# The Quadran+ X

## Rough Graph

*Letters represent points where functions intersect*

## Summary of Plan

We created our roller coaster by first thinking of a design that would meet all of our requirements. We used our experience of riding several different types of roller coasters to come up with a fun, yet realistic design. Also, our knowledge of the different functions we learned throughout the course helped us design each part of the roller coaster with precision. Some difficulties that we experienced were lining up the next function with the previous so that it continued in the location of which the last one ended at, and making a function start at a specific point, and end at another specific point. To solve the first problem, we utilized the method of substitution, by subbing in the final X and Y value of the previous function into the equation of the new function, to find a, k, c, or d values that would make the new function hit the final point of the last one. To solve the second problem, we used the method of elimination to find a function that hit 2 specific points.

## Report

First Function:

Y=10

Restriction on Domain: {x E R |0 ≤ x ≤ 5}

· This represents the beginning of the roller coaster. It starts off at 10 feet, and stays at that height for five seconds.

Second Function:

Y=0.05(x-5)^3+10 OR y=0.05x^3-0.75x^2+3.75x+3.75

Restriction on domain: {x E R |5 ≤ x ≤ 15}

· This polynomial cubic function represents when the roller coaster initially begins to gradually increase in height, to get to 300 feet, and eventually to the drop. Equation has ‘a’ value of 0.05 to make it not steep, and a ‘d’ value of 5 and a ‘c’ value of 10 to make it connect with the end of the previous function.

Third Function:

Y=30.124195x-391.862918

Restriction on domain: {x E R |15 ≤ x} {y E R |300 ≥ y}

· This function represents gradual climb to 300 feet, from 60 feet, leading to the drop. This equation was found through the process of elimination. We wanted a function that hit point (15,60), and (22.9670177, 300), to connect with final point of the previous function, and the first point of the following function.

Fourth Function:

Y=-10(x-22.8670177)^2+300

Restrictions on domain: {x E R|22.9670177901 ≤ x ≤ 27}

· This function represents the drop in the roller coaster, from 300 feet, and for about 4 seconds. The ‘a’ value in the function is -10 because the quadratic function needs to be vertically reflected to represent a drop, and it also needs to be steep. The d value is 22.9670177, and c value is 300, to make the middle of the quad function match up with the final point of the previous function.

Fifth Function:

y=3708.464518/x

Restrictions on domain: {x E R| 27 ≤ x ≤ 35}

· This function follows the drop, and represents the roller coaster getting leveled (gaining balance after drop). This rational function was found by finding an a value that would allow the rational function to hit the previous function’s final point, (27, 137.35054):

· a/27=137.35054, therefore, a=(27)(137.35054), so a=3708.464518

Sixth Function:

Y=45sin(π/5(x-40))+105.956129

Restrictions on domain : {x E R| 35 ≤ x ≤ 50}

· This function represents a wave in the roller coaster, where the cart goes down, up, down, then up. Function was found by making a value 45, which makes the graph steep, but does not make it unrealistically steep, as we wanted the wave to be somewhat exciting. The function was moved 105.956129 feet up, and 40 units right, to make it intersect with the final point of the previous graph. The K value is π/5, because we wanted to make the period 10, so we could make the graph end in 20 seconds, for 2 periods.

Seventh Function:

Y=20log(x-48.01479)+100

Restrictions on domain: {x E R| 50 ≤ x ≤ 55}

· This function represents when the roller coaster gets levelled again, after the wave. ‘a’ value is 20 to vertically stretch long function a bit, and c value is 100 to bring the function around the same level as previous function’s ending. To find ‘d’ value, we subbed in final point of previous function, to see what d value would make them intersect.

Eighth function:

Y=x+61.8835893

Restrictions on Domain: {x E R| 55 ≤ x ≤ 63}

· Function is responsible for leading roller coaster towards the loop, while gaining a little height. The m value is 1, because we did not want the line to be steep at all. The b value was found by subbing in final point of previous function:

· 116.883589=55+b, isolate for b, therefore, b=61.883589

Ninth function:

Y=1.5^(x-61.43838805) +123

Restrictions on Domain: {x E R| 63 ≤ x ≤ 70}

· This function represents beginning of the loop, as cart initially begins to increase height. ‘a’=1.5, because we wanted an exponential function that was a little more steep than normal. C=123 to bring the function close to the same level as the previous function. D value was found by subbing in final point of previous function:

Tenth Function:

Y=59sin(π/5(x+70))+155.182889

Restrictions on domain: {x E R| 70 ≤ x ≤ 75}

· This function represents middle of loop, where cart spins in circle. Speed decreases at top, and increases after coming down. A value is 59 to vertically stretch sin graph, as the loop takes cart high. K=π/5 t0 make period 10, but only use half of it. The c=155.182889, and d=-70 to make graph intersect with final point of previous function.

Eleventh function:

Y=1.5^(-(x-83.56161195)) +123

Restrictions on domain: {x E R| 75 ≤ x ≤ 85}

· Represents end of loop, bringing cart back to steady level. ‘a’ and ‘c’ value match first exponential function used for beginning of loop, and k value is -1 to make end of loop in opposite direction as beginning. D value was found by subbing in final point of previous graph

Twelfth Function:

Y=-5(x-85)^2+123.55810053

Restrictions on domain: {x E R| 85 ≤ x ≤ 89}

· This function represents a drop before coming to an end. The a=-5 to vertically reflect quadratic function, and to stretch the graph by a little bit, as the roller coaster is coming down to 10 feet. C=123.55810053, and d=85, to make quad function intersect with final point of previous function.

Final function:

Y=6452345.494(0.874788014)^x

Restriction on domain: {x E R| 89 ≤ x ≤ 100}

· This function represents end of roller coaster, as this exponential function brings it back to its initial height of 10 feet, and ends at exactly 100 seconds, which represents roller coaster coming back to its beginning. Equation was found by finding an exponential function that hits final point of previous graph, (89, 43.558101) and point (100,10)