Parabola Calculator

A parabola is a symmetrical, U-shaped plane curve where any point is equidistant from a fixed point (the focus) and a fixed straight line (the directrix). This calculator relies on two primary algebraic representations: standard form and vertex form. The standard form is expressed as y = ax² + bx + c (or horizontally as x = ay² + by + c), while the vertex form is written as y = a(x - h)² + k, where (h, k) represents the vertex coordinates. Core computations leverage the discriminant delta formula (Δ = b² - 4ac) to determine root intersections, alongside specialized geometric relations to isolate focal lengths, axes of symmetry, and directional orientations.

Let us analyze a parabola in standard form with coefficients a = 2, b = -8, and c = 6. First, we find the x-coordinate of the vertex using the formula h = -b / (2a), which yields -(-8) / (2 * 2) = 8 / 4 = 2. Next, we substitute x = 2 back into the standard equation to find the y-coordinate (the vertex): k = 2(2)² - 8(2) + 6 = 2(4) - 16 + 6 = -2. Thus, our vertex is at (2, -2). To find the focus, we use the focal parameter p = 1 / (4a), giving 1 / 8 = 0.125. Adding this to the vertex y-coordinate for a vertically oriented upward curve places the focus at (2, -1.875), while the directrix is calculated as y = k - p, resulting in y = -2.125.

Always double-check your input orientation before running calculations; mixing up horizontal and vertical forms will drastically alter your vertex and focus coordinates. When inputting coordinates manually from three distinct points, ensure they are not collinear, as three points lying on a single straight line cannot form a valid quadratic curve. Finally, keep track of your signs when dealing with negative coefficients, particularly during intermediate steps involving squaring values or calculating the discriminant.