There are four basic parabola orientations. "Opening upward" is when the vertex of the parabola points downward and the two branches of the parabola go off to positive infinity. The formula for this type of parabola is aX^2 +bX + c and "a" is positive. The "opening downward" parabolas are the exact opposite; the "a" is negative, the vertex points up and the two branches go off to negative infinity. In the "opening left" and "opening right" parabolas, the Y is squared and the X is not. If the coefficient of the Y^2 is positive, the parabola opens right, and if it is negative, the parabola opens left.
The axis of symmetry is an imaginary line that runs down the center of a parabola. The axis of symmetry is either vertical or horizontal, and so the function for the axis of symmetry has a single variable. For example, if the parabola opens upward and the vertex is at (a, b), the formula for the axis of symmetry is X = a. The reason it is called the axis of symmetry is that the parabola is symmetrical and looks like a mirror-image reflection around the axis of symmetry.
The vertex is the tip of the parabola. It is found by manipulating Y = aX^2 + bX + c into the form Y = a(X - h)^2 + k. Then the vertex is at (h, k). For example, if Y = 2X^2 - 4X + 3 then Y = 2X^2 - 4X + 2 + 1 so Y = 2(X^2 -2X + 1) + 1 or Y = 2(X - 1)^2 + 1. This means that h = k = 1 and the vertex is at (1, 1).
The focus of a parabola is a point on the axis of symmetry that controls the shape of the parabola. The nearer the focus is to the vertex, the narrower the parabola is. To find the focus, put the equation into the form 4p(Y - k) = (X - h) where (h, k) is the vertex and p is distance from the vertex and the focus. For example, we have already seen that Y = 2X^2 -4X + 3 can be transformed into Y = 2(X - h)^2 + k, which means that Y - k = 2(X - h)^2 or (1/2)(K -h) = (X - k)^2 so 4p = 1/2 or p = 1/8. The focus is at (1, 9/8).