Question

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

Answer 1

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).

Answer 2

@SolomonZelman Could you help me again?

Answer 3

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

Answer 4

Alright, thanks!

Answer 5

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.

Answer 6

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"

Answer 7

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: \(\large\color{red}{ \displaystyle f(x)=\lfloor x \rfloor}\)

\(\large\color{royalblue }{ \displaystyle f(-2)=\lfloor -2 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(-2)=-2}\)
\(\large\color{green }{ \displaystyle f(4.5)=\lfloor 4.5 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(-2)=4}\)
\(\large\color{royalblue }{ \displaystyle f(13.9)=\lfloor 13.9 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(13.9)=14}\)
\(\large\color{green }{ \displaystyle f(0)=\lfloor 0 \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(0)=0}\)
\(\large\color{royalblue }{ \displaystyle f(\pi)=\lfloor \pi \rfloor{~~~~~~}\Longrightarrow{~~~~~~}~f(\pi)=3}\)

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

Awesome thanks.

Answer 13

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 \(\large \color{black}{a} \in \mathbb{Z} \)

Answer 14

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:
\(\large\color{black}{ \displaystyle f(x)=\lfloor x-1\rfloor }\)