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}

Certainly! Here's a word problem that utilizes the given piecewise function:

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**Word Problem:**

A large amusement park is analyzing the potential revenue generated from two different types of seasonal promotions. The number of visitors, $ y $, that attend the park depends on the amount of money, $ x $, in thousands of dollars, that the park spends on advertising the promotion. The relationship between the advertising spend and the number of visitors is given by two different quadratic functions, depending on the amount of money spent.

1. **Promotion A**: This promotion is designed to attract families during the spring and is effective when the park spends between $450,000 and $1,550,000. The number of visitors $ y $ that Promotion A attracts is given by the equation:
   \[
   y = -0.001(x - 1000)^2 + 3000 \quad \text{for} \quad 450 \leq x \leq 1550
   \]
   where $ x $ is the advertising expenditure in thousands of dollars.

2. **Promotion B**: This promotion is targeted towards young adults during the summer and is effective when the park spends between $2,450,000 and $3,550,000. The number of visitors $ y $ that Promotion B attracts is modeled by the equation:
   \[
   y = -0.001(x - 3000)^2 + 3000 \quad \text{for} \quad 2450 \leq x \leq 3550
   \]
   where $ x $ is the advertising expenditure in thousands of dollars.

**Questions:**

- What amount of advertising spending will attract the maximum number of visitors for each promotion?
- What is the maximum number of visitors that can be expected for each promotion?
- If the park spends $1,000,000 on Promotion A, how many visitors are expected to attend?
- How many visitors can be expected if $2,500,000 is spent on Promotion B?
- Compare the number of visitors attracted by spending $1,550,000 on Promotion A versus $3,550,000 on Promotion B.
- At what spending level does Promotion B start attracting more visitors than Promotion A?
- What is the minimum number of visitors expected for each promotion within the given range of spending?
- How does the effectiveness of spending on Promotion A change if the park spends just below or just above $1,000,000?

**Tip:** Quadratic functions with negative coefficients open downwards, meaning the maximum value occurs at the vertex.

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Explore how a large amusement park uses piecewise quadratic functions to analyze revenue from seasonal promotions. Learn about the quadratic function forms and their applications in real-world scenarios. Suitable for high school students studying quadratic equations.

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