Graphing Quadratic Equations in Vertex Form

1. \(81\) meters

The height of the hovercraft at the time of takeoff is given by \(h(0)\).

\(\begin{aligned}

h(0) &=-3(0-3)^{2}+108 \\

&=-3(9)+108 \\

&=81

\end{aligned}\)

In conclusion, the height of the hovercraft at the time of takeoff is \(81\) meters.

2. \(5\) dollars

The company's profit is modeled by a quadratic function, whose graph is a parabola.

The maximum profit is reached at the vertex.

So in order to find when that happens, we need to find the vertex's \(x\)-coordinate.

The function \(P(x)\) is given in vertex form.

The vertex of \(-3(x-5)^{2}+12 \text { is at }(5,12)\).

In conclusion, the company will earn a maximum profit when the socks are priced at \(5\) dollars.

3. \(200\) thousand fish

The fish population is modeled by a quadratic function, whose graph is a parabola.

The maximum number of fish is reached at the vertex.

So in order to find the maximum number of fish, we need to find the vertex's \(y\)-coordinate.

The function \(P(x)\) is given in vertex form.

The vertex of \(-2(x-9)^{2}+200\) is at \((9,200)\).

In conclusion, the maximum fish population is \(200\) thousand.

4. Lower current: \(0\) amperes, Higher current: \(6\) amperes

The circuit's power is \(0\) when \(P(c)=0\).

\(\begin{gathered}

P(c)=0 \\

-20(c-3)^{2}+180=0 \\

-20(c-3)^{2}=-180 \\

(c-3)^{2}=9 \\

\sqrt{(c-3)^{2}}=\sqrt{9} \\

c-3=\pm 3 \\

c=\pm 3+3 \\

c=6 \text { or } c=0

\end{gathered}\)

In conclusion, these are the currents that will produce no power:

Lower current: \(0\) amperes

Higher current: \(6\) amperes