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 given by the function E(t) = -t² + 10t + 20, where 't' is the number of hours the machine has been running since the start of a shift. After how many hours will the machine reach its maximum output?
- A5 hours
- B2 hours
- C10 hours
- D20 hours
Show answer & explanationAnswer & explanation
Correct answer: A. 5 hours
The function E(t) = -t² + 10t + 20 is a quadratic function, representing a parabola opening downwards. Its maximum value occurs at the vertex. The x-coordinate (in this case, 't') of the vertex of a parabola in the form ax² + bx + c is given by the formula t = -b / (2a).
Why the other options are wrong
- B. This value does not correspond to the vertex of the parabola.
- C. This value is the coefficient of the linear term, not the vertex.
- D. This value is the constant term, not the vertex.
Vertex of a Parabola (Quadratic Function)
The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function. For a parabola y = ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
- For a < 0, the parabola opens downwards, and the vertex is a maximum.
- For a > 0, the parabola opens upwards, and the vertex is a minimum.
- The x-coordinate of the vertex gives the input value where the max/min occurs.
Memory trick: Vertex 'V'alue is 'V'ery 'V'ital for 'V'aluable 'V'alues.