Draw a roller coaster on a graph with functions.

Question

Hello I have a project to do and I'm really stuck, please help me through it. So I have to design a roller coaster with functions. I have to draw a short roller coaster on graph paper, plot ordered pairs on its path, and determine the slope, or rate of change, along the ride. I only have to draw the side view of the roller coaster. In order to gain speed, the roller coaster should have an initial climb, at least two hills and one loop. I have to label the x and y axes. The value of x measures the horizontal distance from the roller coaster cart to the starting point.And the value of y represents the height of the roller coaster cart. I have to plot ordered pairs on the initial climb and determine the slope. I have to find the equation of the line that represents my initial climb. I also have to find the domain and range of the roller coaster. Please explain

Answer 1

Hi, could you maybe sketch an example of what you would like your roller coaster to look like? Then, from there I could help you find equations to best fit the shapes of your roller coaster.

 

 I believe it is best to break the roller coaster up into many separate parts/shapes. Think about the incline, for example: a typical roller coaster incline could look like a diagonal, straight line. You could use a linear equation to stand for this part of the coaster. An example of this would be y=4x+0. Whatever your y-intercept is will be where any linear equation crosses the y-axis on a graph (in this particular case that number is 0); also, the higher the value for your slope in linear equations, the steeper your line will be. 

 

As far as the "hills" or curved parts of the coaster go, I recommend using quadratics, such as y=2*x^2 + 4. If the coefficient in front of the "x^2" term is positive, the parabola will be concave up / open upwards; likewise, a negative coefficient in front of this term would have the parabola opening downward / concave down. The number +4, as used in this example, shifts the parabolas up or down--up for a positive value, and down for a negative value.

 

When figuring out the equation(s) to use for your loop, think of its shape as a circle. The equation for circles on the origin--point (0,0) when graphing--is x^2+y^2=r^2 where "r" is the radius of the circle. (x-h)^2 + (y-k)^2 = r^2 is also the equation for circles, where "h" is your horizontal shift and "k" is your vertical shift. (h,k) is the vertex (center( of the circle based on where it's plotted on the graph.  

 

After finding all of your equations, I recommend graphing all of them together and then adjusting your equations if need be based on where you need them to "start" and "end" so that your coaster is "continuous." You will need to take into account what your domain (x-axis) and range (y-axis) will be as well. 

 

I recommend that to check out the online Desmos Graphing Calculator if you have never tried to use it before. (You can just search for it on Google, and it should be one of the first links.) I find that it comes in handy when I have a function/shape of a line in mind and want some help finding what the equation would be for it.

 

I hope that all of this will help you get started on this project. Feel free to respond with further questions. Best of luck!