# The Functinator-inator

## Written Report

The linear equation y=29x+10 has a positive slope, that is increasing. As the x values (time) increases the y values (height) also increase. Since this is a roller coaster the domain has to be x>0 because time cannot be negative.

The second equation which is Y= -(x - 12.1)^2+300 is a quadratic function. A negative “a” value shows a reflection in the x axis. This was used to make the roller coaster have a downward turn. As time increases within this equation the height gradually decreases because of the negative “a value”.

The third equation is Y= -2(x – 30)^3 +18 which is a polynomial equation. It is a cubic function that has a negative “a” value. As time increases the roller coaster approaches towards the ground. Therefore the height decreases as the time increases. The fourth equation Y=(x-32)4 + 0.5x – 1.9 which is a quartic function. Since the “a” value is positive height will increases as time increases. This equation makes the roller coaster approach to the maximum height. As time is increasing the height increases significantly.

The fifth equation Y=-[15.5/(x-34.7)] +155 is a reciprocal function. The restrictions on this graph indicate where the roller coaster is turning. As time is increasing the height also increases.

The equations Y=10sin (1/3x) +150 is a sine function. The equation has a maximum value and minimum value. When the function approaches the maximum value the height increases along with the time. When the roller coaster is approaching the minimum value; height is decreasing along with the time.

The equations Y= -[100/(x-82)] +100, Y= -9(x-71.01)1/5 +131, as time is increasing the height is increasing. Y= -9(x-71.01)1/5 +131 is a exponential function therefore will increase as time increases. Y=-12(x-82.5)2 +250 is a quadratic function that is reflected in the x axis. Therefore it will have a maximum point, which indicates the function is decreasing. As time is increasing the height decreases because of the reflection in the x axis. Y= -18log0.8(x-86.4) +30 is a logarithmic function which means it will decrease in height as time passes. The roller coaster is ending therefore the function will decrease in height.

## Creating the Rollercoaster

To form our rollercoaster we began with a linear equation in order to elevate the rollercoaster. Majority of rollercoasters then follow with their highest drop, so we placed a quadratic equation after the linear. We developed a negative quadratic equation so that the parabola would be reflected in the x-axis and would appear as a drop.

Then we added a cubic function, in order to produce more excitement in our rollercoaster, since more successful rollercoasters follow with less higher drops after the initial drop. After, we incorporated a quartic function, because after the previous drop, the rollercoaster would still have enough power and a large enough velocity to go up again.

We then added another quadratic function to provide a smoother ride and to ease into a sinusoidal function. The sinusoidal function is two cycles long and was added in order to offer riders with a break. There was then an exponential function to give riders a sudden drop and to build extreme suspense. After that there was rational function to offer the final drop. Then there was one more.

The challenge of this assignment was to make sure the functions join together to form one overall rollercoaster. Being precise as well as accurate with decimal numbers when developing restrictions on the range and domain ensured that the rollercoaster formed one ride.

The difficulties that we endured during this task was to create a functional and realistic rollercoaster. This meant that the functions needed to have extra restrictions on both the x and y values to fit within a specific time interval of 100 seconds. This also meant that the restrictions needed to be decimal numbers to ensure that the functions connect, as a real rollercoaster would. Also, we incorporated extra functions in between the required functions in order to prevent sharp turns that could probably lead to accidents occurring on real rollercoaster. Moreover, functions that are greatly stretched vertically are more steep and they were placed before functions that are shallow or are more compressed. This is because higher steeps allow for greater acceleration and give the rollercoaster more power to move over a greater horizontal distance.

## y = 250 feet

The roller coaster reaches a height of 250 feet at 3 different points during the ride. The calculations below indicate exactly what time the roller coaster hits this height.

## y = 12 feet

The roller coaster reaches a height of 12 feet at 2 different times during the ride. The calculations below indicate exactly what times the roller coaster hits this height.

## AVERAGE RATE OF CHANGE

The Average Rate Of Change (AROC) is equal to the change in y- values divided by the change in x-values.

## AROC (10-15s)

The calculation of the average rate of change between 10 seconds and 15 seconds is shown below.

## AROC (50-60s)

The calculation for the average rate of change between 50 seconds and 60 seconds.

## IROC (35s)

To calculate the Instantaneous Rate of Change at 35 seconds, a tangent line must be drawn from the x-value and add 0.001, then solve.
Therefore, the Instantaneous Rate of Change at 35 seconds is 108.55401.