore, the final answer is:
\[ y = -\frac{1}{4}(x + 4)^2 + 2 \]
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