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

How does one calculate the value within range $$-1.0$$ to $$1.0$$ to be a number within the range of e.g. $$0$$ to $$200$$, or $$0$$ to $$100$$ etc. ?

• For example $x ↦ 100(x + 1)$? Or do you mean something else? Apr 30 '13 at 12:24

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:

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

• Thanks. What is the name/tag of such a formula? Apr 30 '13 at 13:17
• I'd say it's an affine transformation. Apr 30 '13 at 15:33
• This is brilliant!! I'm surprised I've never seen something like this before. I knew what I was trying to do was simple, yet I couldn't do it mentally. Nov 6 '13 at 4:31
• That's slope intercept form of line equation and that's brilliant indeed. Jul 23 '17 at 13:03
• @Talespin_Kit: By definition, $y=ax+b$ is an affine transformation. Dec 13 '19 at 17:05

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 $y = f(x)$.