Algebra 2: Functions, Equations, Roller Coaster Design Project

This assignment involves the design and analysis of a roller coaster using algebraic concepts. The student drew a roller coaster on graph paper, including an initial climb, hills, and a loop. The student then identified ordered pairs on the roller coaster's path, calculated the slope (rate of change) for different sections, and determined the equation of the initial climb. The roller coaster is analyzed as a function, involving linear, circular, and parabolic components, connected by critical points. The calculations demonstrate the application of algebra in engineering design, specifically in determining the steepness of hills and the overall characteristics of the roller coaster's path. The student also found the domain and range of the function, further illustrating the mathematical concepts used in this project.

1)

2)
3)
Ordered pairs of the initial climb are-
(1,1) and (4,15)
Slope at the beginning of 1st hill= y2− y1
x2−x1
¿ 15−1
4−1
= 14
3

(4,15) and (7.5,4)
Slope at the end of 1st hill= y2− y1
x2−x1
=−9
3.5
=-2.57
(7.5,4) and (10.5,12)
Slope at the beginning of 2nd hill= y2− y1
x2−x1
=8
3
=2.67
(10.5,12) and (13,7)
Slope at the end of 2nd hill = y2− y1
x2−x1
=−5
2.5
=-2
There for the steeper hill is the 1st hill with slope 5.5.
4)
Equation of line = y− y1 =m( x−x1)
Equation of the initial climb = y−1=14
3 (x−1)
3 y=14 x −11
5)
Domain – set of all x values
=(1,∞)
Range – set of all x values
=(1,15]
6) Ordered pairs of the initial climb are-
(1,1) and (4,15)
The equation of this hill from the starting point is 3 y=14 x −11

That is , dy
dx =14
3 (by rate of change of y with respect to x)
This hill has the greatest slope among the three hills.
There fore this hill is steeper among the three.
7)
The roller coaster is a function.
The functions involved are straight lines, circle and parabola.
All these functions are connected by critical points that create a larger function of x.
The roller coaster linearly moves from initial point with slope 14/3, goes along all straight lines and enter
the circular loop of radius 2, gains the speed and enters into the parabolic curve with the energy
obtained from the loop.