https://file.aiquickdraw.com/mathbot/file/17305133670701vrx853d.jpg

draw a set of x-y axes and a regular grid on top of the photograph, and then show the equation of this simple parabola represented by the roller coaster, opening downward. Label the coordinates and axes. Make sure to keep the numbers small and everything as simple as possible, and show the work.

Here's the diagram with a coordinate grid overlay on the roller coaster's arch, showing it as a parabola. The equation we used to model this arch is:

\[
y = -x^2 + 10
\]

### Explanation of the Equation
- This parabola opens downward, modeled by the negative sign in front of \(x^2\).
- The vertex (highest point) is at \((0, 10)\), representing the top of the arch.
- Points \((-5, 0)\) and \((5, 0)\) are positioned symmetrically on the x-axis, where the arch meets the supports at the base.

### Steps Summary
1. **Vertex at \((0, 10)\)**: The topmost point of the arch.
2. **Symmetry and Width**: The points \((-5, 0)\) and \((5, 0)\) align with the arch's width, giving us the shape and scale.
3. **Parabola Equation**: Plugging the vertex into the standard form for a downward parabola, we derived the simple equation above.

Would you like more details on how I derived the equation, or have any specific questions?

### Related Questions
1. How would we adjust the equation if the vertex was at a different height?
2. Can we determine the arch's maximum height and width from this equation?
3. How can we use calculus to find the slope of the parabola at a given point?
4. What changes would occur in the equation if the parabola were stretched wider?
5. How would this model change if the arch followed a different curve, like a sine wave?

### Tip
In general, the vertex form of a parabola, \( y = a(x - h)^2 + k \), is very useful for modeling arches and similar structures in real life because it allows direct placement of the vertex at any point \((h, k)\).

Draw a set of x-y axes and a regular grid on top of the photograph, and then show the equation of this simple parabola represented by the roller coaster, opening downward. Label the coordinates and axes. Make sure to keep the numbers small and everything as simple as possible, and show the work.

Learn how to overlay a grid on a roller coaster arch image and derive the equation for the parabola it forms. This guide covers key concepts in algebra, specifically parabolas and quadratic equations, with a simple example suitable for Grades 8-10.

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