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.
In this section, we will write the equation of a parabola in standard form, as opposed to the equation of a quadratic or second degree polynomial. The language we use when discussing the object is specific.It is true that a quadratic function forms a parabola when graphed in the plane, but here we are using the phrase standard form of the equation of a parabola to indicate that we wish to describe the geometric object. When talking about this object in this context, we would naturally use the equations described below.
How To: Given its focus and directrix, write the equation for a parabola in standard form.
Determine whether the axis of symmetry is the [latex]x[/latex]– or [latex]y[/latex]-axis.
If the given coordinates of the focus have the form [latex]\left(p,0\right)[/latex], then the axis of symmetry is the [latex]x[/latex]-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 [latex]y[/latex]-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.
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]
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].