How to Translate Parabolas

By Maria O'Brien
Translate Parabolas

A lens, an arch and a cross-section of a dome or satellite dish are all examples of parabolas. They have many uses in astronomy, physics and architecture.Algebraically, a parabola is the U-shaped graph of a quadratic equation. In geometry, the same shape is defined as the set of all points in a plane equidistant from a fixed point (the focus, F) and a fixed line (the directrix, L) in that plane.The vertex of a parabola is the point at the tip of the U (regardless of the direction of the parabola: È , Ç , É , Ì ). It is the point of intersection between the parabola and a line drawn through the focus, perpendicular to the directrix. This line is called the axis of symmetry; the parabola is symmetrical around this line.When the vertex of a parabola is the origin, it is easy to write its equation in standard form. Translation of the axes is a way of writing the equation of a parabola in standard form when the vertex is not the origin. Here’s what you need to do.

Familiarize yourself with the standard form of a parabola. That is:y squared = 4px, where the focus is (p.0) and the directrix is x = - porx squared = 4py, where the focus is (0,p) and the directrix is y = - pThe best way to understand what each of the terms means, their relationships and how that shapes a parabola is to graph many parabolas. Be sure to also graph the focus and diretcrix.

Use the equation(y - k)squared = 4p(x - h) OR (x - h)squared = 4p(y - k)to translate the axes. The vertex is (h,k) instead of the origin (0,0). The axes of symmetry must be parallel to either the x or the y axis. The focus is (h + p, k) and the directrix x = h- p for the first equation. In the second equation, the focus is (h, k + p) and the directrix is y = k - p.

Graph whatever information you are given before you start to solve any problem. This will help you visualize the parabola, and will provide a check of your answer when you are finished.

Substitute given values into the equations for the focus and directrix. Solve as for a system of equations.

Plug the calculated values into the equation for a parabola with translated axes (Step 2, above) and simplify.Let‘s look at a sample problem: Write the equation for a parabola in standard form when given that the directrix is y = -1 and the focus is (4,5).Plot this to help you visualize the parabola.Then substitute the given values into the equations for the focus and directrix.y = k - p = -1 for the directrixand(h, k + p) is (4,5) for the focus. Which means that h = 4 and k + p = 5Solve the system of equations k - p = -1 and k + p = 5. One way would be to solve k - p = -1 for k: k = p - 1 and substitute that into k + p = 5. You getp - 1 + p = 52p = 6p = 3 and therefore k = 2.Plug these values into the equation(x - h)squared = 4p(y - k)(x - 4)squared = 4*3(y - 2)(x - 4)squared = 12(y - 2)