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Study Guides > College Algebra CoRequisite Course

Summary: The Parabola

Key Equations

Parabola, vertex at origin, axis of symmetry on x-axis y2=4px{y}^{2}=4px
Parabola, vertex at origin, axis of symmetry on y-axis x2=4py{x}^{2}=4py
Parabola, vertex at (h,k)\left(h,k\right), axis of symmetry on x-axis (y−k)2=4p(x−h){\left(y-k\right)}^{2}=4p\left(x-h\right)
Parabola, vertex at (h,k)\left(h,k\right), axis of symmetry on y-axis (x−h)2=4p(y−k){\left(x-h\right)}^{2}=4p\left(y-k\right)

Key Concepts

  • A parabola is the set of all points (x,y)\left(x,y\right) in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
  • The standard form of a parabola with vertex (0,0)\left(0,0\right) and the x-axis as its axis of symmetry can be used to graph the parabola. If p>0p>0, the parabola opens right. If p<0p<0, the parabola opens left.
  • The standard form of a parabola with vertex (0,0)\left(0,0\right) and the y-axis as its axis of symmetry can be used to graph the parabola. If p>0p>0, the parabola opens up. If p<0p<0, the parabola opens down.
  • When given the focus and directrix of a parabola, we can write its equation in standard form.
  • The standard form of a parabola with vertex (h,k)\left(h,k\right) and axis of symmetry parallel to the x-axis can be used to graph the parabola. If p>0p>0, the parabola opens right. If p<0p<0, the parabola opens left.
  • The standard form of a parabola with vertex (h,k)\left(h,k\right) and axis of symmetry parallel to the y-axis can be used to graph the parabola. If p>0p>0, the parabola opens up. If p<0p<0, the parabola opens down.
  • Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides.

Glossary

directrix a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant focus (of a parabola) a fixed point in the interior of a parabola that lies on the axis of symmetry focal diameter (latus rectum) the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola parabola the set of all points (x,y)\left(x,y\right) in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix

Licenses & Attributions