Consider the following quadratic function. \[ f(x)=3 x^{2}-30 x+71 \] (a) Write the equation in the form \( f(x)=a(x-h)^{2}+k \). Then give the vertex of its graph. Writing in the form specified: \( f(x)= \) Vertex: ( \( \square, \square) \)

To rewrite the quadratic function \( f(x) = 3x^2 - 30x + 71 \) in vertex form, we need to complete the square. First, factor out the 3 from the \( x^2 \) and \( x \) terms: \[ f(x) = 3(x^2 - 10x) + 71 \] Next, complete the square inside the parentheses. To do this, take half of -10 (which is -5), square it (getting 25), and add and subtract it inside the brackets: \[ f(x) = 3(x^2 - 10x + 25 - 25) + 71 \] This simplifies to: \[ f(x) = 3((x - 5)^2 - 25) + 71 \] Distributing the 3 gives: \[ f(x) = 3(x - 5)^2 - 75 + 71 \] Finally, combine the constants: \[ f(x) = 3(x - 5)^2 - 4 \] Thus, the function in vertex form is: \[ f(x) = 3(x - 5)^2 - 4 \] The vertex of the graph is \( (5, -4) \). So the answer is: \[ f(x)= 3(x-5)^{2}-4 \] Vertex: \( (5, -4) \)