By Marie Mulrooney
Updated Aug 30, 2022
Wojciech Gajda/iStock/Getty Images
In everyday life, parabolas model the trajectories of thrown, kicked, or fired objects. They also form the basis of satellite dishes and reflectors, because their shape focuses all incoming rays at a single point inside the curve known as the focus. Mathematically, a parabola is represented by the quadratic function f(x) = ax² + bx + c. To locate its vertex, find the midpoint between the two x‑intercepts; that gives the x‑coordinate, which you can then plug back into the equation to determine the y‑coordinate.
Rewrite the parabola in standard form, f(x) = ax² + bx + c, if it is not already presented that way.
Identify the coefficients a, b, and c in the equation. If b or c are absent, treat them as zero. The coefficient a is always non‑zero; for example, in f(x) = 2x² + 8x, we have a = 2, b = 8, and c = 0.
Calculate the x‑coordinate of the vertex with the midpoint formula: x = −b⁄(2a). In the example above, x = −8⁄4 = −2.
Substitute this x‑value back into the original equation to find the y‑coordinate. Using x = −2: f(−2) = 2(−2)² + 8(−2) = 8 − 16 = −8. Thus the vertex is at (−2, −8).
When a parabola is expressed in vertex form, f(x) = a(x − h)² + k, the values h and k are directly the x‑ and y‑coordinates of the vertex. If k is omitted, it defaults to zero. For example, f(x) = 2(x − 5)² yields the vertex (5, 0), while f(x) = 2(x − 5)² + 2 gives (5, 2).
Pay close attention to the sign of the coefficient a. While x² is always positive, the overall term ax² can be positive or negative depending on a, which affects the parabola’s direction.
