Roller Coaster Assignment

Equation One

The equation of this function is 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

The equation of this function is 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

The equation of this 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.

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.

What is the average rate of change from 10 to 15 seconds?

The average rate of change from 10 to 15 seconds is 0.193135 feet per second.

What is the average rate of change from 50 to 60 seconds?

Therefore, the average rate of change from 50 to 60 seconds is -3.55685 feet per second.

What is the instantaneous rate of change at 35 seconds?

Therefore, the instantaneous rate of change at 35 seconds is -4.92 feet per second.