GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A production manager is analyzing the efficiency of a new machine. The machine's output, in units per hour, is modeled by the function E(t) = -0.5t² + 10t + 50, where t is the number of hours the machine has been running since the start of the shift. What is the maximum output the machine achieves?
- A80 units/hour
- B120 units/hour
- C50 units/hour
- D100 units/hour
Show answer & explanationAnswer & explanation
Correct answer: D. 100 units/hour
The function is a downward-opening parabola, so its maximum value occurs at the vertex. Find the t-coordinate of the vertex using t = -b/(2a) and then substitute this value back into the function to find the maximum output.
Why the other options are wrong
- A. This is a plausible distractor, perhaps from a calculation error.
- B. This is a plausible distractor, perhaps from a calculation error.
- C. This is the y-intercept (initial output), not the maximum output.
Vertex of a Parabola (Maximum/Minimum)
For a quadratic function in the form f(x) = ax² + bx + c, the vertex represents the maximum value if 'a' is negative, or the minimum value if 'a' is positive. The x-coordinate of the vertex is given by -b/(2a).
- Vertex x-coordinate: x = -b / (2a)
- Vertex y-coordinate: f(-b / (2a))
- If a < 0, parabola opens down, vertex is a maximum.
- If a > 0, parabola opens up, vertex is a minimum.
Memory trick: Vertex's Peak or Valley, that's where the extremum lies!