Find a function \( f \) whose graph is a parabola with the given vertex and that passes through the given point. \[ f(x)=\square \text { vertex }(-1,5) \text {; point }(-2,-1) \]

To find the function \( f(x) \) for a parabola with vertex \((-1, 5)\) that passes through the point \((-2, -1)\), we start with the vertex form of a quadratic function, which is: \[ f(x) = a(x - h)^2 + k \] Where \((h, k)\) is the vertex. Here, \(h = -1\) and \(k = 5\), so we can plug those values in: \[ f(x) = a(x + 1)^2 + 5 \] Next, we need to find the value of \(a\) using the point \((-2, -1)\). We substitute \(x = -2\) and \(f(x) = -1\) into the equation: \[ -1 = a(-2 + 1)^2 + 5 \] \[ -1 = a(-1)^2 + 5 \] \[ -1 = a + 5 \] Now, solving for \(a\): \[ a = -1 - 5 = -6 \] Now we can substitute \(a\) back into the vertex form of the function: \[ f(x) = -6(x + 1)^2 + 5 \] This is the function whose graph is a parabola with a vertex at \((-1,5)\) and that passes through the point \((-2, -1)\).