To solve this problem, we first need to identify the vertex form of a parabola. The vertex form of a quadratic function is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola. Since we are given that the parabola has a minimum at the point [tex]\((-3, 9)\)[/tex], we can identify [tex]\(h = -3\)[/tex] and [tex]\(k = 9\)[/tex].
Thus, the equation of the parabola in vertex form is:
[tex]\[ y = a(x + 3)^2 + 9 \][/tex]
Now, we need to determine the value of [tex]\(a\)[/tex]. Since the parabola has a minimum, it opens upwards, meaning that the coefficient [tex]\(a\)[/tex] should be positive.
Let's examine each of the given options:
1. Option A: [tex]\(g(x) = 3(x - 3)^2 + 9\)[/tex]
- The vertex form here is [tex]\(y = 3(x - h)^2 + 9\)[/tex], where [tex]\(h = 3\)[/tex]. This has a vertex at [tex]\((3, 9)\)[/tex], not [tex]\((-3, 9)\)[/tex]. So, this option is incorrect.
2. Option B: [tex]\(g(x) = -\frac{1}{2}(x - 3)^2 + 9\)[/tex]
- The vertex form here is [tex]\(y = -\frac{1}{2}(x - h)^2 + 9\)[/tex], where [tex]\(h = 3\)[/tex]. This has a vertex at [tex]\((3, 9)\)[/tex], not [tex]\((-3, 9)\)[/tex]. Additionally, since [tex]\(a\)[/tex] is negative, the parabola opens downwards, which is not what we want. So, this option is incorrect.
3. Option C: [tex]\(g(x) = 2(x + 3)^2 + 9\)[/tex]
- The vertex form here is [tex]\(y = 2(x - (-3))^2 + 9\)[/tex], where [tex]\(h = -3\)[/tex] and [tex]\(k = 9\)[/tex]. This matches the vertex [tex]\((-3, 9)\)[/tex] and [tex]\(a = 2\)[/tex] is positive, meaning the parabola opens upwards. Therefore, this option is correct.
4. Option D: [tex]\(g(x) = -(x + 3)^2 + 9\)[/tex]
- The vertex form here is [tex]\(y = -(x - (-3))^2 + 9\)[/tex], where [tex]\(h = -3\)[/tex] and [tex]\(k = 9\)[/tex]. This matches the vertex [tex]\((-3, 9)\)[/tex], but since [tex]\(a\)[/tex] is negative, the parabola opens downwards, which is not what we want. So, this option is incorrect.
Therefore, the correct equation that represents the function is:
[tex]\[ C. \quad g(x) = 2(x + 3)^2 + 9 \][/tex]
