Final answer:
The student's question concerning Janet's donut shop revenue revolves around understanding quadratic functions representing the revenue as the price of donuts changes. Two alternative forms of the quadratic equation are given, and the use of graphs is suggested as an approach to find the maximum revenue and break-even points.
Step-by-step explanation:
The question is asking for a detailed explanation of the function of Janet's donut shop revenue, which is given in three different forms of quadratic equations. Each equation represents the same revenue function R(x) and showcases how revenue changes with varying price levels for a single donut.
The first form R(x) = -2x² + 12x - 10 represents a standard quadratic equation. The second R(x) = -2(x-3)² +8 is the vertex form that indicates the maximum revenue point at the vertex (3,8). Lastly, the factored form R(x) = -2(x - 1)(x - 5) reveals the price points where revenue is zero. By examining these forms, a student can understand the relationship between price and revenue, as well as solve for maximum revenue and break-even points.
Using graphs is an alternative to solving the quadratic equations algebraically. Plotting the demand curve P = 8 -0.5Qd and supply curve P = -0.4 +0.2Qs on the same set of axes, as indicated in the supply and demand model, can visualize the equilibrium where supply equals demand at a specific price and quantity.
For example, in the scenario with personal pizzas, where the price is set at $2, the equilibrium occurs at a supply of 12 pizzas, confirming the correctness of the algebraic or graphical solution. This type of analysis can also be applied to Janet's donut shop revenue equations to find the optimal price that maximizes revenue or to find the break-even points.