Need help creating an expression/function for a specific curve

Hello everyone. I'm not sure if this question falls more into the algebra side of things or the trig side of things, but I figured this would be a good place to start.

Here's the problem. I need to find a function for a curve where f(x) = 0 when x = 0 or x = 1.

Now, the obvious solution is f(x) = sin(x * pi)

Here is the problem, a sin curve is too gradual. I need the curve to be close to its peak from....lets say x = .2 to x = .8, arbitrarily speaking.

I've tried taking a basic parabola, reflecting it across the x axis, and trying to adjust it's parameters to make a functional curve, but it hasn't gone so well.

Can any one educate me on creating this function, or maybe if its easier can I simply change the slope of the sin curve? (I don't think I can, but it's worth asking)