12.3 The parabola  (Page 3/11)

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Graph $y^2=\mathrm{-16}x.$ Identify and label the focus, directrix, and endpoints of the latus rectum.

Focus: $\left(-4,0\right);$ Directrix: $x=4;$ Endpoints of the latus rectum: $\left(-4,\pm 8\right)$

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Graphing a parabola with vertex (0, 0) and the y -axis as the axis of symmetry

Graph $x^2=\mathrm{-6}y.$ Identify and label the focus , directrix    , and endpoints of the latus rectum    .

The standard form that applies to the given equation is $x^2=4py.$ Thus, the axis of symmetry is the y -axis. It follows that:

• $-6=4p,$ so $p=-\frac{3}{2}.$ Since $p<0,$ the parabola opens down.
• the coordinates of the focus are $\left(0,p\right)=\left(0,-\frac{3}{2}\right)$
• the equation of the directrix is $y=-p=\frac{3}{2}$
• the endpoints of the latus rectum can be found by substituting $y=\frac{3}{2}$ into the original equation, $\left(\pm 3,-\frac{3}{2}\right)$

Next we plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola    .

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Graph $x^2=8y.$ Identify and label the focus, directrix, and endpoints of the latus rectum.

Focus: $\left(0,2\right);$ Directrix: $y=\mathrm{-2};$ Endpoints of the latus rectum: $\left(\pm 4,2\right).$

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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.

Graphing parabolas with vertices not at the origin

Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated $h$ units horizontally and $k$ units vertically, the vertex will be $\left(h,k\right).$ This translation results in the standard form of the equation we saw previously with $x$ replaced by $\left(x-h\right)$ and $y$ replaced by $\left(y-k\right).$

To graph parabolas with a vertex $\left(h,k\right)$ other than the origin, we use the standard form ${\left(y-k\right)}^2=4p\left(x-h\right)$ for parabolas that have an axis of symmetry parallel to the x -axis, and ${\left(x-h\right)}^2=4p\left(y-k\right)$ 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.

A General Note

Standard forms of parabolas with vertex ( h , k )

[link] and [link] summarize the standard features of parabolas with a vertex at a point $\left(h,k\right).$

Axis of Symmetry	Equation	Focus	Directrix	Endpoints of Latus Rectum
$y=k$	${\left(y-k\right)}^2=4p\left(x-h\right)$	$\left(h+p,k\right)$	$x=h-p$	$\left(h+p,k\pm 2p\right)$
$x=h$	${\left(x-h\right)}^2=4p\left(y-k\right)$	$\left(h,k+p\right)$	$y=k-p$	$\left(h\pm 2p,k+p\right)$