#step functions

What is an integer?

A set of numbers that contains positive whole numbers, negative whole numbers, and zero. So pretty much anything that doesn’t involve decimals or fractions when written out numerically.

4 is an integer.

-1 is an integer.

sqrt(2) is NOT an integer.

π is NOT an integer. (uh. That's supposed to be pi.)

 (Integers encompass the subsets of whole numbers and natural numbers, if you remember talking about that in class. Normally, it’s first-week-of-school stuff.)

What exactly is a Greatest Integer Equation?

(I’ve actually never heard it called a floor graph before. I have heard them referred to as step functions, and they’re also an example of a piece-wise graph. Probably yet another thing America does differently. Or maybe it was just my school. *sigh*)

Let’s take a look at the most basic floor equation: y = [x]

Basically, what it’s telling you is that y needs to be an integer. AKA, y cannot be a decimal. If you have a decimal value for y, well SCREW YOU, WE DON’T CARE. GET RID OF IT! How do you get rid of it? Round down to the closest integer. Examples:

6 would stay 6

3.7 would become 3

2.9999999999999999999999999999 would become 2

2.45345723458234 would become 2

0 would stay 0

-0.5 would become -1 (remember, we’re rounding DOWN)

-1 would stay -1

-3.23874563874 would become -4

 SO.

Llet’s look at the first of the equations you specified: y=[2x]

First things first, let’s make a table.

See how I crossed out -2.6 and 5.4? That’s because of what we were talking about above. Rounding down because

but rather

Now let’s take a look at another one of the equations you specified: y=[x+3]

What’s important to note, is that you’re going to be rounding down whatever’s within the brackets. SO.

See what I did there?

NOTE: You solve whatever’s within the brackets first, THEN round down. So in this case, the computation went

        y = [.5 + 3]

        y = [3.5]

        y = 3

Now let’s go head to head with: y=[x] + 3

I showed a computational example in the pic.

Soooo, these two equations aren’t actually different. BUT the process IS slightly different because you’re rounding down x BEFORE adding 3, since 3 is no longer in the bracket.

I’m going to leave y=2[x] for you to do, but if you still can’t get it, let me know. Because I’ll have you know right here and now, IT’S NOT THE SAME AS y=[2x]. Why? BECAUSE YOU DO WHAT’S IN THE BRACKETS FIRST, THEN ROUND, THEN THE REST OF THE OUTSIDE EQUATION!

Hint example: y = 2 * [1.3]

                   y = 2 * 1

                   y = 2

So now, the moment we've all been waiting for...

GRAPHING TIME!!!

y = [2x]

First things first. What do you always do when you have a graph you have NO idea where to start with? Make a table! We’ve already done that, so the next step is to graph each point.

If you can’t read the graph, there’s no funky numbering system or anything.

Note I used the values that give an integer for y. By doing so, graphing becomes MUCH easier.

Now. Think about that. What I just wrote. I used values that give an integer for y. Because what happens if I don’t? What would that graph as?

So how would the infinite possibilities of x look? Like x=1.15, 1.35, or 1.4999999999999999999999999?

Note the circle I stuck at the end of that line. That’s how you graph a line when all the possible little values of x up until the circled value equals a number, but the circled value is something else, like how 1.4999999999999999999999999999999 is 2, while 1.5 is 3.

SO. The entire graph is

Instead of using my crappy hand-written one, here’s a nice, computerized graph. (Though Wolfram failed me by not sticking the circles at the end. But that’s what Paint is for :P)

And that’s how you graph a floor function. Here are the graphs for the other 3 you asked about, but try to figure them out on your own before peeking :) These are also missing the circles at the right ends of the lines, but I'm not editing them in there because it's late, I have a physics midterm on Monday, and I've procrastinated enough already.

y=2[x] (two graphs for this one! One is more close-up than the other)

y=[x+3]

y=[x] + 3

Also, note that the brackets aren’t actually brackets. It’s supposed to be an L and a backwards L like

 It’s just difficult to computer it, and though I can (like here), it’s time consuming since I lack the proper software.

Hope this helped!