Conic Sections Reference
Parabolas as Conic Sections
A parabola is the set of all points the same distance from a focus and a directrix. This geometric definition explains the focus, directrix, focal width, and latus rectum.
Fact Table
| Form | Opens | Focus and Directrix |
|---|---|---|
| $(x-h)^2=4p(y-k)$ | Up if $p>0$, down if $p<0$. | Focus $(h,k+p)$, directrix $y=k-p$. |
| $(y-k)^2=4p(x-h)$ | Right if $p>0$, left if $p<0$. | Focus $(h+p,k)$, directrix $x=h-p$. |
| Vertex | Turning point. | $(h,k)$ |
| Latus rectum | Segment through the focus perpendicular to the axis. | Length $|4p|$. |
Content Formulas
Vertical Axis
$$(x-h)^2=4p(y-k)$$
Horizontal Axis
$$(y-k)^2=4p(x-h)$$
Focal Width
$$\text{length of latus rectum}=|4p|$$
Function Connection
$$y=a(x-h)^2+k\quad\Longleftrightarrow\quad a=\frac1{4p}$$
The squared variable tells us the direction: if $x$ is squared, the parabola opens up or down; if $y$ is squared, it opens left or right.
Classic Examples
Find Focus and Directrix
Find the focus, directrix, and focal width of $(x-2)^2=12(y+1)$.
Solution StrategyMatch the equation to $(x-h)^2=4p(y-k)$.
$$(x-2)^2=12(y-(-1))$$
$$h=2,\quad k=-1,\quad 4p=12$$
$$p=3$$
$$\text{focus: }(2,2)$$
$$\text{directrix: }y=-4$$
$$\text{focal width: }12$$
Write an Equation
A parabola has vertex $(1,-2)$ and focus $(5,-2)$. Write its equation.
Solution StrategyThe focus is to the right of the vertex, so use the horizontal form.
$$h=1,\quad k=-2,\quad p=4$$
$$(y+2)^2=4(4)(x-1)$$
$$(y+2)^2=16(x-1)$$