I have the equation to produce the second curve I'm using to make a sinusoidal function that smoothly maps from 0 to 1. The range for y= needed to be [0.5, 1] and it seems to be correct.
Wolfram|Alpha tells me this is equivalent to
I looked at all the trig equations on Wikipedia to try refreshing my Trig knowledge and got enough to transpose the basic sine curve into the form I need it. I can't work out the steps to simplify my equation to the answer Wolfram|Alpha provided though.
In the image below, I'm use the Orange curve function for range of x=[0, 0.5) and the Purple curve function for range x=[0.5, 1] to make a composite function.
Grapher file if you are on Mac (Grapher comes free with most versions of OS X):