GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A meteorologist is tracking a weather balloon. The balloon's altitude, h(t), in meters, after t minutes, is given by the function h(t) = -t² + 10t + 20. What is the maximum altitude the weather balloon reaches?
- A20 meters
- B45 meters
- C25 meters
- D50 meters
Show answer & explanationAnswer & explanation
Correct answer: B. 45 meters
The function h(t) = -t² + 10t + 20 is a quadratic equation, which forms a parabola. Since the leading coefficient is negative, the parabola opens downwards, and its vertex represents the maximum point. The x-coordinate of the vertex is given by -b/(2a), and the y-coordinate is the maximum value.
Why the other options are wrong
- A. This is the initial altitude (y-intercept), not the maximum altitude.
- C. This is incorrect; it might be a miscalculation of the vertex's y-coordinate.
- D. This is incorrect; it might result from an arithmetic error or misunderstanding of the vertex formula.
Vertex of a Parabola (Maximum/Minimum)
For a quadratic function in the form f(x) = ax² + bx + c, the vertex is the point where the parabola reaches its maximum or minimum value. If a < 0, it's a maximum; if a > 0, it's a minimum.
- x-coordinate of vertex: -b/(2a)
- y-coordinate of vertex: f(-b/(2a))
- Represents the turning point of the parabola
Memory trick: Vertex is the peak or valley, where the action stops and turns!