Calculus Constructing a smooth roller coaster: f(x) = -(5/64)x2+0.7x ; 0<x<16

[Morgenstern]

Given that my equation for f(x) = -(5/64)x2+0.7x ; 0<x<16, equation for L1 is y=0.7x ;-5<x<0 and equation for L2 y = -1.8x +20; 16<x<20

How do I come up with equations more complex equations which can connect to L1 and L2 to create a roller coaster with smooth transitions (i.e. same tangents)

 

[Deleted member 4993]

Given that my equation for f(x) = -(5/64)x2+0.7x ; 0<x<16, equation for L1 is y=0.7x ;-5<x<0 and equation for L2 y = -1.8x +20; 16<x<20

How do I come up with equations more complex equations which can connect to L1 and L2 to create a roller coaster with smooth transitions (i.e. same tangents)

What is the equation (expression) for the slope of a tangent line to the given curve [f(x) = -(5/64)x2+0.7x]?

In adddition, the curve and the line should have same (x,y) at the transition point.