To determine which equation could represent a parabola with a minimum at the point [tex]\((-3,9)\)[/tex], we need to observe the following characteristics of the function:
1. Vertex Form of a Parabola: The general vertex form of a quadratic function is given by [tex]\( g(x) = a(x-h)^2 + k \)[/tex] where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
2. Minimum Point: The value of [tex]\(a\)[/tex] in the equation will determine if the parabola opens upwards (minimum point) or downwards (maximum point). If [tex]\(a > 0\)[/tex], the parabola opens upwards, making [tex]\((h, k)\)[/tex] a minimum point. If [tex]\(a < 0\)[/tex], the parabola opens downwards, making [tex]\((h, k)\)[/tex] a maximum point.
3. Given vertex: For the point [tex]\((-3, 9)\)[/tex], the vertex form should be [tex]\( g(x) = a(x+3)^2 + 9 \)[/tex], indicating that the vertex [tex]\((h, k)\)[/tex] corresponds to [tex]\(h = -3\)[/tex] and [tex]\(k = 9\)[/tex].
Now, let's analyze the given options:
A. [tex]\( g(x) = 3(x-3)^2 + 9 \)[/tex]
- Here, the vertex form suggests [tex]\(h = 3\)[/tex] and [tex]\(k = 9\)[/tex], which gives a vertex at point [tex]\((3, 9)\)[/tex]. This does not match our given vertex of [tex]\((-3, 9)\)[/tex].
B. [tex]\( g(x) = -(x+3)^2 + 9 \)[/tex]
- Here, the vertex form suggests [tex]\(h = -3\)[/tex] and [tex]\(k = 9\)[/tex], which gives a vertex at point [tex]\((-3, 9)\)[/tex]. However, the coefficient of the quadratic term is negative ([tex]\( -1 \)[/tex]), indicating that the parabola opens downwards and thus the point [tex]\((-3, 9)\)[/tex] is a maximum point. This contradicts the requirement of having a minimum.
C. [tex]\( g(x) = -\frac{1}{2}(x-3)^2 + 9 \)[/tex]
- Here, the vertex form suggests [tex]\(h = 3\)[/tex] and [tex]\(k = 9\)[/tex], which gives a vertex at point [tex]\((3, 9)\)[/tex]. This does not match our given vertex of [tex]\((-3, 9)\)[/tex].
D. [tex]\( g(x) = 2(x+3)^2 + 9 \)[/tex]
- Here, the vertex form suggests [tex]\(h = -3\)[/tex] and [tex]\(k = 9\)[/tex], which gives a vertex at point [tex]\((-3, 9)\)[/tex]. The coefficient of the quadratic term is positive ([tex]\( 2 \)[/tex]), indicating that the parabola opens upwards and hence the point [tex]\((-3, 9)\)[/tex] is a minimum point.
Since [tex]\( g(x) = 2(x+3)^2 + 9 \)[/tex] matches all the given characteristics correctly, the correct answer is:
D. [tex]\( g(x)=2(x+3)^2+9 \)[/tex]
