# Calculating a value inside one range to a value of another range

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

• For example $x ↦ 100(x + 1)$? Or do you mean something else? – k.stm 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? – jarryd Apr 30 '13 at 13:17
• I'd say it's an affine transformation. – Matt L. 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. – Michael Lewis Nov 6 '13 at 4:31
• That's slope intercept form of line equation and that's brilliant indeed. – Ankit singh kushwah Jul 23 '17 at 13:03
• Thanks it works, but why? – Veneet Reddy Aug 9 '17 at 12:14

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 $y = f(x)$.