To determine which equation represents the function of a parabola with a vertex at the point [tex]\((-3, 9)\)[/tex], we need to understand the vertex form of a parabola's equation. The vertex form of a parabolic equation is given by:
[tex]\[ g(x) = a(x-h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. In this case, the given vertex is [tex]\((-3, 9)\)[/tex]. Therefore, substituting the vertex [tex]\((-3, 9)\)[/tex] into the vertex form, we get:
[tex]\[ g(x) = a(x + 3)^2 + 9 \][/tex]
We now analyze each option to see which one appropriately matches this form.
1. Option A: [tex]\( g(x) = -(x + 3)^2 + 9 \)[/tex]
- The equation is in the form [tex]\( g(x) = a(x + 3)^2 + 9 \)[/tex] with [tex]\( a = -1 \)[/tex].
- The vertex [tex]\((-3, 9)\)[/tex] corresponds correctly to the given vertex form.
- The negative sign indicates the parabola opens downwards which is okay.
2. Option B: [tex]\( g(x) = -\frac{1}{2}(x - 3)^2 + 9 \)[/tex]
- The equation is in the form [tex]\( g(x) = -\frac{1}{2}(x - 3)^2 + 9 \)[/tex].
- Here the vertex is [tex]\( (3, 9) \)[/tex], which does not match the given vertex [tex]\((-3, 9)\)[/tex].
3. Option C: [tex]\( g(x) = 3(x - 3)^2 + 9 \)[/tex]
- The equation is in the form [tex]\( g(x) = 3(x - 3)^2 + 9 \)[/tex].
- Here the vertex is [tex]\( (3, 9) \)[/tex], which does not match the given vertex [tex]\((-3, 9)\)[/tex].
4. Option D: [tex]\( g(x) = 2(x + 3)^2 + 9 \)[/tex]
- The equation aligns with [tex]\( g(x) = a(x + 3)^2 + 9 \)[/tex] with [tex]\( a = 2 \)[/tex].
- The vertex [tex]\((-3, 9)\)[/tex] matches the given vertex, and the positive constant [tex]\( a = 2 \)[/tex] means the parabola opens upwards.
Both options A and D have the vertex at [tex]\((-3, 9)\)[/tex] as required, but they differ in their 'a' values which indicate whether the parabola opens upwards or downwards.
Given the correct analysis of the vertex and the quadratic form, the correct answer is:
A. [tex]\( g(x) = -(x + 3)^2 + 9 \)[/tex]
