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 , then the axis of symmetry is the x-axis. Use the standard form .
- If the given coordinates of the focus have the form , then the axis of symmetry is the y-axis. Use the standard form .
- Multiply .
- 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 and directrixSolution
The focus has the form , so the equation will have the form .- Multiplying , we have .
- Substituting for , we have .