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Advanced Math Difficulty: Hard

An auditorium has seats for $1,800$ people. Tickets to attend a show at the auditorium currently cost $dollar sign 4.00$ . For each $dollar sign 1.00$ increase to the ticket price, $100$ fewer tickets will be sold. This situation can be modeled by the equation $y = - 100 x^{2} + 1,400 x + 7,200$ , where $x$ represents the increase in ticket price, in dollars, and $y$ represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of $x$ is the maximum of the graph?

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Explanation

Choice B is correct. It’s given that the situation can be modeled by the equation $y = - 100 x^{2} + 1,400 x + 7,200$ , where $x$ represents the increase in ticket price, in dollars, and $y$ represents the revenue, in dollars, from ticket sales. Since the coefficient of the $x^{2}$ term is negative, the graph of this equation in the xy-plane opens downward and reaches its maximum value at its vertex. If a quadratic equation in the form $y = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants, is graphed in the xy-plane, the x-coordinate of the vertex is equal to $- \frac{b}{2 a}$ . For the equation $y = - 100 x^{2} + 1,400 x + 7,200$ , $a = - 100$ , $b equals 1,400$ , and $c equals 7,200$ . It follows that the x-coordinate of the vertex is $- \frac{1,400}{2 (- 100)}$ , or $7$ . Therefore, if the given equation is graphed in the xy-plane, the maximum of the graph occurs at an x-value of $7$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.