Questions: (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(x) = -(1/3)(x - 1/2)^2 - 5/2 (a) The vertex is (Simplify your answer. Type an ordered pair, using integers or fractions.)

Solution Steps

The given quadratic function is: \[ f(x) = -\frac{1}{3}\left(x-\frac{1}{2}\right)^{2}-\frac{5}{2} \] The vertex form of a quadratic function is: \[ f(x) = a(x-h)^2 + k \] where \((h, k)\) is the vertex. Comparing the given function with the vertex form, we have: \[ h = \frac{1}{2}, \quad k = -\frac{5}{2} \] Thus, the vertex is: \[ \left( \frac{1}{2}, -\frac{5}{2} \right) \]

The axis of symmetry for a quadratic function in vertex form \(f(x) = a(x-h)^2 + k\) is given by \(x = h\). Therefore, the axis of symmetry is: \[ x = \frac{1}{2} \]

The concavity of the quadratic function is determined by the sign of the coefficient \(a\). In this case, \(a = -\frac{1}{3}\), which is negative. Therefore, the graph is concave down.

Final Answer