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 [latex]\left(p,0\right)[/latex], then the axis of symmetry is the x-axis. Use the standard form [latex]{y}^{2}=4px[/latex].
    • If the given coordinates of the focus have the form [latex]\left(0,p\right)[/latex], then the axis of symmetry is the y-axis. Use the standard form [latex]{x}^{2}=4py[/latex].
  • Multiply [latex]4p[/latex].
  • 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 [latex]\left(-\frac{1}{2},0\right)[/latex] and directrix [latex]x=\frac{1}{2}?[/latex]

Solution

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

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

Therefore, the equation for the parabola is [latex]{y}^{2}=-2x[/latex].

Try It 4

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

Solution