Final answer:
To find a function with the same shape as g(x) = (2/3)x^2 and a vertex at (-7, 9), we use the vertex form of a quadratic equation, resulting in f(x) = (2/3)(x + 7)^2 + 9, which is Option C.
Step-by-step explanation:
To write an equation that represents a function with the same shape as the graph of g(x) = (2/3)x^2 with a vertex at (-7, 9), we need to use the vertex form of a quadratic function, which is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, and 'a' determines the width and direction of the parabola.
Since the original function g(x) has a coefficient of (2/3) before x^2, and we want to maintain the same shape, our 'a' value will also be (2/3). The vertex of our function is at (-7, 9), so 'h' is -7 and 'k' is 9.
Substituting these values into the vertex form gives us f(x) = (2/3)(x + 7)^2 + 9, which corresponds to Option C