Key Equations

Key Concepts

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