Question

The path followed by a roller coaster as it climbs up and descends down from a peak can be modeled by a quadratic function, where h(x) is the height, in feet, and x is the horizontal distance, also in feet. The path begins and ends at the same height, covers a total horizontal distance of 100 feet, and reaches a maximum height of 250 feet. Which of the functions could be used to model this situation?

Answer 1

`y = -0.1x^2 + 250 ft`

Explanation:

Because this quadratic equation would have the curve-down form of:

`y = -ax^2 + b`

where a and b are positive coefficient.

If we let the peak (250 ft) of the curve be at x = 0. Then

`y = -a0^2 + b = 250`

`b = 250`

Also at the begins and ends, thats where y = 0, the 2 points are separated by 100 ft. So let the begin at -50 ft and the end at 50ft. We have

`-a(\pm 50)^2 + 250 = 0`

`-a2500 = -250`

`a = 250/2500 = 0.1`

Therefore, the model quadratic equation of our path would be

`y = -0.1x^2 + 250 ft`