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Study Guides > College Algebra

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 (p,0)\left(p,0\right), then the axis of symmetry is the x-axis. Use the standard form y2=4px{y}^{2}=4px.
    • If the given coordinates of the focus have the form (0,p)\left(0,p\right), then the axis of symmetry is the y-axis. Use the standard form x2=4py{x}^{2}=4py.
  • Multiply 4p4p.
  • 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 (12,0)\left(-\frac{1}{2},0\right) and directrix x=12?x=\frac{1}{2}?

Solution

The focus has the form (p,0)\left(p,0\right), so the equation will have the form y2=4px{y}^{2}=4px.
  • Multiplying 4p4p, we have 4p=4(12)=24p=4\left(-\frac{1}{2}\right)=-2.
  • Substituting for 4p4p, we have y2=4px=2x{y}^{2}=4px=-2x.
Therefore, the equation for the parabola is y2=2x{y}^{2}=-2x.

Try It 4

What is the equation for the parabola with focus (0,72)\left(0,\frac{7}{2}\right) and directrix y=72?y=-\frac{7}{2}? Solution

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