Write the given expression in the form \( f(x)=a(x-h)^{2}+k \). Identify the vertex. \( f(x)=2 x^{2}-5 x+1 \) Choose the equation below. A. \( f(x)=2\left(x+\frac{5}{4}\right)^{2}-\frac{17}{8} \) C. \( f(x)=2\left(x-\frac{5}{2}\right)^{2}-\frac{23}{2} \) \( f(x)=2\left(x-\frac{5}{4}\right)^{2}-\frac{17}{8} \)

To write \[ f(x)=2x^2-5x+1 \] in the form \[ f(x)=a(x-h)^2+k, \] we complete the square. 1. Factor out the coefficient of \(x^2\) from the quadratic and linear terms: \[ f(x)=2\left(x^2-\frac{5}{2}x\right)+1. \] 2. To complete the square for \[ x^2-\frac{5}{2}x, \] take half of \(-\frac{5}{2}\) and square it: \[ \left(-\frac{5}{4}\right)^2=\frac{25}{16}. \] Write: \[ x^2-\frac{5}{2}x=\left(x-\frac{5}{4}\right)^2-\frac{25}{16}. \] 3. Substitute back into the function: \[ f(x)=2\left[\left(x-\frac{5}{4}\right)^2-\frac{25}{16}\right]+1. \] 4. Distribute \(2\): \[ f(x)=2\left(x-\frac{5}{4}\right)^2-\frac{50}{16}+1. \] 5. Simplify \(\frac{50}{16}\) to \(\frac{25}{8}\) and express \(1\) as \(\frac{8}{8}\): \[ f(x)=2\left(x-\frac{5}{4}\right)^2-\frac{25}{8}+\frac{8}{8}, \] \[ f(x)=2\left(x-\frac{5}{4}\right)^2-\frac{17}{8}. \] Thus, the vertex form is \[ f(x)=2\left(x-\frac{5}{4}\right)^2-\frac{17}{8}, \] with the vertex at \[ \left(\frac{5}{4},-\frac{17}{8}\right). \] Comparing with the given options, the correct choice is \[ f(x)=2\left(x-\frac{5}{4}\right)^2-\frac{17}{8}. \]