Digital SATMath: Advanced MathMedium
A financial analyst is modeling the quarterly profit of a startup using the function P(x) = -2x^2 + 20x - 30, where P(x) is the profit in thousands of dollars and x is the quarter number (x ≥ 1). What is the maximum profit the startup can achieve?
- A$50,000
- B$20,000
- C$10,000
- D$30,000
Show answer & explanationAnswer & explanation
Correct answer: B. $20,000
The profit function is a downward-opening parabola, so the maximum profit occurs at the vertex. The x-coordinate of the vertex is -b/(2a) = -20/(2 * -2) = -20/-4 = 5. Substituting x=5 into the profit function: P(5) = -2(5)^2 + 20(5) - 30 = -2(25) + 100 - 30 = -50 + 100 - 30 = 20. So the maximum profit is $20,000.
Why the other options are wrong
- A. This is incorrect; it might be a miscalculation of the vertex or a misunderstanding of the function.
- C. This is incorrect; it might be a miscalculation or selecting the x-coordinate instead of the y-coordinate.
- D. This is the y-intercept, which is not the maximum profit.
Vertex of a Parabola
For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex is the point (h, k) where h = -b/(2a) and k = f(h). It represents the maximum or minimum value of the function.
- If a > 0, the parabola opens upwards, and the vertex is a minimum.
- If a < 0, the parabola opens downwards, and the vertex is a maximum.
- The x-coordinate of the vertex gives the input value at which the max/min occurs.
Memory trick: Vertex is the 'top' or 'bottom' of the 'V' or 'A' shape.