Going from a value inside $[-1,1]$ to a value in another range

If you have numbers $x$ in the range $[a,b]$ and you want to transform them to numbers $y$ in the range $[c,d]$ you need to do this:

$$y=(x-a)\frac{d-c}{b-a}+c$$

A short proof of Matt L.'s answer:

We want a function $f: [a, b] \rightarrow [c, d]$ such that

$$ \begin{alignat}{2} f&(&a) &= c \\ f&(&b) &= d. \end{alignat} $$

If we assume the function is to be linear (that is, the output scales as the input does), then

$$\dfrac{d - c}{b - a} = \dfrac{f(x) - f(a)}{x - a}.$$ Simplifying yields the desired formula for $y = f(x)$.