GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A technician is monitoring the temperature of a chemical reaction. The temperature, T (in degrees Celsius), over time, t (in minutes), is given by the function T(t) = -t^2 + 10t + 20. At what time does the reaction reach its maximum temperature?
- A10 minutes
- B20 minutes
- C25 minutes
- D5 minutes
Show answer & explanationAnswer & explanation
Correct answer: D. 5 minutes
The function T(t) = -t^2 + 10t + 20 is a quadratic equation in the form of at^2 + bt + c. The maximum value of a parabola opening downwards (a < 0) occurs at its vertex. The t-coordinate of the vertex is given by the formula t = -b / (2a). Here, a = -1 and b = 10, so t = -10 / (2 * -1) = -10 / -2 = 5 minutes.
Why the other options are wrong
- A. This is the coefficient of 't', not the time of maximum temperature.
- B. This is the y-intercept (initial temperature), not the time of maximum temperature.
- C. This is a distractor, possibly related to the maximum temperature value itself (T(5) = -25 + 50 + 20 = 45).
Vertex of a Parabola (Maximum/Minimum)
For a quadratic function f(x) = ax^2 + bx + c, the vertex is the point where the parabola reaches its maximum or minimum value. The x-coordinate of the vertex is given by x = -b/(2a).
- If a > 0, the parabola opens upwards, and the vertex is a minimum.
- If a < 0, the parabola opens downwards, and the vertex is a maximum.
- The y-coordinate of the vertex is f(-b/(2a)).
Memory trick: Vertex is the peak or valley, find it with negative b over two a.