Consider the following quadratic function. \[ g(x)=-2 x^{2}+8 x-11 \] (a) Write the equation in the form \( g(x)=a(x-h)^{2}+k \). Then give the vertex of its graph. Writing in the form specified: \( g(x)=\square_{0} \) vertex: \( (\square) \)

We start with the quadratic function \[ g(x)=-2x^2+8x-11. \] **Step 1: Factor out the coefficient of \(x^2\) from the first two terms.** Factor \(-2\) from \(x^2-4x\): \[ g(x)=-2\left(x^2-4x\right)-11. \] **Step 2: Complete the square.** Inside the parentheses, complete the square for \(x^2-4x\). Recall that \[ x^2-4x = \left(x-2\right)^2-4. \] Substitute this back into the equation: \[ g(x)=-2\left[\left(x-2\right)^2-4\right]-11. \] **Step 3: Distribute and simplify.** Distribute \(-2\) across the bracket: \[ g(x)=-2\left(x-2\right)^2 + 8 - 11. \] Combine the constant terms: \[ 8 - 11 = -3. \] Thus, the function in vertex form is: \[ g(x)=-2\left(x-2\right)^2 - 3. \] **Step 4: Identify the vertex.** In the vertex form \(g(x)=a(x-h)^2+k\), the vertex is \((h,k)\). Here we have: \[ h=2, \quad k=-3. \] Therefore, the vertex is: \[ (2,-3). \] **Final Answer:** Writing in the form specified: \[ g(x)=-2(x-2)^2-3 \] Vertex: \[ (2,-3) \]