Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
A project manager is analyzing task completion times. The time T (in hours) to complete a certain task is modeled by the function T(x) = 2x^2 - 12x + 25, where x is the number of workers assigned to the task. What is the minimum time required to complete the task?
- A12 hours
- B7 hours
- C3 hours
- D25 hours
Show answer & explanationAnswer & explanation
Correct answer: B. 7 hours
The function is a quadratic equation whose graph is a parabola opening upwards (since the leading coefficient is positive). The minimum value occurs at the vertex. Find the x-coordinate of the vertex using x = -b/(2a) and then substitute this value back into the function to find the minimum time.
Why the other options are wrong
- A. Incorrectly identified -b as the minimum value, not considering 2a.
- C. This is the x-coordinate of the vertex (number of workers), not the minimum time.
- D. This is the y-intercept, not the minimum value of the parabola.
Minimum of Quadratic Function
For a quadratic function f(x) = ax^2 + bx + c with a > 0, the minimum 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 minimum y-value.
- If a < 0, the vertex represents a maximum value instead.
Memory trick: Vertex is the turn, -b/2a is the key to learn.