Word Problem:

A city is planning a grand fireworks display to celebrate a major event. The fireworks will be launched from two different locations: Location A and Location B. The height $y$ (in meters) of the fireworks at time $x$ (in seconds) after launch is modeled by two different equations depending on the time range.

Location A:

• The fireworks are launched at time $x = 1000$ seconds, and the height is modeled by the equation: $y = -0.001(x - 1000)^2 + 3000$ This equation is valid for the time interval $450 \leq x \leq 1550$ seconds.

Location B:

• The fireworks are launched at time $x = 3000$ seconds, and the height is modeled by the equation: $y = -0.001(x - 3000)^2 + 3000$ This equation is valid for the time interval $2450 \leq x \leq 3550$ seconds.

Piecewise Function:

The height $y$ of the fireworks as a function of time $x$ can be written as a piecewise function:

\begin{cases} -0.001(x - 1000)^2 + 3000, & \text{for } 450 \leq x \leq 1550 \\ -0.001(x - 3000)^2 + 3000, & \text{for } 2450 \leq x \leq 3550 \end{cases}$$ This function describes how the height of the fireworks changes over time for each location. ### Questions: 1. What is the maximum height the fireworks reach at Location A? 2. At what time does the maximum height occur for the fireworks launched at Location A? 3. What is the height of the fireworks at Location B when $$x = 3100$$ seconds? 4. How long does it take for the fireworks at Location A to reach their maximum height? 5. What is the time duration during which the fireworks from Location A are in the air? 6. How does the shape of the graph change if the coefficient of $$x^2$$ is modified? 7. What would happen if the coefficient of $$x^2$$ in the equations was positive instead of negative? 8. What is the significance of the value 3000 in the equations? **Tip:** In a piecewise function, make sure to clearly define the intervals for each equation to avoid confusion about when each equation applies.