Make a word problem piecewise function using
y=−0.001 (x−1000)² + 3000 {450≤x≤1550}
y=−0.001(x−3000)² +3000 {2450≤x≤3550}

Let's create a word problem that involves a scenario where we can use piecewise functions to model real-world behavior. 

### Word Problem

**Scenario:**  
A new theme park is launching two different roller coasters that operate at different times of the day. Each roller coaster has a different level of excitement, which can be represented by a piecewise function. The excitement level \( y \) depends on the position \( x \) on the track. 

**Roller Coaster A:**  
The excitement level \( y \) of Roller Coaster A can be modeled by the equation:
\[ 
y = -0.001 (x - 1000)^2 + 3000 \quad \text{for} \quad 450 \leq x \leq 1550 
\]

**Roller Coaster B:**  
The excitement level \( y \) of Roller Coaster B can be modeled by the equation:
\[ 
y = -0.001 (x - 3000)^2 + 3000 \quad \text{for} \quad 2450 \leq x \leq 3550 
\]

The park management wants to know at which positions the excitement levels are at their maximum and how they vary across different parts of the tracks.

### Questions

1. **What is the maximum excitement level for Roller Coaster A, and at what position does it occur?**
   - *Hint:* Determine the vertex of the parabola described by the function for Roller Coaster A.

2. **What is the maximum excitement level for Roller Coaster B, and at what position does it occur?**
   - *Hint:* Determine the vertex of the parabola described by the function for Roller Coaster B.

3. **At what positions are the excitement levels of both roller coasters equal to 2900?**
   - *Hint:* Solve \( y = 2900 \) for both piecewise functions.

4. **How do the excitement levels of both roller coasters compare at the endpoints of their respective domains?**
   - *Hint:* Calculate \( y \) for the endpoints of the domains for both functions.

5. **Describe the behavior of the excitement level as a rider moves from \( x = 450 \) to \( x = 1550 \) on Roller Coaster A.**

6. **Describe the behavior of the excitement level as a rider moves from \( x = 2450 \) to \( x = 3550 \) on Roller Coaster B.**

7. **Is there any position where the excitement level for both roller coasters is the same, considering both domains?**

8. **How does the excitement level change at the midpoint of each roller coaster track?**
   - *Hint:* Calculate the excitement level at \( x = 1000 \) for Roller Coaster A and \( x = 3000 \) for Roller Coaster B.

### Tip
To find the maximum or minimum of a quadratic function, identify the vertex of the parabola. In the function \( y = a(x-h)^2 + k \), the vertex is at \( (h, k) \), which gives the maximum or minimum value depending on whether the parabola opens upwards or downwards.

Explore a word problem involving roller coaster excitement levels modeled using piecewise quadratic functions. Discover maximum excitement levels, points of equality, and behavior along different parts of the roller coaster tracks.

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