Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A financial analyst is modeling the quarterly profit of a company. The profit P (in thousands of dollars) from selling x units of a product is given by the function P(x) = -0.5x^2 + 20x - 150. What is the maximum profit the company can achieve?
- A$200,000
- B$100,000
- C$150,000
- D$50,000
Show answer & explanationAnswer & explanation
Correct answer: D. $50,000
The profit function is a quadratic equation with a negative leading coefficient, meaning its graph is a downward-opening parabola. The maximum profit occurs at the vertex. Calculate the x-coordinate of the vertex using x = -b/(2a), then substitute this value back into the profit function to find the maximum profit.
Why the other options are wrong
- A. Incorrectly calculated the vertex, potentially doubling the correct answer.
- B. Made an error in calculation for the vertex y-coordinate.
- C. Incorrectly used the constant term or made calculation errors.
Maximum of Quadratic Function
For a quadratic function f(x) = ax^2 + bx + c with a < 0, the maximum value occurs at the vertex.
- The x-coordinate of the vertex is given by -b/(2a).
- Substitute this x-value into the function to find the maximum y-value.
- If a > 0, the vertex represents a minimum value instead.
Memory trick: Vertex is the peak or pit, -b/2a is the right fit.