The graph of a company's profit function is shown. Use the graph to answer the following questions about points A, B, and C. The x y-coordinate plane is given. The horizontal axis is labeled Units. The vertical axis is labeled Dollars. The curve P(x) enters the window on the y-axis at y = −2000, goes up and right becoming less steep, passes through point A at approximately (30, −800), crosses the x-axis at x = 50, passes through point B at approximately (200, 3500), passes through point C at approximately (350, 5000), and exits the window in the first quadrant. (a) Rank from smallest to largest the amounts of profit received at these points. a. A, B, C b. A, C, B c. B, A, C d. B, C, A e. C, A, B f. C, B, A

To solve this problem, we need to interpret the graph of the profit function \( P(x) \) based on the given descriptions of points A, B, and C. The y-values (Dollars) represent the company's profit at each point, while the x-values (Units) represent the quantity sold or produced.

### Coordinates:
- Point A: \( (30, -800) \)
- Point B: \( (200, 3500) \)
- Point C: \( (350, 5000) \)

The y-coordinates at each of these points tell us the profit at those points:
- At A, the profit is \( -800 \) dollars.
- At B, the profit is \( 3500 \) dollars.
- At C, the profit is \( 5000 \) dollars.

### Ranking the profits:
From smallest to largest:
1. Point A has the smallest profit, \( -800 \) dollars.
2. Point B has a profit of \( 3500 \) dollars.
3. Point C has the largest profit, \( 5000 \) dollars.

Thus, the correct ranking from smallest to largest profit is **A, B, C**, corresponding to choice **a**.

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Would you like further details on how to interpret graphs of functions, or any other questions related to this?

### Related Questions:
1. What does the slope of the profit function represent in this context?
2. How do we determine the maximum profit from a profit function?
3. What happens at the x-intercept of the profit function, and why is it important?
4. Can we use derivatives to identify points like A, B, and C on this profit graph?
5. How does the shape of the curve indicate changes in profit over time?

### Tip:
In economics, the x-axis often represents quantity, while the y-axis represents values like revenue, cost, or profit. Understanding how these relate visually helps in decision-making.

Analyze the company's profit function graph and rank the profits at points A, B, and C. This problem covers key concepts in coordinate geometry and profit function interpretation, providing step-by-step guidance for students in Grades 9-12.

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