What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 8)?
+2666 By the distance formula, the distance from the origin to the graph is \(\sqrt{(y-0)^2 + (x-0)^2} = \sqrt{y^2 + x^2}\)
We know that \(y = {1 \over 2}x^2 - 4\), meaning \({1 \over 2}x^2 = y + 4\), so \(x^2 = 2y + 8\)
Substituting this gives us \(\sqrt{y^2 + 2y + 8}\).
The minimum value occurs at \(y = -{b \over 2a} \), which is -1.
This means the minimum is \(\sqrt{(-1)^2 + 2(-1) + 8} = \color{brown}\boxed{\sqrt{7}}\)
