\( 16 \quad \) (a) Find the vertex and axis of symmetry of the quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. \( f(\mathrm{x})=-\frac{1}{3}\left(\mathrm{x}-\frac{1}{2}\right)^{2}-\frac{7}{2} \) (a) The vertex is \( \square \). (Simplify your answer. Type an ordered pair, using integers or fractions.)

To find the vertex of the quadratic function \( f(x) = -\frac{1}{3}\left(x - \frac{1}{2}\right)^{2} - \frac{7}{2} \), we can use the vertex form of the quadratic. The vertex is given directly in the function's form as \( \left(h, k\right) \), where \( h = \frac{1}{2} \) and \( k = -\frac{7}{2} \). Therefore, the vertex is \( \left(\frac{1}{2}, -\frac{7}{2}\right) \). The axis of symmetry is a vertical line that passes through the vertex, expressed as \( x = \frac{1}{2} \). Now to determine whether the graph is concave up or down: since the coefficient of the \((x - \frac{1}{2})^{2}\) term is negative (specifically, \(-\frac{1}{3}\)), the graph is concave down. Lastly, graphing this function would show a downward-opening parabola with the vertex at \( \left(\frac{1}{2}, -\frac{7}{2}\right) \), illustrating the maximum point of the parabola. Enjoy sketching a beautiful parabola that dips down like a roller coaster!