Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I approached graphing $2^{4-x}$ by simplifying the function to $2^{-x}$ (i.e. the reflection in the y-axis of $2^{x}$), and then applying what I've learned that adding a constant to $x$ translates the graph to the left by $x$ units. My answer was wrong. The correct answer is $2^{-x}$ shifted to the right by 4 units.

I hope someone can provide a clear rule, or show how I should've thought about this properly. I figure it's because the coefficient on $x$ is negative that reverses the rule, but there's probably a cleaner way to think about it.

share|cite|improve this question
To get $2^{4-x}$ from $2^{-\color{maroon}x}$ you subtract $4$: $2^{-(\color{maroon}{x-4})}=2^{-x+4}=2^{4-x}$ (note the parentheses here). So essentially, what you said about the negative reversing the rules is correct. – David Mitra Dec 11 '12 at 12:46
How about looking at it as $2^{4} / 2^x$? Does that help you? What you did was also not wrong in theory, you were just using $-x$ (subtracting $4$ from $x$). – Amzoti Dec 11 '12 at 12:53
up vote 3 down vote accepted

Had you been adding $4$ to the $x$, you'd indeed have been shifting to the left. However, in this case, you had $-x$. Thus, $$4-x=-x+4=-(x-4),$$ so in effect, you were subtracting $4$ from the $x$, when all was said and done. The negative did, indeed, reverse the rule.

Put another way, if we have two functions $f(x)$ and $g(x)$ and a positive constant $a$, then $g(x)$ is obtained from shifting $f(x)$ left by $a$ if $g(x)=f(x+a)$, and right by $a$ if $g(x)=f(x-a)$. In this case, $f(x)=2^{-x}$, $g(x)=2^{4-x}$ and $a=4$. $$f(x+4)=2^{-(x+4)}=2^{-4-x}\neq g(x),$$ but $$f(x-4)=2^{-(x-4)}=2^{4-x}=g(x).$$

share|cite|improve this answer
I appreciate the thorough answer. I see now that if I had considered $2^{-x}$ as $2^{-(x)}$ then I would've subtracted the 4 from $x$ and got $2^{4-x}$ – PeteUK Dec 11 '12 at 15:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.