I recently came across the following question:
When $h(x)=3^{x-b}$, how does $b$ change the graph of $f(x)$ (where $f(x)=3^x$)?
Answers provided were as follows:
If $b>0$, then this corresponds to a horizontal translation to the right $b$ units.
If $b<0$, then this corresponds to a horizontal translation to the left $b$ units.
Shouldn't the second answer technically be "[...] horizontal translation to the left $|b|$ units"? Otherwise, if $b=-2$ for example, then you would be saying the graph is translated to the left $-2$ units, and transferring or shifting anything negative units in one direction is not generally a well-defined concept unlike a negative angle. On the real number line, one speaks of $-4$ meaning $4$ units to the left and $+4$ as $4$ units to the right. Something like $-(-4)$ does not even make sense on the real number line--we know $-(-4)=4$ by the existence and uniqueness of an additive inverse in $\mathbb{R}$, whereby we could plausibly say that $-(-4)=4$ corresponds to $+4$ or $4$ units to the right (but only after using the facts that the additive inverse exists and is unique).
A question such as the following would then really not make much sense: Which of the following functions transforms the graph of $y=a^x$ by reflecting it over the $x$-axis, shifting it $b$ units to the left and $c$ units up: $(a)^{-x+b}+c$, $-(a)^{x+b}$, $(-a)^{x-b}+c$, $y=-(a)^{x-b}+c$, or none of these?
Clearly, the second choice is the intended answer, but does this not assume that $b$ and $c$ are nonnegative? If $b$ were negative and $c$ were positive, then $y=-(a)^{x-b}+c$ is the graph of $y=a^x$ reflected over the $x$-axis, shifted $|b|$ units to the left, and $c$ units up. The second choice, $y=-(a)^{x+b}+c$, would not seem to make sense if $b$ were negative because then we would run into the ill-defined concept of moving negative units in a direction.
What would rigorously be the best answer here and/or clarification in general concerning the perceived issues brought up above?
For any function $f(x)$, the new function $g(x)=f(x+a)$ will we:
$1)$ horizontal translation of $f$ by a factor $|a|$ to the right if $a<0$ and;
$2)$ horizontal translation of $f$ by a factor $|a|$ to the left if $a>0$.
Just to make that clear:
Suppose that $a>0$ and $x_0$ is such that $f(x_0)=k$.
So $g(x_0-a)=f(x_0-a+a)=f(x_0)=k$ and then we can see that the $g$ will became $k$ at $x_0-a$ and that will happen for every point. Once $a>0$ we went to the left of $x_0$ by a factor $|a|$.
For $a<0$ just use a similar idea.