Tuesday, March 4, 2008

Reciprocals; the Mathematical Flip

Well hello again, it's me Justus, being that thing called scribe again (thanks to Benofschool >_<;). Anyways lets get this show on the road, as I haven't got much time until my basketball practice -_-; *Note* Sorry this was put out so late guys, I tried to get it done before I had to leave for my practice, but I ended up falling just short and having to finish it, well, now, once I got home. So yeah, I apologize for that. >.<; So onward! Mr.K started off the period by finishing up and reviewing the inverse function work/question we had from yesterdays class. That question can be seen on the first slide (not including the cover slide with the skater on the front.) To solve these questions (as seen on the slide) we began by graphing the original function, f(x) = √(x+9) -2. Now because some people weren't quite sure what the graph of √x looked like. Mr. K went over it and some of its properties with us. First he mentioned that the graph of √x looks like a parabola rotated -pi/2 radians (or flipped on its side, opening to the right). He also mentioned that the graph of √x is split into two parts, the top half (also known as the principle function) and the bottom half. This division of the graph occurs because of the fact that a square root may be positive or negative. The bottom half of the graph comes from the possible - value of the square root. Thus, the graph on the slide, only looks like half the parabola. After going over that, and having Francis write what he got as the Inverse function on the Smart Board, Mr.K quickly went over the rest of the answers for the questions.

Things to Remember from this opening rant/the opening minutes of class
- The graph of √x looks like a parabola on it's side, opening left, minus the bottom half
- -b/2a is the x coordinate of the vertex
- The X intercept of the inverse functions graph, equals the Y intercept of the original functions graph.
- The Inverse graph should reflect along the line y=x
- And finally, Inverse undoes what the original function does AND THAT'S ALL (hence why on the slide (#3), there is a piece of the inverse graph is marked out in green.)

SO. After finishing that kinda longish review of yesterdays opening mind bender we started today's lesson, involving reciprocals. Our opening slide had two sets of numbers, the first going 1, 2, 4, 10, 100, 1 000, 1 000 000, and the second going, 1, 0.5, 0.25, 0.1, 0.01, 0.001, 0.000 001. The text at the top of this slide, read, "Find the reciprocal of each of these numbers. If it is a decimal number, convert it to a fraction first. Now we all know from previous mathematics courses, that the reciprocal of any number, is simply the fractional form of that number, "flipped." That is to say, the numerator and denominator have switched places so that the denominator is now the numerator and vice versa. (ex. for the fraction, 3/8 the reciprocal would be, 8/3. The reciprocal of the number 5, would be 1/5.) By now we had realized that the relationship between to two sets of numbers was obvious, they were in fact reciprocals of each other. However this wasn't the point, the point my friends, I am about to reveal to you, so pay attention. The main point of writing those numbers, and finding their reciprocals (as seen on the slide), was so that we'd understand the following.

As the original numbers get larger, their RECIPROCALS get smaller.

OR as the reciprocal of a number gets larger, the original value of said number gets smaller.

(*note* the terms biggering and smallering are suitable for use in these statements, as replacements for the phrases, "getting bigger, and getting smaller")

In case you were wondering about negative numbers fear not, Mr. K had thought of that too, and had coined a term specifically for this. In the instance of negative values (-1, -2, -4, -10, etc...) their reciprocal values are said to be Biggering Negatively (notice the reciprocal values here are -1, -1/2, -1/4, -1/10, which are all larger then their reciprocal cousins, as they are closer to/moving towards "0"). It should also be mentioned that the reverse case is also true, should the values be moving towards 0, their reciprocals would be, "Smallering Negatively" as seen in the following case (-1, -1/2, -1/4, -1/10 reciprocal values are, -1, -2, -4, -10 respectively)

Mr. K said that if you understood these concepts, then you understood the lesson for today.

So equipped with our new found knowledge we set off to graph these things called reciprocals. To do this we started off as follows.

"Taking the graph of x+2, graph 1/f(x)"

At first, these appeared to be a daunting task, I mean, telling someone the reciprocal was one thing, but graphing it? A whole 'nother matter indeed. However Mr. K once again came to our rescue and showed us a couple of steps to aid us in our reciprocal graphing woes.

Step.1) Graph the original function.
- Now this may seem like kind of a no-brainer, but in the second example we received we saw that we would not always be giving the original function first, and would have to figure out what it was first, THEN graph it, THEN move on to the next step.

Step.2) Find every point on the original graph where y= ±1.
- This step is necessary because as we found in our numbers (shown above starting with 1 and ending with 1 000 000) the reciprocal of 1 is always 1, and the reciprocal of -1 is always -1. THUS, these points will always be on the reciprocal graphs. *note* These points are called Invariant Points

*Vocabulary Skill + 5!*

Step 3.) Look at where your original graph has roots.
-This step is necessary because where ever your original graph has roots, the reciprocal graph will have an asymptote, due to the fact that the reciprocal of 0 is undefined.

Step 4.) Now the final piece of the puzzle, putting it all together and drawing/sketching out the reciprocal graph. To do this, we must simply follow what we found out before about reciprocals. When the original value is biggering, the reciprocal value is smallering. Keeping this in mind, we may begin construction of the line. To do this, look at the line of the graph, and see which direction it's going. For example, looking at the graph of X+2, and it's reciprocal, 1/X+2

Click here for the graph!

So looking at the graph you should notice the two green dots. Those are out invariant points, and it is from these that we will be working. Since it's normally a good idea to start with the extremes and work from there, that's what we'll be doing today.

*note* when doing these, you'll determine the graph direction by going towards the asymptotes. This may make more sense in a moment.

So starting at the point (-1,1) and working towards the asymptote, we find the graph smallering. Thus by our "mantra" so to speak, we know that the reciprocal graph line, must be biggering. Thus we draw our first point. Next we go to the other extreme, at the point (-3, -1). here, the graphs behaviour towards the asymptote is to smaller negatively (which is technically biggering.) Thus, the reciprocal line graph must bigger negatively (which is thus smallering.) Finally we are left to do the final lines, those not seen on the extremes, the ones curving up vertically, near the asymptote. Much like the previous ones, the solution lies in the biggering and smallering, using the invariant points as our start point. Looking at the graph from the green point and going towards the asymptote, we see that it is still smallering, and thus the reciprocal line much be biggering, this time, up the y axis. The same is true for the other point, except it is biggering negatively, as it is in "quad 3" and thus negative in value.

I think this for the most part concludes this blog entry. There is one more thing though, as we put together another graph in our examples.

Click Here for the Second Graph!

Plotting the reciprocal graph for this function was the same as the other, except there were two asymptotes instead of one. Following the steps you may eventually come to a point where your wondering what to do with the vertex of this graph. Because it is a y intercept, you can figure out where it will be on the y-axis in the reciprocal graph (in this case the y intercept is at -4, so the reciprocal point is at -1/4) Since this is a smooth graph, all that remains to be done is join the point at (0, -1/4) to the rest of the graph. In the image above, this step is colored in orange (sorry if it's hard to see). So yes, the part everyone usually scrolls down the whole thing to see, whose the next scribe -_-;

Alright, I believe that really and truly raps it up for today.Again I'm sorry the blog came out late, and maybe slightly all over the place. Hopefully its understandable, cause this is basically how I interpreted the lesson, and I've been told my way of doing things is a little bit weird at times :p If anyone has any questions about my post, or needs me to clarify something, feel free to post it. Also if anyone has anything to add, or knows I forgot something, feel free to tell me (and PLEASE DO TELL ME) so that I can fix it, or add it in or whatever. Also on the flip side if you feel so inclined you can add it in yourself :p

Well since I'm too tired to make a big huge crazy skill testing question to decipher to find out who the next scribe is, I'll probably just come out and tell you it's Richard. Kinda like how I just did...

OH also, homework was exercise #10, for anyone who missed that.

One more thing, don't forget about Mr.K lying during class :p It seems to me we've forgotten about it already! Lets be sure to catch him in the act tomorrow eh? :D

Finally, the word you've all been waiting for.

REMEMBER! When A number is biggering, it's reciprocal is smallering!


Anonymous said...

Hi Justus!

I'm thinking this scribe is a tremendous help to you and your classmates-- good explanations, color to stress biggering and smallering, graphs to illustrate, and a closing reminder of the main concept! Thanks too for including the word of the day!

Is catching Mr. K lying a challenge to all of you that involves math specifically or his comments in general?

Great scribe,

Skyline said...

I'm pretty sure its Mr. K's comments in general, although many of them so far have been math related.

Ex, when doing phase shifts of trig functions he said, "Because the Value of C is -4, the graph moves 4 units to the right." when it SHOULD have been, 4 units to the left.

Thanks for the comments Lani, I was worried I hadn't included enough with my scribe post ^-^;