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 \(y=−\frac{7}{2}\)?