Find a function $f$ whose graph is a parabola with the given vertex and that passes through the given point.
\[
f(x)=\text { vertex }(-1,5) \text {; point }(-2,-3)
\]
Answer
The function \(f\) whose graph is a parabola with the given vertex and that passes through the given point is \(f(x) = \boxed{5 - 8(x + 1)^2}\).
Step 1 ：The general form of a parabola is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. Given the vertex is \((-1, 5)\), we can substitute \(h = -1\) and \(k = 5\) into the equation to get \(f(x) = a(x+1)^2 + 5\).
Step 2 ：We also know that the parabola passes through the point \((-2, -3)\). We can substitute \(x = -2\) and \(f(x) = -3\) into the equation to solve for \(a\).
Step 3 ：Solving the equation gives us \(a = -8\).
Step 4 ：Substituting \(a = -8\) into the equation \(f(x) = a(x+1)^2 + 5\) gives us the final function.
Step 5 ：The function \(f\) whose graph is a parabola with the given vertex and that passes through the given point is \(f(x) = \boxed{5 - 8(x + 1)^2}\).
