[EDITED TO CORRECT LEFT/RIGHT CONFUSION]
- Let $f$ be a function defined by $f(x)=mx$.
It is possible to produce a horizontal shift of $a$ units to the right ( provided $a$ is positive) by defining a function $g$ such that $g(x)=f(x-a)$.
This horizontal shift also produces a vertical shift, but the formula works inasmuch as the intended horizontal shift occurs.
- However, the intended shift does not occur, or occurs not in the intended way, when one defines a function $h$ as follows : $h(x)= f(x-a)+b$ .
For in that case, the vertical shift perturbates, so to say, the intended horizontal shift.
- Hence my question: should one say that the rule " $h(x)=f(x-a)+b$ produces a horizontal shift of $a$ units ( to the left or to the right, depending on the sign of $a$) and a vertical shift of $b$ units ( up or down) " only works for non-linear functions?
Note : my question does not deal with the vertical shift , which , apparently, always occurs in the intended way.
Below, an xample, with an intended shift of $4$ units ( to the right) and an actual shift of $5$ units.
https://www.desmos.com/calculator/nd7uu4wnug
Best Answer
The simple fact is that if $f$ is linear, you can't retrieve the horizontal and vertical components of the shift just by looking at the graph. But we can still describe a transormation in terms of horizontal and vertical shifts; it's just that this description is not uniquely determined. There is no paradox here.