- The \(x\)-coordinate of the vertex represents the optimal selling price for maximum profit (\(\boxed{D}\)).
- The \(y\)-coordinate of the vertex represents the maximum daily profit itself (\(\boxed{B}\)).
- The \(x\)-intercepts represent prices where profit is zero, known as break-even points (\(\boxed{C}\)).
- The \(y\)-intercept represents the financial outcome (a loss) if the T-shirts are given away for free (\(\boxed{A}\)).
Explanation
- Understanding the Profit Function Model
The profit function \(y = -10x^2 + 160x - 430\) is a quadratic equation that describes the relationship between the selling price of T-shirts \(x\) and the daily profit \(y\). The graph of this function is a parabola that opens downward because the coefficient of \(x^2\) is negative (\(-10 < 0\)). This tells us that there is a maximum profit possible at the vertex of the parabola. We need to identify which components of this parabola correspond to specific business scenarios: the break-even prices, the maximum profit, the optimal selling price, and the profit when the item is free.
- Analyzing the Vertex of the Parabola
The vertex of a parabola \(f(x) = ax^2 + bx + c\) is found at an \(x\)-coordinate of \(x = -\frac{b}{2a}\). In this scenario, the \(x\)-coordinate represents the price that leads to the highest possible value for the profit function. Thus, the \(x\)-coordinate of the vertex corresponds to the selling price that yields the maximum daily profit, which is description D. The \(y\)-coordinate at this vertex, \(f(x_{vertex})\), represents the highest output of the function, which in this case is the maximum possible daily profit. This matches description B.
- Analyzing the Intercepts
An \(x\)-intercept of a function occurs where the output (in this case, profit \(y\)) is zero. If the profit is zero, the business is neither making money nor losing money, which is known as a break-even point. Therefore, an \(x\)-intercept represents a selling price that results in no profit, which is description C. The \(y\)-intercept occurs when the input \(x\) is zero (price is \$0). Substituting \(x = 0\) into the function, we find \(y = -430\). This represents the daily loss the company faces if they give the T-shirts away for free. This matches description A.
- Conclusion and Mapping Results
Combining our findings, we map the properties of the parabola to the given descriptions:
- The \(x\)-coordinate of the vertex represents the price for maximum profit (D).
- The \(y\)-coordinate of the vertex represents the maximum profit itself (B).
- An \(x\)-intercept represents a break-even price (C).
- The \(y\)-intercept represents the profit when the items are free (A).
Thus, the final assignments are:
\(\boxed{D}\) \(x\)-coordinate of the vertex of the function
\(\boxed{B}\) \(y\)-coordinate of the vertex of the function
\(\boxed{C}\) an \(x\)-intercept of the function
\(\boxed{A}\) a \(y\)-intercept of the function
Examples
In business and economics, quadratic models are frequently used to optimize outcomes like profit or revenue. For example, a bakery might find that increasing the price of a loaf of bread helps profit up to a certain point, after which customers buy less, causing total profit to decline. By modeling their daily profit as a quadratic function of price, the bakery owner can mathematically determine the "sweet spot"—the price that maximizes their daily earnings—ensuring they operate as efficiently as possible. Understanding how the vertex and intercepts of these models relate to real-world variables like 'price' and 'profit' is a powerful tool for any business owner looking to make data-driven decisions.