Write the given expression in the form \( f(x)=a(x-h)^{2}+k \). Identify the vertex. \( f(x)=2-15 x-3 x^{2} \)

To rewrite the expression \( f(x) = 2 - 15x - 3x^2 \) in the form \( f(x) = a(x-h)^2 + k \), we will first rearrange the terms and then complete the square. 1. Rearrange the expression: \[ f(x) = -3x^2 - 15x + 2 \] 2. Factor out \(-3\) from the quadratic terms: \[ f(x) = -3(x^2 + 5x) + 2 \] 3. Complete the square for the expression inside the parentheses. To do this, take half of the coefficient of \(x\) (which is \(5\)), square it, and add and subtract it inside the parentheses: - Half of \(5\) is \(\frac{5}{2}\). - Squaring it gives \(\left(\frac{5}{2}\right)^2 = \frac{25}{4}\). 4. Add and subtract \(\frac{25}{4}\) inside the parentheses: \[ f(x) = -3\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) + 2 \] \[ = -3\left((x + \frac{5}{2})^2 - \frac{25}{4}\right) + 2 \] 5. Distribute \(-3\): \[ f(x) = -3(x + \frac{5}{2})^2 + \frac{75}{4} + 2 \] 6. Convert \(2\) to a fraction with a denominator of \(4\): \[ 2 = \frac{8}{4} \] So, \[ f(x) = -3(x + \frac{5}{2})^2 + \frac{75}{4} + \frac{8}{4} \] \[ = -3(x + \frac{5}{2})^2 + \frac{83}{4} \] Now, we have the expression in the form \( f(x) = a(x-h)^2 + k \): \[ f(x) = -3\left(x + \frac{5}{2}\right)^2 + \frac{83}{4} \] From this, we can identify: - \( a = -3 \) - \( h = -\frac{5}{2} \) - \( k = \frac{83}{4} \) The vertex of the parabola is given by the point \((h, k)\): \[ \text{Vertex} = \left(-\frac{5}{2}, \frac{83}{4}\right) \] Thus, the final answer is: \[ f(x) = -3\left(x + \frac{5}{2}\right)^2 + \frac{83}{4}, \quad \text{Vertex} = \left(-\frac{5}{2}, \frac{83}{4}\right) \]