Writing Equations of Parabolas in Standard Form

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.

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

Example 4: Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix

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

Solution

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

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

Therefore, the equation for the parabola is ${y}^{2}=-2x$ .

Try It 4

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

Solution

The Parabola