Parabola
A parabola is the U-shaped curve that graphs a quadratic function such as y = ax² + bx + c. It is symmetric about a vertical line through its vertex, the point where the curve turns around.
A parabola is the graph of a quadratic function, y = ax² + bx + c with a ≠ 0. Every parabola has a turning point called the vertex, an axis of symmetry running through it, and a consistent opening direction: upward when a > 0, downward when a < 0. Larger |a| makes the curve narrower; smaller |a| makes it wider. Geometrically, a parabola is also defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
The key features come straight from the equation. The axis of symmetry is x = −b / (2a), and plugging that x-value back in gives the vertex. The y-intercept is c, and the x-intercepts — where y = 0 — are the roots of the quadratic, found by factoring, the quadratic formula, or completing the square. A parabola can cross the x-axis twice, touch it once, or miss it entirely, matching the sign of the discriminant b² − 4ac.
Vertex form, y = a(x − h)² + k, rewrites the same curve so the vertex (h, k) can be read directly, which makes translations easy to see: replacing x with x − h shifts the graph right h units, and adding k shifts it up k units.
Parabolas are a fixture of standardized test math. The SAT tests quadratics and function translations in its Advanced Math domain, the GRE asks about graphing parabolas and identifying vertices and intercepts, and the CLT covers parabolas through its quadratics questions.
Key takeaways
- A parabola is the graph of a quadratic function y = ax² + bx + c.
- It opens upward when a > 0 and downward when a < 0, with the vertex as its maximum or minimum point.
- The axis of symmetry is x = −b / (2a), and the vertex lies on it.
- The discriminant b² − 4ac tells how many times the parabola crosses the x-axis (two, one, or zero).
- The SAT, GRE, and CLT all test finding a parabola's vertex, intercepts, and translations.