Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A company's profit P (in thousands of dollars) from selling x hundreds of units is modeled by the function P(x) = -2x^2 + 20x - 30. What is the maximum profit the company can achieve?
- A$50,000
- B$30,000
- C$20,000
- D$10,000
Show answer & explanationAnswer & explanation
Correct answer: C. $20,000
This is a quadratic function with a negative leading coefficient (-2), meaning its graph is a parabola opening downward, and its vertex represents the maximum profit. The x-coordinate of the vertex is -b/(2a). For P(x) = -2x^2 + 20x - 30, a=-2 and b=20. So, x = -20 / (2 * -2) = -20 / -4 = 5. Now substitute x=5 into the profit function: P(5) = -2(5)^2 + 20(5) - 30 = -2(25) + 100 - 30 = -50 + 100 - 30 = 20. Since P is in thousands of dollars, the maximum profit is $20,000.
Why the other options are wrong
- A. Incorrect. This value is too high and likely results from a major error in calculation.
- B. Incorrect. This could be a miscalculation or selecting the constant term.
- D. Incorrect. This might be a result of a calculation error after finding the vertex.
Quadratic Max/Min Applications
Using the vertex of a quadratic function to find the maximum or minimum value in real-world scenarios, such as profit maximization or cost minimization.
- If 'a' (leading coefficient) > 0, vertex is a minimum.
- If 'a' < 0, vertex is a maximum.
- x-coordinate of vertex: -b/(2a).
- y-coordinate of vertex: substitute x-value into function.
Memory trick: Downward parabola, profit's peak; upward, cost's lowest streak.