Summary: The Parabola

Key Equations

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

Key Concepts

  • A parabola is the set of all points [latex]\left(x,y\right)[/latex] 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 [latex]\left(0,0\right)[/latex] and the x-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens right. If [latex]p<0[/latex], the parabola opens left.
  • The standard form of a parabola with vertex [latex]\left(0,0\right)[/latex] and the y-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens up. If [latex]p<0[/latex], 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 [latex]\left(h,k\right)[/latex] and axis of symmetry parallel to the x-axis can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens right. If [latex]p<0[/latex], the parabola opens left.
  • The standard form of a parabola with vertex [latex]\left(h,k\right)[/latex] and axis of symmetry parallel to the y-axis can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens up. If [latex]p<0[/latex], 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 [latex]\left(x,y\right)[/latex] 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