So, I was recently reviewing slope. While doing some problems with the slope formula $m = {y_2 - y_1\over x_2 - x_1}$, I started thinking why it was that and not $m = {y_1 - y_2\over x_1 - x_2}$. Because point one and two are interchangeable, this should work. Is it doors work, why did they decide to put two first, then one?
It seems like it would be more reasonable to put numbers in there natural order if you had the chance.
On
When you start at a point like $x$ and introduce some increment $h$ you get to the point $x+h$ so your change in $x$ is $$x_2 - x_1 = (x+h)-x =h$$
As a result of the change from $x$ to $x+h$, we get some change in $y$ namely we go from $f(x)$ to $f(x+h)$, so our change in $y$ is $$f(x+h)-f(x) =y_2-y_1$$
This it is natural to define the average rate of change as $$\frac {y_2- y_1}{x_2-x_1}$$
They are the same. That is, $\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_1 - y_2}{x_1 - x_2}$
So why is it convention to say $\frac{y_2 - y_1}{x_2 - x_1}$?
Typically, we choose the points $(x_1,y_1), (x_2, y_2)$ such that on the graph of the line $(x_1,y_1)$ is to the left of $(x_2, y_2)$. That is $x_1 < x_2$ and $x_2 - x_1 > 0$ and the denominator of $\frac{y_2 - y_1}{x_2 - x_1}$ is postive. And, if the slope is postive, the numerator is also positive.
Also, if I wanted to describe a vector from point A to point B, I would say $B-A.$ So, $\frac{y_2 - y_1}{x_2 - x_1}$ suggests to me that we are measuring from $(x_1, y_1)$ to $(x_2,y_2)$