Answer: The equation h(x) = -4(x + 2)(x - 18) models the height of the rocket (in meters) as a function of time (x) in seconds after the launch. Let's break down the equation to understand its key characteristics:
1. The rocket is launched from a height of h(0) = -4(0 + 2)(0 - 18) = -4(2)(-18) = 144 meters above the ground. This is the initial height at the time of the launch.
2. The term "x + 2" represents the time elapsed since the launch in seconds. When x = 0 (at the time of launch), this term is 2, and when x = 18 seconds, it becomes 20.
3. The term "x - 18" represents the time elapsed since 18 seconds. When x = 0 (at the time of launch), this term is -18, and when x = 18 seconds, it becomes 0.
4. The product of these two terms (-4(x + 2)(x - 18)) models the rocket's height at any given time x seconds after the launch.
5. The -4 in the equation is a scaling factor, which can affect the overall shape of the curve but doesn't change the basic characteristics described above.
The function represents the rocket's trajectory. It starts 144 meters above the ground, and its height changes as time passes. As the time increases, the height changes due to the quadratic nature of the function, with the height reaching zero when the term (x + 2)(x - 18) becomes zero, which happens at x = 18 seconds. After this time, the rocket will be at or below ground level.
The graph of this function would be a downward-opening parabola, and it describes the rocket's height as it rises and then descends after 18 seconds.