Find the focus and the directrix of \(y = -\frac{1}{2}x^2 + x - \frac{1}{2}\text{.}\)
Solution.
We begin by completing the square:
\begin{equation*}
y = -\frac{1}{2}x^2 + x - \frac{1}{2} = -\frac{1}{2}(x-1)^2,
\qquad\text{i.e.}\qquad (x-1)^2 = -2y.
\end{equation*}
Comparing with \(x^2 = 4py\) gives \(4p = -2\text{,}\) so \(p = -\frac{1}{2}\text{.}\) The graph is the standard parabola shifted right by one unit, so the vertex is at \((1,0)\text{.}\) Therefore the focus is at
\begin{equation*}
(1,\, p) = \left(1,\, -\tfrac{1}{2}\right),
\end{equation*}
and the directrix is the horizontal line \(y = -p = \tfrac{1}{2}\text{.}\)
