Parabolas Centered at the Origin

In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

HOW TO

Given its focus and directrix, write the equation for a parabola in standard form.

1. Determine whether the axis of symmetry is the $x$- or $y$-axis.
   1. If the given coordinates of the focus have the form $(p,0)$, then the axis of symmetry is the $x$-axis. Use the standard form $y^2=4px$.
   2. If the given coordinates of the focus have the form $(0,p)$, then the axis of symmetry is the $y$-axis. Use the standard form $x^2=4py$.
2. Multiply $4p$.
3. Substitute the value from Step 2 into the equation determined in Step 1.

EXAMPLE 3

Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix

What is the equation for the parabola with focus $(−\frac{1}{2},0)$ and directrix $x=\frac{1}{2}$?

Solution

The focus has the form $(p,0)$, so the equation will have the form $y^2=4px$.

• Multiplying $4p$, we have $4p=4(−\frac{1}{2})=−2$.
• Substituting for $4p$, we have $y^2=4px=−2x$.

Therefore, the equation for the parabola is $y^2=−2x$.

TRY IT #3

What is the equation for the parabola with focus $(0, \frac{7}{2})$ and directrix $y=−\frac{7}{2}$?