profit vs. price\nwhich function represents profit, $p(x)$, as a function of $x$, the price per music…

profit vs. price\nwhich function represents profit, $p(x)$, as a function of $x$, the price per music class?\n$p(x)=-(x - 6)^2+116$\n$p(x)=(x + 6)^2+116$\n$p(x)=-2(80^x)$\n$p(x)=-80(2^x)$

profit vs. price\nwhich function represents profit, $p(x)$, as a function of $x$, the price per music class?\n$p(x)=-(x - 6)^2+116$\n$p(x)=(x + 6)^2+116$\n$p(x)=-2(80^x)$\n$p(x)=-80(2^x)$

Answer

Explanation:

Step1: Analyze the 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. Looking at the graph, the vertex (the maximum point of the profit - price relationship) is at the point ((6,116)), so (h = 6) and (k=116). The function should be of the form (P(x)=a(x - 6)^2+116). Since the parabola opens downwards (the profit decreases as the price moves away from (x = 6)), (a<0).

Step2: Check the options

Option 1: (P(x)=-(x - 6)^2+116) is in the correct vertex - form with (a=- 1<0), (h = 6) and (k = 116). Option 2: (P(x)=(x + 6)^2+116) has the wrong (h) value ((h=-6) instead of (h = 6)) and the parabola opens upwards ((a = 1>0)). Option 3: (P(x)=-2(80^x)) is an exponential function, and the graph of an exponential function (y = ab^x) ((b>0,b\neq1)) is not a parabola. Option 4: (P(x)=-80(2^x)) is also an exponential function, and its graph is not a parabola.

Answer:

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