Write down the standard form of the parabola equation. The standard form of the equation for a vertically orientated parabola is (x - a)^2 = 4*p*(y - b), while that for a horizontally orientated parabola is (y - b)^2 = 4*p*(x - a).
In these equations, (a, b) is the coordinate of the vertex and p is the distance between the vertex and the focus along the axis.
Calculate the unknown values "a," "b" and "p." This is easiest if the coordinates of the vertex and focus are given because both "a" and "b" are then known and "p" is the linear distance between the vertex and focus, calculated as follows:
p = sqrt[(a - c)^2 + (b - d)^2]
In the equation, (a, b) represents the coordinates of the vertex and (c, d) represents the focus, while "sqrt" stands for "square root."
If the coordinates of the vertex but not the focus are given, it is necessary to know the coordinates of one other point on the parabola. In this case, "a" and "b" are once again known. Substitute "a," "b" and the coordinates of the point on the parabola into the standard equation and solve for "p."
If the vertex coordinates are not given, three points on the parabola are needed. Substitute each of these points into the standard equation to generate three equations. Solve these equations simultaneously for the unknowns "a," "b" and "p."
Generate the specific parabola equation. Substitute "a," "b" and "p" from Step 2 into the standard parabola equation.
Solve for the y-intercept. The y-intercept is the point where the parabola intersects the y-axis. The y-axis has the property that x = 0 everywhere. Calculate the y-intercept by setting x = 0 in the equation from Step 3 and solving for "y."