Digital SATMath: Advanced MathMedium
A financial analyst is modeling the quarterly profit P(x) of a company, in thousands of dollars, as a function of the number of items sold, x, in hundreds. The model is given by P(x) = -2x^2 + 12x - 10. What is the maximum profit the company can achieve?
- A$18,000
- B$10,000
- C$8,000
- D$12,000
Show answer & explanationAnswer & explanation
Correct answer: C. $8,000
The maximum profit occurs at the vertex of the parabola. For a quadratic function in the form ax^2 + bx + c, the x-coordinate of the vertex is -b/(2a). Substitute this x-value back into the function to find the maximum P(x).
Why the other options are wrong
- A. This would be an error in calculation, possibly forgetting the negative sign of a or miscalculating the square.
- B. This is the y-intercept (when x=0), not the maximum profit.
- D. This is the value of 12x if x=1, but not the maximum profit.
Vertex of a Parabola (Maximum/Minimum)
For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex represents the maximum point if a < 0 (parabola opens down) or the minimum point if a > 0 (parabola opens up).
- The x-coordinate of the vertex is given by x = -b / (2a).
- Substitute this x-value back into the function to find the y-coordinate (maximum/minimum value).
- The vertex is the turning point of the parabola.
Memory trick: Find the peak, with -b over 2a, that's the trick!