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Assume a function called $f(x)$ , Then all of us know that of we draw $f(x+a)$ it will be a transformation to the left or right and $f(x)+b$ to up or down.

But when I drew floor function on Desmos online graphing I found something a little bit different.

https://imgur.com/bHUeSZ5

As you see that the black function is exactly on the purple one and that confusing me , Here $f(x+a)=f(x)+a$ , And we are just rising the function up .

To be more specific I need you to explain these question :

$1.$ Why when we add a number in the floor function notation $f(x+a)$ or $[\frac {x}{2}+1]$ it rise the function.

$2.$How can transform the function to the left just a unit.

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The fact that $\lfloor f(x) + 1 \rfloor = \lfloor f(x) \rfloor + 1$ for any function $f$ should be fairly obvious, if you think about what the floor function does: round down to nearest integer. (You can replace "$+1$" here with "$+n$" for any integer $n$, but it doesn't work for non-integers.)

So for your particular example $f(x) = \frac{x}{2}$, you get $\lfloor \frac{x}{2} + 1 \rfloor = \lfloor \frac{x}{2} \rfloor + 1$, and the graph is moved one unit upwards.

But that's also equal to $\lfloor \frac{x}{2} + 1 \rfloor = \lfloor \frac{x+2}{2} \rfloor = \lfloor f(x+2) \rfloor$, so it's the same thing as moving the graph two units to the left.

If you want to move the graph one unit to the left, take $\lfloor f(x+1) \rfloor = \lfloor \frac{x+1}{2} \rfloor = \lfloor \frac{x}{2} + \frac{1}{2} \rfloor$ instead.

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  • $\begingroup$ When we say $[\frac {x}{2}+1]$ is this trassfomation to left or up ? $\endgroup$ – Mohammad Alshareef May 7 at 10:51
  • $\begingroup$ @MohammadAlshareef: Both! As I wrote, it's the same thing (for this particular graph). $\endgroup$ – Hans Lundmark May 7 at 11:38

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