Questions and Answers
What is the polynomial for a coaster that starts with an x-intercept at x = 500 and rises to a maximum before crossing the x-axis?
y = ax(x - 500)(x - 1000)
What is the polynomial for a coaster that crosses at x = 300 and 700, and rises before it drops?
y = -ax(x - 300)(x - 700)(x - 1000)
What is the polynomial for a simple coaster that 'bumps' the axis at x = 500 and rises before it falls?
y = -ax(x - 500)^2(x - 1000)
What is the polynomial for a coaster that crosses at x = 500 after rising and falls smoothly at x = 1000?
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What is the polynomial for a coaster that includes a swoop down at x = 300, crosses upward at x = 700, and ends smoothly at x = 1000?
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Study Notes
Basic Coaster Design
- Build a polynomial with a specific structure: y = ax(x - 500)(x - 1000).
- Features an x-intercept at x = 500, rising to a maximum, then falling and rising again towards x = 1000.
Enhanced Coaster with Extra Rise and Fall
- Design a coaster with crossings at x = 300 and x = 700.
- The polynomial function is structured as y = -ax(x - 300)(x - 700)(x - 1000).
- Ensure the graph rises prior to dropping through the axis.
Simple Bump Design
- Create a coaster that "bumps" the x-axis at x = 500.
- Use the polynomial y = -ax(x - 500)^2(x - 1000).
- Prioritize an initial rise before the fall.
Realistic Coaster with Smooth Ending
- Develop a coaster that crosses the x-axis at x = 500 after an initial rise and fall.
- Employ a more realistic approach, allowing a smooth transition at x = 1000 with the polynomial y = -ax(x - 500)(x - 1000)^2.
Complex Coaster Design
- Design a coaster that incorporates multiple features:
- A "swoop" down at x = 300.
- An upward crossing at x = 700.
- A smooth ending at x = 1000 utilizing an exponent of 2.
- The polynomial function is expressed as y = ax(x - 300)^3(x - 700)(x - 1000)^2.
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Description
Explore the intricacies of designing roller coasters through polynomial functions. This quiz covers various coaster designs featuring x-intercepts, bumps, and smooth transitions. Engage with how polynomial shapes influence the coaster's path and structure.
