(PreCalculus) Greatest integer function graph variations

[katieighty]

Hi!
I'm having trouble with variations with the greatest intefer function graph. Normally, I think they kind of resemble a staircase, but the ones I've been asked to graph look nothing like steps. (I know this because I've been given an answer key, but I want to know how they are getting their answer. Parroting answers that mean nothing to me will get me nowhere, you know what I mean?) I've found nothing useful or even comparable on the internet, and I was hoping you could help. Because it is a graph, and I can't physically show you the page I'm looking at, I'm going to give you the coordinates of the points and lines. I hope it helps.

There's five questions that I've been struggling with.
The first problem y = x - [x] has points on (-3,0) (-2,0) (-1,0) (0,0) (1,0) and (2,0). Each of these points is attached to a line that leads up and to the right, so that the end of the line (a hole) is directly over the adjacent coordinate. For example, the line attached to (-3,0) ends in a hole on (-2,1). The line attached to (-2,0) ends in a hole directly over (-1,1). And so on.

The second problem is G(x) = [x squared]. The answering graph has a points on (0,0) (-1,1) (-1 1/3, 2) (-1 2/3,3) (-2,4) on the left side and points (1,1) (1 1/3,2) (1 2/3,3) (2,4) on the right. Now each of those points (save -2,4 and 2,4 and 0,0) is attached to a line and a hole positioned 1/3 lower or higher than the original point. So, the hole attached to the line attached to (-1,1) is on (-1 1/3,1); and the hole of (1 2/3,3) is on (2,3). Each on of the holes falls directly under the next whole coordinate. And then for (0,0) the lines stretch out in both directions so that there are holes on points (0,-1) and (0, 1). Does that make sense? It's not something that you can easily visualize.

The third, fourth, and fifth problems look similar in that the graphs appear to be completely random. Maybe if I can get help with one, I'll be able to sort the next two out on my own. Yeah, maybe.

The third question is H(x) = [x] over x. It's graph involves both points attached to lines attached to holes and and points attached to rays (there's only one, but it's on a curve. I don't know where they're pulling these things out of.) All of the points end in 1. So, there's (-3,1) (-2,1) (-1,1) (1,1) (2,1) and (3,1). The lines attached to them go upward in the 4th quadrant and down in the 1st quadrant.

The fourth question is H(x) = x over [x]. It's graph is similar to question 3, except there is no ray. All of the points are on (x,1). Lines go down in the 4th quadrant and up in the 1st. Each of the lines ends in a hole. The lengths of the lines do seem to vary though.

The fifth appears to be the hardest. The question is F(x) = x squared - [x]. It's graph involves holes and rays. All of the lines are curved (very gradually, but curved nontheless).There does not appear to be any order in this graph other than the lines still go up in the 4th quadrant and down in the 1st quadrant.

Hopefully, I remembered the correct order for the quadrants. In the event that I didn't, the 4th quadrant is the upper left, and the 1st is the upper right. And I also hope that I put this in the right category. I didn't see anything that looked explicitely PreCalculus... I hope you calculus geniuses can get this. The curriculum I'm working with is beyond outdated. Nothing is adequately explained. Sorry, that's not really your problem, is it?

I've been stuck on these questions for days. I can't thank you enough for any help (however great or small) offered! Thank you so much!

 

[pka]

What exactly is your question?
You have clearly and correctly described the graphs.
What can we do for you?

 

[katieighty]

OK.... I think the best kind of help I could get is a very detailed walk-through of (each?!) the questions.

Also, I don't understand why it was necessary to curve some of the lines (in problems 3 and 5). Because these questions do differ from the standard step graph, I need to know what else is different. If anything. Thanks.

 

[pka]

We are able to give that kind of instruction on this sight.
This is a help service not a tutorial site.

However, I can give this bit of help or insight.
The greatest integer, [x], also known as the ‘floor’ function is constant between any two integers. Thus the graph of the function [x]/x looks like k/x between any two integers. Graph 2/x on a graphing utility. The is what [x]/x looks like between 2 & 3.
The graph \(\displaystyle x^2 -\[ x \]\) looks like \(\displaystyle x^2 - 2\) between 2 & 3.

 

[soroban]

Hello, katieighty!

Let's baby-talk through the first one . . .

\(\displaystyle 1)\;\;y \:=\:x\,-\,[x]\)

Consider what happens on some interval, say, \(\displaystyle [2,\,3]\)

Find the \(\displaystyle y\)-value for various values of \(\displaystyle x\).

\(\displaystyle \begin{Bmatrix}x\,=\,2.0 & \Rightarrow & y \:=\:2.0\,-\,[2.0] \:=\:2.0\,-\,2\:=\:0.0 \\
x\,=\,2.1 & \;\Rightarrow\; & y \:=\:2.1\,-\,[2.1] \;=\;2.1\,-\,2\:=\:0.1 \\
x\,=\,2.5 & \;\Rightarrow\; & y\:=\:2.5\,-\,[2.5]\:=\:2.5\,-\,2\:=\:0.5 \\
x\,=\,2.9 & \;\Rightarrow\; & y\:=\:2.9\,-\,[2.9] \:=\:2.9\,-\,2\:=\:0.9 \\
x\,=\,3.0 & \;\Rightarrow\; & y \:=\:3.0\,-\,[3.0]\:=\:3.0\,-\,3\:=\:0.0
\end{Bmatrix}\)

The graph looks like this:

                            o
                          *
                        *
                      *
                    *
      --+---------*---------*---
        1         2         3

Since this happens over every unit interval,
. . the graph looks like this:

                |
                o         o         o         o
              * |       *         *         *
            *   |     *         *         *
          *     |   *         *         *
        *       | *         *         *
    - * - - - - * - - - - * - - - - * - - - - - -
     -1         0         1         2         3

 

[katieighty]

Thank you both for your help. Greatly appreciated.