GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A chemist is observing a chemical reaction where the concentration of a certain substance, C(t), in moles per liter, after t minutes, is given by the function C(t) = 0.5t² - 4t + 10. During which time interval is the concentration of the substance decreasing?
- A0 < t < 4
- B0 < t < 8
- Ct > 8
- Dt > 4
Show answer & explanationAnswer & explanation
Correct answer: A. 0 < t < 4
This is a parabola opening upwards (because a=0.5 > 0). The concentration decreases until it reaches its minimum at the vertex, then increases. The x-coordinate of the vertex is t = -b/(2a).
Why the other options are wrong
- B. This interval includes some time where the concentration is increasing.
- C. This interval is entirely within the increasing portion of the function.
- D. The concentration increases for t > 4.
Interpreting Quadratic Graphs (Increasing/Decreasing)
For a parabola, the function decreases until it reaches its vertex (if it opens upwards) or increases until it reaches its vertex (if it opens downwards). The axis of symmetry (-b/2a) marks the turning point.
- Parabola opens up (a>0): Decreases then increases.
- Parabola opens down (a<0): Increases then decreases.
- Vertex x-coordinate: -b/(2a).
- Intervals are defined relative to the vertex's x-coordinate.
Memory trick: Vertex is the Turn, 'a' tells the Trend!