Consider the following quadratic function. \[ g(x)=-3 x^{2}-24 x-47 \] (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: \( g(x)=\square \) Vertex: ( \( \square . \square \) )

To write the equation in the form \( g(x) = a(x-h)^2 + k \), we can complete the square. The function is: \[ g(x) = -3(x^2 + 8x) - 47. \] Now, to complete the square on the expression \( x^2 + 8x \): 1. Take half of the coefficient of \( x \) (which is 8), square it, and add and subtract it inside the parentheses: \[ 8/2 = 4 \Rightarrow 4^2 = 16. \] 2. Rewrite the expression: \[ g(x) = -3(x^2 + 8x + 16 - 16) - 47 \] \[ = -3((x + 4)^2 - 16) - 47 \] \[ = -3(x + 4)^2 + 48 - 47 \] \[ = -3(x + 4)^2 + 1. \] So, in the specified form, we have: \[ g(x) = -3(x + 4)^2 + 1. \] The vertex of the graph, given by the point \( (h, k) \), is \( (-4, 1) \). Thus, the final results are: Writing in the form specified: \( g(x)=-3(x+4)^{2}+1 \) Vertex: ( \( -4, 1 \) )