function for fitting a curve to 3 points

Hi MHF community! Hi. I develop audio synthesis and sequencing application, and am currently exploring a new synthesis technique. As part of this, I'll be specifying three [x,y] co-ordinates. two of them will represent the 'base' of a waveform ie [a, b] and [c, b]. the third will represent the 'peak point' above the 'base' ie [d, e] where a<d<c and e>b. Now, I need a function with a user-specified parameter: 'skinniness' (s for short). When s is a low value, the function should draw a smooth curve from [a,b], with a peak at [d, e], then down back to [c,b]. The low s value means the curve should 'bow out' as much as possible. When s is an intermediate value, the curve should essentially look similar to a triangle ie a straight line from [a,b] to [d,e] and a straight line from [d,e] to [c,b]. When s is high, the curves between the points should be as 'sucked in' as possible; something that looks line an exponential from [a,b] to [d,e], and a reversed exponential from [d,e] back down to [c,b]. Assistance please to meet this objective?