Parabolas as Conic Sections

Example 1:

Find the focus and directrix of the parabola $y=-2x^2$ .

Since the $x$ -term is squared, the axis is vertical, and the standard form is

$x^2=4py$

Rewriting the equation in standard form:

$x^2=-\frac{1}{2}y$

Comparing the equation with the standard form:

$\begin{array}{l}4p=-\frac{1}{2}\\ p=-\frac{1}{8}\end{array}$

The focus of the parabola which is in standard form $x^2=4py$ , is $\left(0,p\right)$ .

So, the focus of the equation is $\left(0,-\frac{1}{8}\right)$ .

The directrix of the parabola which is in standard form $x^2=4py$ , is $y=-p$ .

So, the directrix of the equation is $y=\frac{1}{8}$ .

Example 2:

Graph the equation and then find the focus and directrix of the parabola $y^2=-3x$ .

The equation is of the form $y^2=4px$ where $4p=-3$ .

Solving for $p$ :

$p=-\frac{3}{4}$

The focus of the parabola which is in standard form $y^2=4px$ , is $\left(p,0\right)$ .

So, the focus of the equation is $\left(-\frac{3}{4},0\right)$ .

The directrix of the parabola which is in standard form $y^2=4px$ , is $x=-p$ .

So, the directrix of the equation is $x=\frac{3}{4}$ .

Since the variable $y$ is squared, the axis of symmetry is horizontal. Also the value of $p$ is less than $0$ , the parabola opens to the left.

Choose negative $x$ -values and make a table.

$X$	$-1$	$-2$	$-3$	$-4$
$Y$	$1.73$	$2.45$	$3$	$3.46$

Plot the points and draw a parabola through the points.