Friday, February 29, 2008

Odd and Even Functions

Hello, fellow classmates. My name is Paul, and I will be your scribe for today, and I will be writing about "odd" and "even" functions in an overview of Friday's class.

So our class today started with a review of yesterday's class (slide 2), which basically stated that:

In the formula y = af[b(x-c)]+d:
- If a > 1, the graph is stretched vertically
- If 0 < |a| <> 1, the graph is compressed horizontally
- The y-coordinates of f are multiplied by a
- If b > 1, the graph is compressed horizontally ("speeds up")
- If 0 < |b| < 1, the graph is stretched horizontally ("slows down")
- The x-coordinates of f are multiplied by (1/b)

Following the review, we spent some time practicing graphing functions, which can be seen on slide 3.

Stretches before translations!

Here's a quick rundown of the formula and what each variable does:

y = af[b(x-c)]+d
a -> Vertical stretch/compression
d -> Vertical shift
b -> Horizontal stretch/compression
c -> Horizontal shift

Note: Remember that you have to stretch the graph first, but it doesn't have to be both stretches first (abcd/abdc)! You can stretch the x axis and then shift it before stretching the y axis(adbc), the important part is that the stretch for each axis must happen before the shift for its respective axis. This was discussed on slide 5, where we wrote the possible orders you can apply transformations.

After the practice with drawing graphs, we were introduced to images, which is the graph after it has been shifted/stretched (slides 4 and 6).

Also, when talking about an image, the "formula" is:

(image) is the image of (original coordinates) under (function).

So an example would be:

(0,7) is the image of (-2,-2) under the function y=-3f[1/2(x-4)]+1

We then proceeded to learn all about the online bookmark service and website
I won't really go into detail about this, since it's pretty self-explanatory. However in the interest of thoroughness, I've made a quick image here:

And with that, we proceeded to our next topic of reflections (slide 7):

- Basically, a vertical reflection is a reflection along the x-axis which occurs when the y-coordinates of any function f(x) are multiplied by (-1).
- Conversely, a horizontal reflection is a reflection along the y-axis which occurs when the x-coordinates of any function f(x) are multiplied by (-1).

Slide 8 contains a picture of a reflected sine wave, neato.

Inverses are also introduced on slide 7:

As stated by the slide, the inverse of f(x) is f^-1(x) [f to the power (-1) then/applied to (x)]. Which looks like this (image follows):

Thank you, Paint, for that excellent demonstration.


The truth is that
(can you handle this?)... [f(x)]^-1 (note how the power is on the outside), is the one that is equal to [1/f(x)].

So in summary:

[f(x)]^-1 = [1/f(x)] √ Correct
f^-1(x) = [1/f(x)] X Wrong

Its also pronounced "eff inverse." And finally, f^-1(x) undoes what f(x) does.

Lastly, we started talking about "odd" and "even" functions (slides 10 and 9 respectively):

A function is "even" if it is symmetrical about the y-axis, which occurs only when f(x) = f(-x). If you look, a cosine curve is a nice example of an "even" function.

A function is "odd" if it is symmetrical about the origin, which occurs only when f(-x) = -f(x) (the negative of the whole function). If you look, a sine curve is a nice example of an "odd" function.

And that concludes my scribe post summarizing the topics we covered during our class last Friday. Now about the scribe...

Since some people actually want to be scribe (unlike myself who was hoping to be scribe for pi day), I hereby declare that benofschool has the honour of being the next scribe for Monday's class.

And don't forget to get a Delicious (mmm, delicious) account if you haven't already! Here's a direct link for you who are too lazy to type it (shame on you), or like convenient things.

So I bid you adieu and offer my sincerest apologies for the lateness of my post. Unfortunately, I had work Sunday and was busy the majority of Saturday. I did infact start this post on Friday, but did not manage to finish it until today. I am highly aware of the time I am posting this at, especially considering I had resolved to post it first thing on Friday.

Good night, and I'll see you tomorrow.

P.S. I would like to offer the following advice to the following scribes:

Blogger's draft system is not without its flaws, and I have had to rewrite the second paragraph (involving the overview of the a and b variables) numerous times because the draft system confuses my use of the arrow signs (<>) as parts of html tags and deletes everything inbetween. Thus, if you think you're not going to finish your post, just save yourself some hassle and put it in a notepad document instead of using the draft system.

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