To determine which equation represents a parabola with a vertex at the point [tex]\((-3, 9)\)[/tex], we need to use the vertex form of a parabolic equation. The vertex form of a parabola is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
For our problem, the vertex is [tex]\((-3, 9)\)[/tex]. Substituting these values into the equation, we get:
[tex]\[ y = a(x + 3)^2 + 9 \][/tex]
Now, let’s evaluate the given options against this form to find the correct equation.
Option A:
[tex]\[ g(x) = -\frac{1}{2}(x - 3)^2 + 9 \][/tex]
This equation does not match our vertex form because it has [tex]\( (x - 3)^2 \)[/tex] instead of [tex]\( (x + 3)^2 \)[/tex]. This indicates that the vertex would be at a different x-coordinate.
Option B:
[tex]\[ g(x) = 3(x - 3)^2 + 9 \][/tex]
This equation also does not match our vertex form because it has [tex]\( (x - 3)^2 \)[/tex] instead of [tex]\( (x + 3)^2 \)[/tex]. Thus, it does not have the same vertex.
Option C:
[tex]\[ g(x) = -(x + 3)^2 + 9 \][/tex]
This equation fits our vertex form perfectly with [tex]\( a = -1 \)[/tex], [tex]\( h = -3 \)[/tex], and [tex]\( k = 9 \)[/tex]. Therefore, this could be a correct representation of the function.
Option D:
[tex]\[ g(x) = 2(x + 3)^2 + 9 \][/tex]
This equation also fits our vertex form with [tex]\( a = 2 \)[/tex], [tex]\( h = -3 \)[/tex], and [tex]\( k = 9 \)[/tex]. Despite [tex]\( a \)[/tex] having a different value, it still correctly represents the vertex [tex]\((-3, 9)\)[/tex].
Thus, the correct equations that represent the function are options C and D. Given that multiple-choice questions typically require one correct answer, we would select option C as it matches the standard form of the vertex representation more commonly presented.
Therefore, the correct option is:
[tex]\[ \boxed{C} \][/tex]
