I have some confusion about shifts concerning exponential functions. I can best describe my question with an example.

Take $y = e^{-(x-3)}$. This graph has a reflection over the $y$-axis and is shifted right $3$ units.

Why right instead of left, though. Considering that this is equivalent to $y=e^{(-x+3)}$ I thought that the shift would be opposite of the sign being that it is in parentheses.

Take $y = e^{x+3}$ for example. The graph of this function is shifted left $3$ because of the parentheses.

How am I supposed to figure out the shifts in the graphs? Am I missing a detail here?


All you have written is correct. You only have to take care on the order of the transformations. For this, ask: 'What happens to $x$?' and reverse the order and the operations.

In the case of $e^{-(x-3)}$, $x$ is first decreased by $3$, then multiplied by $-1$. If we reverse these operations, we see that first we have to reflect the graph of $e^x$ along the $y$-axis and then shift it to the right by $3$ (shift it to the left by $-3$).

For the same $e^{-x+3}$, we find that $x$ is first multiplied by $-1$ then the gotten expression is increased by $3$, so, reversing these, we first shift, indeed to the left, and then reflect.


The transformation for $e^{-(x-3)}$ corresponds to the substitions: let $u:=x-3$. First, from $u\mapsto e^u$ we go to $u\mapsto e^{-u}$ by reflecting the original graph on the $y$ axis. Then making the substition $x\mapsto x-3$ i.e. $x\mapsto u$ in the variable will give us the second step. You will be convinced if you plug in (enough) concrete values of $x$: e.g. if $x=3$ then $u=0$ and then $e^{-(x-3)}=e^{-u}=1$. If $x=4$ then $u=1$, and so on..

In general, the graph of $g(x)=f(x-3)$ is shifted to the right (to the left by $-3$) compared to the graph of $f(x)$, because $$g(x+3)=f(x)\,.$$

  • $\begingroup$ Thanks. I understand the last line, about first shifting and then reflecting. The line above that is still confusing to me. Why is the order different between the two equations and why do is the graph shifted to the right, not left in the first one. $\endgroup$ – foobar512 Sep 18 '13 at 11:39
  • $\begingroup$ I have updated. You have to draw enough (simple) examples to understand it deeply. Use the weapon of substition for easy-to-compute concrete values.. $\endgroup$ – Berci Sep 18 '13 at 16:08

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