Question

The surface of a pedestrian bridge forms a parabola. Let the surface at one side of the bridge be represented by the origin (0,0) and the surface at the other side be represented by (16,0). The center of the bridge is 2 feet higher than each side and can be represented by a vertex of (8,2). Write a function in vertex form that models the surface of the bridge.

Please help.

Answer 1

Check the picture below. So the bridge more or less looks like so.

since we know the vertex, we'll use that, and we also know a point on the parabola as well, namely (16,0).

`\bf ~~~~~~\textit{parabola vertex form} \\\\ \begin{array}{llll} \stackrel{\textit{we'll use this one}}{y=a(x- h)^2+ k}\\\\ x=a(y- k)^2+ h \end{array} \qquad\qquad vertex~~(\stackrel{}{ h},\stackrel{}{ k}) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{cases} h=8\\ k=2 \end{cases}\implies y=a(x-8)^2+2\qquad (16,0)~~ \begin{cases} x=16\\ y=0 \end{cases}`

`\bf 0=a(16-8)^2+2\implies -2 = a(8)^2\implies -2=64a \\\\\\ \cfrac{-2}{64}=a\implies \cfrac{-1}{32}=a \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill y=-\cfrac{1}{32}(x-8)^2+2~\hfill`