Parabolas
You met parabolas in College Algebra as graphs of quadratic functions, like ${y=x^2}$ (see Quadratic functions). This page gives the geometric definition, which explains why parabolas are used for satellite dishes, car headlights, and solar cookers.
The definition
Take a point ${F}$, called the focus, and a line, called the directrix. A parabola is the set of all points that are the same distance from the focus as from the directrix.
The point halfway between the focus and the directrix is on the parabola. It is the vertex. The line through the focus and the vertex is the axis; the parabola is symmetric about it.
Standard equations
Put the vertex at the origin and the focus at ${(0,p)}$. Then the directrix is the line ${y=-p}$. A point ${(x,y)}$ is on the parabola when its distance to the focus equals its distance to the directrix:
${\sqrt{x^2+(y-p)^2}=|y+p|}$
Square both sides, and simplify:
$\begin{align*}x^2+y^2-2py+p^2&=y^2+2py+p^2\\x^2&=4py\end{align*}$
Swapping the roles of ${x}$ and ${y}$ gives a parabola that opens sideways.
${x^2=4py}$: focus ${(0,p)}$, directrix ${y=-p}$
Opens up if ${p>0}$, down if ${p<0}$.
${y^2=4px}$: focus ${(p,0)}$, directrix ${x=-p}$
Opens right if ${p>0}$, left if ${p<0}$.
The number ${p}$ is the distance from the vertex to the focus (and from the vertex to the directrix).
Example 1: Find the focus and directrix of ${x^2=8y}$, and sketch the parabola.
Solution:
Match ${x^2=8y}$ with ${x^2=4py}$: ${4p=8}$, so ${p=2}$. The focus is ${(0,2)}$ and the directrix is ${y=-2}$. Since ${p>0}$, it opens upward.
To help sketch it, find the points level with the focus. Put ${y=2}$: ${x^2=16}$, so ${x=\pm 4}$. The points ${(\pm 4,2)}$ are on the parabola.
Example 2: Find the equation of the parabola with vertex at the origin and focus ${(-3,0)}$.
Solution:
The focus is on the ${x}$-axis, so use ${y^2=4px}$ with ${p=-3}$:
${y^2=-12x}$
It opens to the left, away from the directrix ${x=3}$.
Shifted parabolas
If the vertex is at ${(h,k)}$ instead of the origin, replace ${x}$ by ${x-h}$ and ${y}$ by ${y-k}$:
${(x-h)^2=4p(y-k)}$${(y-k)^2=4p(x-h)}$
To find the vertex from an expanded equation, complete the square.
Example 3: Find the vertex, focus, and directrix of ${y^2-4y-8x+28=0}$.
Solution:
Keep the ${y}$ terms on the left, and complete the square. Half of ${-4}$ is ${-2}$, and ${(-2)^2=4}$:
$\begin{align*}y^2-4y&=8x-28\\y^2-4y+4&=8x-24\\(y-2)^2&=8(x-3)\end{align*}$
So the vertex is ${(3,2)}$ and ${4p=8}$, so ${p=2}$. The parabola opens to the right. The focus is ${2}$ units right of the vertex, at ${(5,2)}$. The directrix is ${2}$ units left, the line ${x=1}$.
The reflecting property
A parabola has a remarkable property: any ray coming in parallel to its axis bounces off the curve straight through the focus.
That is why a satellite dish is shaped like a parabola, with the receiver at the focus: all the weak signals arriving from far away are collected at one point. Run in reverse, a light bulb at the focus of a parabolic mirror sends out a straight beam, as in a flashlight.
Example 4: A satellite dish is ${4}$ ft across and ${1}$ ft deep at its center. Where should the receiver be placed?
Solution:
Put the vertex of the dish at the origin, opening up: ${x^2=4py}$. The rim is ${2}$ ft from the center and ${1}$ ft up, so the point ${(2,1)}$ is on the parabola:
${2^2=4p(1)}$, so ${p=1}$
The receiver goes at the focus, ${1}$ ft above the center of the dish.
Summary
- A parabola is the set of points equally far from a focus and a directrix.
- ${x^2=4py}$ has focus ${(0,p)}$ and directrix ${y=-p}$. ${y^2=4px}$ has focus ${(p,0)}$ and directrix ${x=-p}$.
- With vertex ${(h,k)}$: ${(x-h)^2=4p(y-k)}$ or ${(y-k)^2=4p(x-h)}$. Complete the square to find ${(h,k)}$.
- Rays parallel to the axis reflect through the focus.