To determine which equation represents a parabola with a minimum point at [tex]\((-3, 9)\)[/tex], we start by considering the vertex form of a quadratic function, which is given by:
[tex]\[ g(x) = a(x - h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola. For our specific problem, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-3, 9)\)[/tex]. This means our equation should look like:
[tex]\[ g(x) = a(x + 3)^2 + 9 \][/tex]
Let's analyze each given option to see which one fits this form:
Option A: [tex]\[ g(x) = 2(x + 3)^2 + 9 \][/tex]
- This option is in the correct form [tex]\((x + 3)^2 + 9\)[/tex], which reflects a vertex at [tex]\((-3, 9)\)[/tex].
- The coefficient [tex]\(a = 2\)[/tex] is positive, indicating that the parabola opens upwards, thereby giving a minimum point at the vertex [tex]\((-3, 9)\)[/tex].
Option B: [tex]\[ g(x) = -\frac{1}{2}(x - 3)^2 + 9 \][/tex]
- This option has the form [tex]\((x - 3)^2 + 9\)[/tex], which reflects a vertex at [tex]\((3, 9)\)[/tex], not [tex]\((-3, 9)\)[/tex].
- Hence, this does not represent the given parabola.
Option C: [tex]\[ g(x) = -(x + 3)^2 + 9 \][/tex]
- This option is close as it has the [tex]\((x + 3)^2 + 9\)[/tex] term and a vertex of [tex]\((-3, 9)\)[/tex].
- However, the coefficient [tex]\(a = -1\)[/tex] is negative, making the parabola open downwards, resulting in a maximum point at the vertex [tex]\((-3, 9)\)[/tex], not a minimum.
Option D: [tex]\[ g(x) = 3(x - 3)^2 + 9 \][/tex]
- This option has the form [tex]\((x - 3)^2 + 9\)[/tex], indicating a vertex at [tex]\((3, 9)\)[/tex] instead of [tex]\((-3, 9)\)[/tex].
- Therefore, this does not fit our criteria.
From this analysis, we see that the equation representing a parabola with a minimum at the point [tex]\((-3, 9)\)[/tex] is:
[tex]\[ \boxed{g(x) = 2(x + 3)^2 + 9} \][/tex]
Thus, the correct answer is option A.
