12. Which function shows \( f(x)=-3 x^{2}+30 x-73 \) in vertex form? \( \begin{array}{l}\text { (A) } f(x)=-3(x-5)^{2}-148 \\ \text { (B) } f(x)=-3(x-5)^{2}-98 \\ \text { (C) } f(x)=-3(x-5)^{2}-73 \\ \text { (D) } f(x)=-3(x-5)^{2}+2\end{array} \)

To express the quadratic function \( f(x) = -3x^2 + 30x - 73 \) in vertex form, we need to complete the square. The vertex form of a quadratic function is: \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. **Step 1: Factor out the coefficient of \(x^2\) from the first two terms.** \[ f(x) = -3(x^2 - 10x) - 73 \] **Step 2: Complete the square inside the parentheses.** Take half of the coefficient of \(x\) (which is \(-10\)), square it, and add and subtract this value inside the parentheses: \[ \text{Half of } -10 = -5 \\ (-5)^2 = 25 \] \[ f(x) = -3(x^2 - 10x + 25 - 25) - 73 \\ f(x) = -3\left((x - 5)^2 - 25\right) - 73 \] **Step 3: Simplify the expression.** \[ f(x) = -3(x - 5)^2 + 75 - 73 \\ f(x) = -3(x - 5)^2 + 2 \] So, the vertex form of the function is: \[ f(x) = -3(x - 5)^2 + 2 \] This corresponds to **option (D)**. **Answer:** (D) \( f(x)=-3(x-5)^{2}+2 \)