Why don't we use the proper terminology for linear and affine?

Question

Why do we call y = mx + b linear (in high school)? Why don't we call it affine?

and linear what is actually linear

$$f(x+y)=f(x)+f(y)$$ $$f(ax)=af(x)$$

Answer 1

As written in den comments, you have to consider the context. There are also other definitions which are not consistent in mathematics. An other example is the set of natural numbers. Some authors say that $0$ is a natural number but other say that the natural numbers start with $1$.

In high school you call $f:\mathbb R\to\mathbb R$, $f(x)=mx+b$ a linear function and sometime we still use it in analysis for example like this:

Define $f:\mathbb R\to\mathbb R$ continuously by $$ f(x)=\begin{cases}0 & x\leq n\\1 & x\geq n+1\\linear & x\in[n,n+1] \end{cases}. $$ Here obviously $f$ is not linear in sense of linear algebra on $[n,n+1]$, but everyone knows how to understand it.

On the other hand, if you consider vector spaces $V$ and $W$ and a linear function $F:V\to W$ then it is obvious, that $F$ has to be linear in sense of linear algebra. So from the context there is no misunderstanding.