https://file.aiquickdraw.com/mathbot/file/17305129830219sktqxui.jpg

 draw a set of x-y axes and a regular grid on top of the photograph, and then show the equation of this parabola represented by the roller coaster, opening downward.

I've added an \( x \)-\( y \) axis and grid lines to the roller coaster image, which will allow us to approximate the parabolic path mathematically.

Next, I'll estimate the equation of this downward-opening parabola by identifying some key points along the track (like the vertex and points on either side). I'll use these points to fit the parabolic equation \( y = ax^2 + bx + c \) and provide an approximation.The approximate equation of the parabolic path followed by the roller coaster is:

\[
y = -0.00375x^2 - 150
\]

This equation represents a parabola that opens downward, with its vertex near the highest point of the track. Here, \( x \) and \( y \) are centered on the middle of the image, and the values are scaled based on the grid.

Would you like further details on how to use this equation, or have any other questions?

### Additional Related Questions
1. How would we find the maximum height reached by the roller coaster?
2. Can we calculate the horizontal distance between two points on the track using this equation?
3. How can we use derivatives to find the slope of the roller coaster at various points?
4. What would be the implications if the parabola were steeper (larger absolute value of \( a \))?
5. How could we convert this coordinate system to a real-world scale, such as meters?

### Tip
For parabolas opening downward, the coefficient of \( x^2 \) (here, \(-0.00375\)) dictates the "width" or steepness of the curve: a smaller absolute value makes the parabola wider, while a larger one makes it steeper.

Draw a set of x-y axes and a regular grid on top of the photograph, and then show the equation of this parabola represented by the roller coaster, opening downward.

Learn how to derive the equation of a downward-opening parabola to model the path of a roller coaster in a photograph. This process involves adding x-y axes, creating a grid, and using key points along the track to fit a parabolic equation. Suitable for high school students in Grades 10-12.

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