To find the correct equation of a parabola with a vertex at the point (2, -5) and a line of symmetry at \(x = 2\), we can start by recalling the vertex form of a parabola:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex. In this case, our vertex \((h, k)\) is \((2, -5)\). Thus, we can write the equation as:
\[ y = a(x - 2)^2 - 5 \]
Next, we need to identify the correct option from the list given. We also notice that the value of \(a\) will affect the direction and width of the parabola but will not impact the vertex or line of symmetry.
Let's look at the options provided:
A. \( y = \frac{1}{2}(x + 5)^2 - 2 \)
B. \( y = \frac{1}{2}(x - 5)^2 + 2 \)
C. \( y = \frac{1}{2}(x + 2)^2 - 5 \)
D. \( y = \frac{1}{2}(x - 2)^2 - 5 \)
E. \( y = \frac{1}{2}(x - 5)^2 + 5 \)
From the equation we formed, we see that the correct form should have \((x - 2)\) due to the vertex being 2, and it should also have \(-5\) as the constant.
Looking at the options:
- Option D: \( y = \frac{1}{2}(x - 2)^2 - 5 \) matches the required format, with the correct vertex at (2, -5).
The other options either do not have the correct vertex \( (2, -5) \) or do not have the correct line of symmetry.
Therefore, the answer is:
D. \( y = \frac{1}{2}(x - 2)^2 - 5 \)