- A company's profit function is graphed on a coordinate plane. The graph shows the function crosses the x-axis at x=2 and x=8, has a y-intercept of −24, and approaches negative infinity as x approaches both positive and negative infinity. Based on this information, what can be concluded about the company's break-even points and long-term profitability?
Answer: The company breaks even at 2 and 8 units, but will experience losses for very large production volumes
The x-intercepts at x = 2 and x = 8 represent break-even points where profit = 0. The y-intercept of -24 means the company has a loss of $24 when producing zero units (perhaps due to fixed costs). The end behavior (approaching negative infinity as x approaches ±∞) indicates the company will experience losses for very large production volumes, contradicting the idea of long-term profitability through economies of scale.
- A demand function D(p) is graphed with price p on the horizontal axis and quantity demanded on the vertical axis. The graph shows a y-intercept of 80, an x-intercept of 40, and the function is linear. If the company currently prices their product at $25, what is the slope of this demand function and what does it represent economically?
Answer: The slope is -2, indicating that for each $1 increase in price, demand decreases by 2 units on average
With points (0, 80) and (40, 0), the slope is (0-80)/(40-0) = -2. This means for each $1 increase in price, quantity demanded decreases by 2 units. Choice B confuses slope with elasticity. Choice C incorrectly interprets slope as marginal revenue. Choice D incorrectly relates slope to profit margin.
- An investment value function shows the following characteristics: y-intercept at 1000, no x-intercepts in the visible domain, decreasing slope over time, and approaches a horizontal asymptote at y=3000 as time approaches infinity. If the slope at t=2 years is 400 dollars per year, what does this information suggest about the investment's performance and risk profile?
Answer: Initial investment of $1000 grows toward $3000 maximum value; the decreasing slope indicates diminishing returns but no risk of loss
The y-intercept of 1000 is the initial investment, the asymptote at 3000 represents the maximum value the investment approaches, and no x-intercepts means the value never reaches zero (no loss of principal). The decreasing slope indicates diminishing returns over time, but since the function never crosses the x-axis, there's no risk of losing the initial investment. Choice B incorrectly interprets the slope as an interest rate. Choices C and D incorrectly associate decreasing slope with increasing risk.
