Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A company's monthly profit P(x) (in thousands of dollars) from selling 'x' units of a product is given by the function P(x) = -0.5x^2 + 10x - 12. At what number of units sold will the company achieve its maximum profit?
- A12 units
- B20 units
- C5 units
- D10 units
Show answer & explanationAnswer & explanation
Correct answer: D. 10 units
For a quadratic function in the form f(x) = ax^2 + bx + c, the x-coordinate of the vertex (which represents the maximum or minimum) is given by x = -b / (2a). Here, a = -0.5 and b = 10. So, x = -10 / (2 * -0.5) = -10 / -1 = 10 units.
Why the other options are wrong
- A. This is the y-intercept, representing profit when 0 units are sold, not the maximum profit point.
- B. This might be a result of incorrectly dividing by 'a' instead of '2a' or a sign error.
- C. This might come from miscalculating -b/2a, or assuming it's related to the constant term.
Vertex of a Parabola (Quadratic)
The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function.
- For f(x) = ax^2 + bx + c, the x-coordinate of the vertex is x = -b / (2a).
- If a > 0, the parabola opens upward, and the vertex is a minimum.
- If a < 0, the parabola opens downward, and the vertex is a maximum.
- The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.
Memory trick: Vertex formula: x equals negative b over two a, finds the peak!