## anonymous one year ago Greatest integer parent function: grt int(x). What is the function for grt int(x) shifted down one unit?

1. anonymous

I tried graphing grt int(x) + 1 (in calc: floor(x) + 1) but it looks like the graph for shifting horizontally (grt int (x + 1).

2. anonymous

@SolomonZelman Could you help me again?

3. SolomonZelman

|dw:1441668972134:dw|

4. SolomonZelman

I am lagginging, i have to refresh. I will just say that to shift C unit down, subtract C from the entire function

5. anonymous

Alright, thanks!

6. anonymous

I'm an idiot, I just realized since it's steps it just elongates each step a bit, nevermind, that's why it looks like the other graph. Thanks for the help.

7. jim_thompson5910

https://www.desmos.com/calculator/6odyay0pdt notice how the red parent function floor(x) gets shifted down 1 unit to get to floor(x)-1 the "floor" function is another way to state the "greatest integer function"

8. SolomonZelman

The greatest integer (parent) function, also known as the floor function {of x}, is often denoted by: $$\large\color{blue}{ \displaystyle f(x)=\lfloor x \rfloor}$$ Or basically, that when you plug in x-values that are on the interval $$\bf [0,1)$$, $$($$including 0, and not including 1$$)$$, then you get 0. When you plug in x-values that are on the interval $$\bf [1,2)$$, $$($$including 1, and not including 2$$)$$, then you get 1. And so it is true that when you plug in x-values from some {and including an} integer $$\bf C$$, and till {but, not including $${\bf C}+1$$}, then you get $$\bf C$$. ------------------------------------------ here, are some examples: In a case where: $$\large\color{red}{ \displaystyle f(x)=\lfloor x \rfloor}$$ $$\large\color{royalblue }{ \displaystyle f(-2)=\lfloor -2 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(-2)=-2}$$ $$\large\color{green }{ \displaystyle f(4.5)=\lfloor 4.5 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(-2)=4}$$ $$\large\color{royalblue }{ \displaystyle f(13.9)=\lfloor 13.9 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(13.9)=14}$$ $$\large\color{green }{ \displaystyle f(0)=\lfloor 0 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(0)=0}$$ $$\large\color{royalblue }{ \displaystyle f(\pi)=\lfloor \pi \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(\pi)=3}$$

9. SolomonZelman

oh, in the third example, I wrote that it is equal to 14. I WAS WRONG it is 13.

10. SolomonZelman

because the greatest integer that is in 13.9 is 13. (Not 14, as I said)

11. SolomonZelman

If you want to use something interesting, $$\large\color{brown }{ \displaystyle f({~}\rm i^i{~})=\lfloor {~}\rm i^i{~}\rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f({~}\rm i^i{~})=0}$$

12. SolomonZelman

((if you have learned about imaginary number i, thatis $$i=\sqrt{-1}$$ ))

13. anonymous

Awesome thanks.

14. SolomonZelman

yes, just in case, verifying, that if you want to shift it C units up/down, right/left then it follows regular rules (And just like by a line, shift right =shift down, and shift left = shift up) Like I mean that: $$\large\color{black}{ \displaystyle f(x)=\lfloor x+a\rfloor\ }$$ is same as $$\large\color{black}{ \displaystyle f(x)=\lfloor x\rfloor\ +a}$$ where $$\large \color{black}{a} \in \mathbb{Z}$$

15. SolomonZelman

So a parent greatest integer function $$\lfloor x\rfloor$$ that is shifted one unit down, you can either right as: $$\large\color{black}{ \displaystyle f(x)=\lfloor x\rfloor-1 }$$ Or, you can re-write it as: $$\large\color{black}{ \displaystyle f(x)=\lfloor x-1\rfloor }$$