Analyzing the graphs of Greatest Integer Functions

[opus]

Homework Statement

Consider $u\left(x\right)=2\left[\frac{-x}{4}\right]$
(a) Find the length of the individual line segments of the function,
(b) Find the positive vertical separation between line segments.

Homework Equations

The output of Greatest Integer Functions are always integers.

The Attempt at a Solution

I'm honestly confused about this whole situation.

Length:
The text states that the coefficient of x within the greatest integer symbols is the length of the individual line segments of the graph.
In $u\left(x\right)=2\left[\frac{-x}{4}\right]$, the coefficient of x is $\frac{-1}{4}$.
However, the solution for the length of the graph states that length=4.
It explains this by stating that there's a decrease of 1 for every increase of 4 in the variable x.
This would make sense if we were talking about the slope of a line, but it doesn't make any sense at all in this context.
And since we're talking about the length of a line segment, does the negation matter?

Vertical Separation:
The text states that the coefficient of the greatest integer function is the positive vertical separation between line segments.
This is a straight forward statement, and the vertical separation=2, but I don't see why this leading coefficient determines this.

Can anyone help me get a better idea of what is going on with the graphs of these functions?

 

[andrewkirk]

I haven't seen the notation you are using but I presume that the square brackets [...] denote the function that gives the greatest integer that is less than or equal to the value of the contents of the brackets. In my experience that is usually indicated by $\lfloor...\rfloor$ and is called the 'floor' function.

If that is the case then the function value is piecewise constant and steps down at each multiple of 4, ie at ...,-8, -4, 0, 4, 8, ...
So each line segment is horizontal and extends for the period that x takes to increase by 4. So its length is 4.

How much does it step down by each time? Well $\lfloor\frac{-x}4\rfloor$ always has an integer value and decreases by 1 each time $x$ reaches a new, higher multiple of 4. So that's a step size of 1, but then it is multiplied by 2 - the leading coefficient - so the step size (the vertical separation) is 2.

The impact of the negation is to make the steps go down as $x$ increases, rather than up. But it doesn't affect the step length or height.

 

[opus]

Yes we are talking about the same function. My text has the it denoted as something similar to but not exactly like [[x]]. But I looked up on Wikipedia the "floor function" that you stated and we're talking about the same thing.

That was a great explanation, thank you.

Let me see if I understand:
The length is 4 because if we were to create a table of values (leaving the leading coefficient of 2 out for simplicity's sake), it would take 4 integer x inputs for the line segment to be complete and move up or down to the next segment? For example, with an input of x = -8 gives an output of 2, and inputs x = -7, -6, -5, -4 all give an output of 1. Then when x = -3, the output is zero so there's a new line segment vertically shifted.

For the vertical displacement: the greatest integer function output is always an integer. So when we take the output of the function's argument, and multiply it by 2, we are vertically shifting by that value 2.

 

[andrewkirk]

Yes that is correct!

 

[opus]

Thank you for the great explanation!