Is $x+1$ translated horizontally by -1 or is it translated vertically by 1?

Question

I know that if you have $y=f(x+a)$, that shifts everything to the left by $a$ (which I tend to think of in terms that you are "tricking" the function by giving it an input that isn't actually $x$ but $x+a$. If you have $y=f(x)+a$, that shifts everything up by $a$ (because $y$ is equal to all of the outputs, $f(x)$, increased by the $+ a$).

What I find a bit confusing is whether or not $x+1$ would be more like the first, in the sense that you are modifying the input to an identity function, or the second where you are modifying the results of an identity function.

Answer 1

$$ \DeclareMathOperator{id}{id} y = x + 1 = \id(x+1) = \id(x) + 1 $$ The first case can be seen as $\id$ (the identity function) translated one unit to the left, the second as $\id$ translated one unit upwards. So both interpretations are fine.

Answer 2

Hint :

Try to draw the graph of identity function $(y=x)$, and then first shift it vertically up by $1$ unit and then second time, shift it horizontally backwards by $1$ unit.

The result is the same. Both graphs are coincident to each other.Therefore, both the ways result in the same transformation.

Answer 3

When you translate $y=x$ up by $1$ unit, you are also translating it to the left hand side by $1$ unit.

It is not always like that, but in this case, moving left and moving up the graph shows two translated ones to be coincident.

Actually, it should not be considered as "I translated, so the function is added by $1$" but "I added by $1$ so that the function is translated", as the former one may mislead to some double-reasoning to your problem.

Well, consider a particular point on the function (as trying to do it in a discrete way), you may find that the new points are not coincident after the two different way of translation. It might be the real reason why you were stuck on that. But when we consider it as a whole graph, it is coincident.

Hope it helps your understanding.

Answer 4

you can safely think about it both ways. it's just the way you want to interpret it.