Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A robotics engineer is programming a robot's movement. The robot's height h (in cm) above the ground at time t (in seconds) is given by the function h(t) = -t^2 + 6t + 7. At what time does the robot reach its maximum height?
- A9 seconds
- B6 seconds
- C3 seconds
- D7 seconds
Show answer & explanationAnswer & explanation
Correct answer: C. 3 seconds
The function h(t) = -t^2 + 6t + 7 is a quadratic function representing a parabola opening downwards. The maximum height occurs at the vertex. The x-coordinate (or t-coordinate in this case) of the vertex is given by t = -b / (2a). Here, a = -1 and b = 6. So, t = -6 / (2 * -1) = -6 / -2 = 3 seconds.
Why the other options are wrong
- A. Incorrect. This might be the maximum height itself (h(3) = -9 + 18 + 7 = 16), or a calculation error.
- B. Incorrect. This is the 'b' coefficient, not the vertex time.
- D. Incorrect. This is the y-intercept (initial height), not the time of maximum height.
Vertex of a Parabola (Time)
For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, which gives the time or input value at which the maximum or minimum occurs, is found using the formula x = -b / (2a).
- Vertex is the turning point of a parabola.
- Formula: x = -b / (2a).
- If a < 0, vertex is a maximum; if a > 0, vertex is a minimum.
Memory trick: Remember 'negative B over 2A' is the secret key to the peak (or valley) time!