# Roller Coaster Assignment

### Making a Rollercoaster out of Math Equations

## Our Rollercoaster

## Equation One

**f(x)=10.5x+10**.

This equation is a **linear function**. The __height over time for this equation increases at a constant rate as it approaches 2 seconds__. This equation's domain restriction is **x must be greater than or equal to zero and x must be equal to or less than 2**. ** ** **{0 </= x </= 2}**. This equation has been shifted up 10 feet and horizontally compressed by a factor of 1/10.5.

## Equation Two

The equation of this quadratic function is **f(x) = -67.25 (x-4)^2 +300**.

The **equation's domain restrictions are x must be greater than or equal to 2 and less than or equal to 6**. The equation has a vertical stretch by a factor of 67.25, is reflected along the x-axis (making it open down), shifted 4 feet to the right and 300 feet up. This makes the equation have a vertex at (4, 300). The function's height over time __increases as it passes 2 seconds and then decreases as it passes 4 seconds until it hits and stops at the the point (6, 31).__

## Equation Three

**f(x) = 0.5 (0.25(x-13))^5 + 22.793457**. As this equation continues from 6 seconds to 9 seconds

__, the height decreases exponentially__. The equations is transformed as its

**shifted 13 feet to the right**and

**shifted 22.793457 feet up**. The

**restrictions of the domain are x must be equal to or greater than 6 and x must be less than or equal to 9. {6</= x </= 9}**

## Equation Four

The equation of this function is **f(x) = 2^(x-15) + 23.284375**.

This equation is an **exponential** function. The transformations present on this equation are: being shifted 15 feet to the right and shifted up 23.284375 feet.

The equation's domain restrictions is **x must be greater than or equal to 9 and less than or equal to 20** (9</= x </=20). The function's __height over time increases exponentially (rapidly) as it passes 9 seconds and continues its growth until it reaches 20 seconds.__

## Equation Five

**sinusoidal function**is

**f(x) = 24sin(0.25(x-20))+55.28**

The transformations of this equation are a vertical stretch by 24, a horizontal stretch by a factor of 4, a phase shift of 20 feet to the right, and also 55.28 feet up. __As time increases, the height increases until 26.28 seconds, then decreases until 38.85 seconds, then increases until 51.42 seconds, decreases until 63.98 and then begins to increase again until stopping at a height of 53.69 at 70 seconds (the function increases and decreases so many time because it is oscillating)__. The domains restrictions are **x must be greater than or equal to 20 and less than or equal to 70**. ** {20 </= x </= 70}**

## Equation Six

The equation of this** logarithmic** function is **f(x) = 10log[-(x - 93.38837)] + 40.**

The equation's domain restrictions are **x must be greater than or equal to 70 and less than or equal to 90 (70</= x </= 90). **The equation has a vertical stretch by a factor of 10, a reflection along the y-axis, and it is shifted 93.38837 feet to the right and 40 feet up. The function's height over time __decreases as it passes 70 seconds and stops at 90 seconds.__

## Equation Seven

The equation of this function is **f(x) = 50.3333x / - (x - 85)(x - 110)** or, with an expanded denominator, **f(x) = 50.3333x / - x^2 + 195x + 9350**. This equation is a rational function (the numerator and the denominator are polynomial functions that make a fraction).

The equation's domain restrictions is **x must be greater than or equal to 90 but less than or equal to 100** (90 </= x </= 100). The function's __height over time decreases as it passes 90 seconds, but as time passes 96 seconds, the height begins to increase, then finally stop at 100 seconds.__

The equation has a vertical stretch by a factor of 50.3333. The equation has been reflected in the y axis. The equation, unrestricted, has a vertical asymptote at x = 85 and x = 110, and a horizontal asymptote at y = 0.

## Rollercoaster Equation Calculations

## Equation One This linear equation is the beginning of the roller coaster. It has a slope of 10.5 because we didn't want the first equation to take up too much time of our total time of 100 seconds, so we horizontally compressed it. | ## Equation Two To get this equation, we substituted the endpoint (2, 31) from equation 1. The reason why we picked (2, 31) was just because it is easier to cut the equations off at whole x values. After the equation was substituted, we got an a value of -67.25. The reason why we shifted it 4 to the left was just so it would connect to equation 1. The reason why we shifted it 300 feet up was so we could meet the requirement of the rollercoaster having a maximum point of 300. | ## Equation Three This polynomial function is a small part of the roller coaster. We made the domain for this equation from six to nine seconds so that the rollercoaster could have a steady drop after being so high for so long from our quadratic equation. We transformed our equation logically so that the function would be near the previous equation's end coordinate, then we substituted the point (6, 31) into our new equation to find a "c" value that would allow the two equation to meet up. |

## Equation One

## Equation Two

## Equation Three

## Equation Four We used base two for our exponential equation so the increase would not be too large as time passed, and then shifted it 15 feet to the right because we wanted the exponential curve of the equation to begin around 9 seconds to make the roller coaster have a smooth climb upwards. To find the "c" value to make sure that this equation would start at the end of the previous equation, we substituted the point (9, 23.3) into the equation and then solved for "c" to get our final equation. | ## Equation Five For the sinusoidal equation, we made a vertical stretch by a factor of 24 and a horizontal stretch by a factor of 4 because we wanted this portion of the roller coaster to take up a lot of time. There are no calculations for this equation because of our knowledge of transformations. The previous equation ended at point (20, 55.28). Sine functions begin at point (0, 0), so if we transform the graph by making the phase shift the x value of the last equation's end coordinate and the axis of symmetry the y value of the last equation's end coordinate, both equations would connect. | ## Equation Six For this equation we started off with f(x) = 10log[-(x-d)] + 40. We stretched it by a factor of 10 just so it wouldn't take up too much space and we shifted it up 40 feet to get it closer to the previous graph. We then substituted the point (90, 45.3) into the equation and got the 'd' value that would connect both equations. |

## Equation Four

## Equation Five

## Equation Six

## Equation Seven We wanted this equation to connect to the previous equation and end at 100 seconds without the rollercoaster moving in a strange or extreme way. So we made our last equation a rational equation with asymptotes near the previous equation's ending point and the other one near 100 seconds, and also reflected it in the y axis so the equation would decrease as it connected to the previous. We substituted point (90, 45.3) to find an "a" value that would connect the two equations which allowed us to find out final equation. |

## Equation Seven

## How we Made Our Rollercoaster and the Difficulties we Faced

In order to create a roller coaster that fulfilled the requirements of the assignment, many trial and errors were made. The first equation was created keeping in mind that our rollercoaster had to have a minimum height of 10 feet high. Once the first equation was made, we just chose a random equation to create the next part of our roller coaster. Using our knowledge of transformations, we logically played with the translations and stretch and compression of the next equation so that the equation would land in the general area of the previous equation's end coordinate, and then substituted that end coordinate into our equation to find the missing value (either a, d, c, or k) that would connect the two equations together. The point we substituted into the equation was the coordinate where we cut off the previous equation to ensure a point of intersection between the two equations. We then continued this method until we reached 100 seconds making sure that the roller coaster met all the necessary requirements. The complications we faced were making sure the right transformations were applied so that the graphs were near where they needed to be placed (for example, how much it needed to shifted or stretched or if it needs a reflection) in order for it to connect to the previous equation. Another complication was making sure that the roller coaster path being modelled using the equations did not make any extreme turns or changes that would cause the roller coaster too look increasingly strange or completely unrealistic.