Parabola
High School
Definition
A two-dimensional u-shaped curve. A parabola is made up of the focus (a point) and a directrix (a line). The focus does not lie on the directrix and the locus of points in the plane is such that the distance to the focus is the same as the distance to the directrix.
Worked examples
\(y = x^2\)
The simplest parabola opens upward with vertex at the origin; each point is equidistant from focus and directrix.
\(y = -2(x - 3)^2 + 1\)
This parabola opens downward, vertex at \((3, 1)\), and is narrower because of the coefficient \(-2\).
\(x = y^2\)
A horizontal parabola opening to the right; \(x\) and \(y\) roles are swapped from the standard vertical form.
Common mistakes
- \(y = x^2 + 3\) shifts the parabola right 3 units → \(y = x^2 + 3\) shifts the parabola up 3 units Adding outside the square moves vertically; \(y = (x - 3)^2\) shifts right.
- All parabolas open upward → Parabolas can open up, down, left, or right The sign of the leading coefficient and which variable is squared determine direction.
- The vertex is always at \((0, 0)\) → The vertex can be at any point \((h, k)\) Vertex form \(y = a(x - h)^2 + k\) places the vertex at \((h, k)\).
Where you'll use it next
Parabolas appear in quadratic functions and equations, conic sections, optimization problems, and physics (projectile motion, satellite dishes, suspension bridges).
Found in 1 StudyPug lesson
Conics - Parabola
12th Grade12thGrade 12 Math
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026