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