Let's assume the maximum temperature during a month is $86 F ^\circ$ of all days and the minimum temperature during a month of all days is $74 F ^ \circ $. The range of temperature is the difference of the two, so $12 F ^\circ$.
The formula for the temperature is $$C=\frac{5}{9}(F-32)$$.
Why isn't the corresponding range of temperature in that particular month in centigrade
$$C=\frac{5}{9}(12-32)$$
but rather around
$$7 C ^\circ$$?
On
Suppose the two temperatures in F are $F_1$ and $F_2$. Then $F_1$ in C, which we call $C_1$, is given by $$ C_1 = \frac{5}{9}(F_1-32), $$ and likewise, $$ C_2 = \frac{5}{9}(F_2-32). $$ Then the difference between $C_1$ and $C_2$ is $$ C_1-C_2 = \frac{5}{9}(F_1-32) - \frac{5}{9}(F_2-32) = \frac{5}{9}(F_1-F_2) : $$ the 32 terms cancel out, because it is the same shift applied to both. (It's the same if you measured on a ruler starting from $1$ instead of $0$: the difference between two measurements would be the same, even though they are both $1$ larger than measuring from the beginning.)
That because the difference of two values of an affine function is the value of the associated linear map at the difference.
In formula: $$\bigl(a(x+\Delta x)+b\bigr)-(ax+b)=a\mkern 2mu\Delta x, \quad\text{not }\; a\mkern 2mu\Delta x+b.$$