The vertex is halfway between focus and directrix, so we know the focus is at (-5,5). Points on the parabola have the form (x, f(x)). They all have equal distance to the focus and the directrix, so they satisfy the following equation: f(x) - (-1) = sqrt( ( x - (-5) )^2 + (f(x) - 5)^2) (Left side is distance to directrix, only depends on the y-coordinate of the point. Right side is the distance between point an focus computed using the pythoagorean theorem.) Squaring both sides and collecting the minuses gives (f(x) +1)^2 = ( x + 5 )^2 + (f(x) - 5)^2 Multiply out the brackets using the binomial theorem: f(x)^2 + 2*f(x) + 1 = x^2 + 10x + 25 + f(x)^2 - 10f(x) + 25 Simplify by subtracting f(x)^2. Then bring +1 to the right side and -10f(x) to the left side. This gives 12*f(x) = x^2 + 10 x + 49 Divide by 12 and you get the final function: f(x) = 1/12 *x^2 + 10/12 * x + 49/12
