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Graph $\text{\hspace{0.17em}}{y}^{2}=\mathrm{-16}x.\text{\hspace{0.17em}}$ Identify and label the focus, directrix, and endpoints of the latus rectum.
Focus: $\text{\hspace{0.17em}}\left(-4,0\right);\text{\hspace{0.17em}}$ Directrix: $\text{\hspace{0.17em}}x=4;\text{\hspace{0.17em}}$ Endpoints of the latus rectum: $\text{\hspace{0.17em}}\left(-4,\pm 8\right)$
Graph $\text{\hspace{0.17em}}{x}^{2}=\mathrm{-6}y.\text{\hspace{0.17em}}$ Identify and label the focus , directrix , and endpoints of the latus rectum .
The standard form that applies to the given equation is $\text{\hspace{0.17em}}{x}^{2}=4py.\text{\hspace{0.17em}}$ Thus, the axis of symmetry is the y -axis. It follows that:
Next we plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola .
Graph $\text{\hspace{0.17em}}{x}^{2}=8y.\text{\hspace{0.17em}}$ Identify and label the focus, directrix, and endpoints of the latus rectum.
Focus: $\text{\hspace{0.17em}}\left(0,2\right);\text{\hspace{0.17em}}$ Directrix: $\text{\hspace{0.17em}}y=\mathrm{-2};\text{\hspace{0.17em}}$ Endpoints of the latus rectum: $\text{\hspace{0.17em}}\left(\pm 4,2\right).$
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.
Given its focus and directrix, write the equation for a parabola in standard form.
What is the equation for the parabola with focus $\text{\hspace{0.17em}}\left(-\frac{1}{2},0\right)\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}x=\frac{1}{2}?$
The focus has the form $\text{\hspace{0.17em}}\left(p,0\right),$ so the equation will have the form $\text{\hspace{0.17em}}{y}^{2}=4px.$
Therefore, the equation for the parabola is $\text{\hspace{0.17em}}{y}^{2}=\mathrm{-2}x.$
What is the equation for the parabola with focus $\text{\hspace{0.17em}}\left(0,\frac{7}{2}\right)\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}y=-\frac{7}{2}?$
${x}^{2}=14y.$
Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ units horizontally and $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ units vertically, the vertex will be $\text{\hspace{0.17em}}\left(h,k\right).\text{\hspace{0.17em}}$ This translation results in the standard form of the equation we saw previously with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ replaced by $\text{\hspace{0.17em}}\left(x-h\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ replaced by $\text{\hspace{0.17em}}\left(y-k\right).$
To graph parabolas with a vertex $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ other than the origin, we use the standard form $\text{\hspace{0.17em}}{\left(y-k\right)}^{2}=4p\left(x-h\right)\text{\hspace{0.17em}}$ for parabolas that have an axis of symmetry parallel to the x -axis, and $\text{\hspace{0.17em}}{\left(x-h\right)}^{2}=4p\left(y-k\right)\text{\hspace{0.17em}}$ for parabolas that have an axis of symmetry parallel to the y -axis. These standard forms are given below, along with their general graphs and key features.
[link] and [link] summarize the standard features of parabolas with a vertex at a point $\text{\hspace{0.17em}}\left(h,k\right).$
Axis of Symmetry | Equation | Focus | Directrix | Endpoints of Latus Rectum |
$y=k$ | ${\left(y-k\right)}^{2}=4p\left(x-h\right)$ | $\left(h+p,\text{}k\right)$ | $x=h-p$ | $\left(h+p,\text{}k\pm 2p\right)$ |
$x=h$ | ${\left(x-h\right)}^{2}=4p\left(y-k\right)$ | $\left(h,\text{}k+p\right)$ | $y=k-p$ | $\left(h\pm 2p,\text{}k+p\right)$ |
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