One of the most common functions around in the bowl function z =
x2 +
y2. It's surface plot is shown on the left. The reason
it's so common is that near a maximum or minimum, all functions
look sort of like this bowl or an upside down bowl. To flip the bowl
upside down, we simply reflect it: z = -(x2 +
y2). This is pictured at the right.
By subtracting a constant from the function, we can shift the whole thing
down (left). To center the vertex of the bowl at the point (x0, y0) we simply
translate the function to be z = (x -
x0)2 + (y -
y0)2 (right).
