How to form functions similar to $\frac 12 - \frac12 \cos \pi x$

Question

I wanted to know how to form curves similar to that of $y = \frac12 - \frac12\cos(\pi x)$. What I need is a function that forms a smooth curve been $x = 0$ to $x = 1$. As you see with the $y = \frac12 - \frac12\cos(\pi x)$ function, between $x = 0$ and $1$, the $y$ value smoothly goes from $y = 0$ to $y = 1$.

However I was wondering how I would go about augmenting this function such that the curve between $0$ and $1$ is more harsh (e.g. the curve goes from $y = 0$ to $y = 1$ jumps up more abruptly) or more gradual (e.g. the curve resembles something that looks like $y = x$)

Answer 1

...I was wondering how I would go about augmenting this function such that the curve between $0$ and $1$ is more harsh (e.g. the curve goes from $y = 0$ to $y = 1$ jumps up more abruptly) or more gradual (e.g. the curve resembles something that looks like $y = x$)

it seems to me that what you want is something like what computer graphics practitioners call a "bias" function.

The first version of this is due to Ken Perlin, he of the award-winning noise function ubiquitous in CG. The definition he uses is

$$\mathtt{bias}(b,t) = t^{-\frac{\log b}{\log 2}}$$

Since transcendental functions are involved, the evaluation of this can get expensive; Schlick thus devised an alternative rational function that can be used for the same purpose:

$$\mathtt{bias}(b,t) = \frac{t}{(1/b-2)(1-t)+1}$$

In both instances, $b$ is an adjustable parameter. You can tweak it to get the desired manner of increase.

Answer 2

The following function is a good choice to satisfy your requirement:

$$ y(x)=x^{t}(2-x)^{t}$$

You may vary the parameter $t$ to regulate steep or slow rise of the curve from 0 to 1.