Digital SATMath: Advanced MathMedium
A robotics engineer is programming a robot arm. The path of the arm's gripper can be described by the function h(t) = -t^2 + 6t - 5, where h(t) is the height in meters and t is the time in seconds. At what times will the gripper be at a height of 0 meters?
- At = 0 and t = 6 seconds
- Bt = 1 and t = 6 seconds
- Ct = 2 and t = 3 seconds
- Dt = 1 and t = 5 seconds
Show answer & explanationAnswer & explanation
Correct answer: D. t = 1 and t = 5 seconds
To find when the height is 0, set h(t) = 0: -t^2 + 6t - 5 = 0. Multiply by -1 to get t^2 - 6t + 5 = 0. This quadratic equation can be factored as (t - 1)(t - 5) = 0. The solutions are t = 1 and t = 5.
Why the other options are wrong
- A. These are the roots if the equation were -t^2 + 6t = 0.
- B. Incorrect factorization; one root is correct, but the other is not.
- C. Incorrect factorization or use of the quadratic formula.
Finding X-intercepts of a Parabola
The x-intercepts (or roots/zeros) of a parabolic function are the points where the graph crosses the x-axis, meaning the y-value (or function output) is zero.
- Set f(x) = 0 and solve for x.
- Can be found by factoring, using the quadratic formula, or completing the square.
- A parabola can have zero, one, or two real x-intercepts.
Memory trick: Where the path meets the ground, solutions are found.