A woodworker makes wooden checkerboards. Her profit is a function of the price she charges. This graph shows her total profits, y, based on the sales price, x of each checkerboard. Identify any zeros of the function, and interpret what the zeros mean in terms of the situation. A. Zero: y=1512.5. This is the maximum amount of profit she can make. B. Zeros: x=15 and x=70. These are the prices at which she neither makes nor loses money. C. Zero: x=42.5. This is the selling price that will give her the greatest profit. D. Zeros: x=15 and x=70. These are the prices at which she makes the least and greatest amounts of profit.
The Correct Answer and Explanation is:
Correct Answer: B. Zeros: x = 15 and x = 70. These are the prices at which she neither makes nor loses money.
Explanation:
In mathematics, the zeros of a function are the values of the independent variable (in this case, the price per checkerboard, x) for which the dependent variable (the profit, y) equals zero. Graphically, these are the points where the graph of the function crosses the x-axis. At those points, the profit is neither positive nor negative—in other words, the woodworker breaks even.
Given the problem, we are told that the profit (y) is graphed as a function of the selling price (x). If the graph touches the x-axis at x = 15 and x = 70, these are the zeros of the function. This tells us that if the woodworker charges either $15 or $70 per checkerboard, she will make zero profit—her revenue will just cover her costs, with no gain or loss.
This situation typically arises when a business sets a price too low (like $15), making many sales but not enough margin to cover costs, or too high (like $70), where high prices discourage buyers, resulting in too few sales. In both cases, profit drops to zero.
By contrast:
- Option A is incorrect because y = 1512.5 is not a zero of the function; it’s likely the maximum profit, which occurs at the vertex of the parabola.
- Option C is incorrect because x = 42.5 is the x-value that gives maximum profit, not a zero.
- Option D misinterprets the meaning of zeros—x = 15 and x = 70 represent points where profit is zero, not the lowest and highest profits.
Thus, option B correctly identifies the zeros and explains their meaning within the real-world context: break-even prices.