For this task, Year 11 math method students have to design a rollercoaster in Desmos using at least five different functions. Their design must include a minimum of one trigonometric function, one exponential function, and one logarithmic function, with the remaining functions chosen to create a realistic track shape (preferrably easier function). The coaster needs to be smooth the entire way through, meaning every segment must connect cleanly without sharp corners, and also be verified for smoothness (continuity). They also have to make sure the steepness never exceeds 70 over any section, so derivatives are used to verify that each part of the ride is safe. For each segment, thry explain the function, the domain, why it fits that part of the coaster, and show the derivative to prove the gradient stays within limits. We also include screenshots from Desmos as evidence. The goal is basically to build a realistic rollercoaster profile, justify the maths behind it, and demonstrate smooth, safe transitions throughout the design. I'd like the first segment to start as an exponential, so that it gradually grows and then the rest can be done by what is asked.