## Key Equations

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

## 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
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