which function represents profit, p(x), as a function of x, the price per music class? p(x)=-(x - 6)^2+116…

Explanation:

Step1: Recall vertex - form of a quadratic function

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. If $a<0$, the parabola opens downwards.

Step2: Identify the vertex of the profit - price graph

From the graph of profit vs. price, the vertex (the maximum point of the profit function) is at the point $(6,116)$. So, $h = 6$ and $k=116$. Since the profit function has a maximum (the parabola opens downwards), $a<0$.

Step3: Check the given functions

For the function $P(x)=-(x - 6)^2+116$, it is in the vertex - form $y=a(x - h)^2+k$ with $a=- 1$, $h = 6$, and $k = 116$. For the function $P(x)=(x + 6)^2+116$, the vertex is $(-6,116)$ which is not correct according to the graph. The functions $P(x)=-2(80^{x})$ and $P(x)=-80(2^{x})$ are exponential functions. The graph of an exponential function $y = ab^{x}$ is not a parabola. The graph of profit vs. price appears to be a parabola.

Answer:

$P(x)=-(x - 6)^2+116$