To write the equation of the quadratic function in vertex form \( y = a(x - h)^2 + k \), we need to identify the vertex \((h, k)\) and the value of \(a\), which indicates the direction of the parabola and its width.
From the question options and given choices, it seems that the vertex of the quadratic function is at \( (3, 2) \) since the choices are centered around this point.
Next, we have to determine the value of \( a \):
- If the parabola opens upwards, \( a > 0 \),
- If the parabola opens downwards, \( a < 0 \).
Since the equation options all seem to have a negative \( a \), we can conclude that the parabola opens downwards.
Without seeing the actual graph, I can't definitively identify whether the value of \( a \) is \( -12 \) or \( -\frac{1}{2} \) or any other value. However, based on your options, we assume:
- The vertex is at \( (3, 2) \).
- The coefficient \( a \) determines the width and orientation of the parabola.
Assuming the parabola is narrow and opens downwards, we can pick \( a = -12 \).
Consequently, if we adopt these assumptions, the equation in vertex form would be:
\[ y = -12(x - 3)^2 + 2 \]
Thus, based on the choices provided and assuming a downward-opening parabola centered on the vertex you mentioned, the equation is:
−12(x−3)²+2