Graph the equation \(x^{2}=16y\).

Solution  The graph of \(x^{2}=16y\) is a parabola of the form \[ x^{2}=4ay \]

where \(a=4\). Look at Figure 37(c). The graph will open up, with focus at\( ( 0,4) \). The vertex is at \(( 0,0) \). To graph the parabola, we let \(y=a;\) that is, let \(y=4\). (Any other positive number will also work.) \begin{eqnarray*} x^{2} &=&16y \underset{\underset{\color{#0066A7}{\hbox{\(y=a=4\)}}}{\color{#0066A7}{\displaystyle\uparrow }}}{=} 16( 4) =64 \\[-5.6pt] x &=&-8\qquad\hbox{ or }\qquad x=8 \end{eqnarray*}

image \(x^{2}=16y\)

The points\( ( -8,4)\) and\( ( 8,4)\) are on the parabola and establish its opening. See Figure 38.

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