Bobbing on Shwards and Things

So, the infamous conic sections unit was completed late last week, and this would be a bob for said unit.

Where to begin? the good? the bad? the ugly? Lets start with the ugly

THE UGLY *dun dun dun*

-GRAPHS! Yes I'm terrible with graphs but I am getting better at them, thanks to a certain "perspective." More on that later though.

The Bad

Not to much here this time, thankfully.

-One of the most difficult parts of this unit is how most of the equations, have another equation which is nearly identical. This made memorizing/remembering which was which difficult at times.
-Going from standard to general form, and remembering which is which was often a pain.

The Good

-Short unit
-Easily understood by used of "visual" tricks (what I meant in the ugly section. By thinking of what the graph of some equation looks like, the question is often made much easier.
-Overall the concepts in this unit, and the difficultly level of the questions was not very demanding.

So yeah, there is it, my small and not very impressive bob post. :P Now I'm back to studying haha.

Ciao!

Hyperbolae Unveiled...Sorta

Alright guys, no shenanigans this time, I'm tired, just got off work, and would like to sleep. So I apologize if I'm a little bit frank, and a little less funny an animated in my scribe post compared to my usual ones. Hopefully the quality is still there though, if not, I'll fix it up tomorrow or w/e. Alrighty, here we go.

*note* links added under appropriate images for full sized goodness. Like I mean, it wont matter on the smartboard, since that things huge, but at home I doubt everyone has like 70 inch moniters, so ja. Thats why there there. Why arent they hotlinked or w/e its called? Cause its 1:24am haha.

Morning Class

So this mornings class wasn't much of anything special, which is kinda different from the norm (kinda paradoxical I know.) Anyways we started going through Mr. K's slides, and we solved the little equation for one of the friendly Ellipses which can be seen on the following slide.

As always, fully more large sized versions of these slides can be found in the slides themselves. Which Mr.K posts quite consistently :P

Moving on. After that we filled out a health survey. That basically took up the rest of the morning class. However, we did have a moment to do a quiz, which was marked the following afternoon class. If you were not there *cough cough* then talk to Mr.K, see if you can write it. The solutions are on the slides. If you were here, then you don't really need an explanation right? Cool.

Moving on.

Afternoon Class


The afternoon class was slightly more work oriented, and was not interrupted by some unnecessary forms/paperwork.

We began by correcting the quiz thinger from the first period.

Some usefull things to take out of that include the following.

How to distinguish Various conics from their equations.


-If the Equation has an x² term but no y² term it is a Horizontal Parabola
-If the Equation has an y² term but no x² term it is a Vertical Parabola
-If the Equation has an y² AND an x² term, with coefficients that are THE SAME, it is a Circle.
-If the Equation has an y² AND an x² term, with coefficients that are DIFFERENT, it is an Ellipse.

Remember those :)

Next we took out our previous nights homework (that being measuring, and finding the difference, of the lengths from one point on a branch of the hyperbola, to both foci. That will make more sense when the image comes up in a moment.)

The results of taking up that homework are shown as follows.

Okay, I'm going to try to explain this as best as I can, but like I said before, I just got off work, I'm crazy tired, not running on anywhere near enough sleep and a host of other not so great things. So if I phail (yes thats a "ph" fail. That doesn't mean an acidic fail but rather, means a really bad fail for all you en zero zero be's out there.)

So, the homework was to pick a point on one of the branches of the hyperbola, and measure that distance from the point, to both foci (that is to say, measure from say, point P, to F₁, and then from point P to F₂). We then too those values, and found the difference, (PF₁-PF₂). Voila! homework done. By now it should have come as no surprise that the result were values which should have been basically the same.

Once we did that, things started to get tricky. The next thing we did, was find the Vertices (sp?) of the hyperbola. To do this, you take your ruler, and line it up so it goes through both focus points. The POINTS at which the line made by the ruler intersects with the curve of the hyperbola make up it's vertices.

Okay, so if you followed my last paragraph's directions correctly, you should end up with two points at what are the vertices of the hyperbola. The next step is to connect those two points with your handy dandy ultra straight ruler. This creates what we called the TRANSVERSE AXIS. This axis can be described as the line which connects the two vertices of a hyperbola, and is seen AND labeled in green on the slide crop up there. Got it? I hope so. Moving along.

Next, we bisected this transverse axis.

"How do we do that Mr.k?"
"Well I'll show you."

Thats a rough translation of what probably everyone was thinking. Except the Ogre 1 and 2 of our class. They knew the answer already, (Outside joke.)

So to bisect the transverse axis, we folded our papers for one more time. This time, we folded one focus point, directly on top of the other. Thus we obtained a crease which defined not only the bisector, but the perpendicular Bisector of the Transverse Axis. We see this as a vertical red line on the image above. Lets step things up a notch (I know Mr. K did.)

So, do you remember the Transverse axis? and how we just cut it in two? Were going to take this line, and modify it a little bit. Using our rulers again, we measured the line from one of the FOCI (in this case F₂) to that vertical line (which we now call the CONJUGATE AXIS.) This gave us some length.

Now this next bit is tricky, we took the length we measured from the PREVIOUS STEP (that being from the conjugate axis, to Focal point F₂) and moved it so that one end of it was on the (whatever the singular of vertices is. Verticii?) of the nearest branch, and so that the other end of the length lies on the Conjugate axis. Since I bet your saying "what the frig" I made a quick little paint mockup to get the point across, or rather, to help you "see" what I mean.


http://i24.photobucket.com/albums/c21/skyL193/Slide3.jpg

Okay, so as I wrote on the picture, the light green is the length Conjugate Axis to F₂. Then by moving it like I mentioned before, we get the makings of a triangle.

Step 3. Measure the distance from the point where the light green line meets the conjugate axis, to where the dark green axis meets the conjugate axis. This we can call length b for explanations sake. The next step is to take this length, and go from the intersection of the Transverse axis and the conjugate axis, (aka the cross in the centre) and "b" units down. I'll show you what I mean again.


http://i24.photobucket.com/albums/c21/skyL193/Slide4.jpg

So, as I wrote again, the light blue, and dark blue lines are the ones I was talking about before, and they are of equal length. Thus we get those points G1 and G2. This brings us to our next step.

By drawing perfectly 90 degree lines from the vertices (going through them, vertically) and from the end points of the light and dark blue lines (going through points G1 and G2, horizontally) we can construct a box of sorts. This is seen once again on the slides.


http://i24.photobucket.com/albums/c21/skyL193/Slide5.jpg

So you see the box right? The black dotted lines there are actually it :P So yeah after that we took our ruler again, and drew that orange cross that goes far off the page kinda. We drew that cross by going from one corner of the box to the other (and beyond.) After we had done this Mr.K revealed to us that those orange lines, are actually asymptotes for the graph. In other words, neither branch of the hyperbola will actually ever touch those orange crossey lines.

The final slide here, is the equation of the hyperbola, along with some similarities, and differences, which we came up with together, to help us remember the equation, and keep it distinct from the others.


http://i24.photobucket.com/albums/c21/skyL193/FinalSlide.jpg

To Conclude: The basic parts within the anatomy of a hyperbola
- Transverse Axis.
- Conjugate Axis
- Focus Points
- The Asymptotes
- And finally, you should know the equation, which is shown in the above slide image.

Alrighty, I think that about wraps this blog post up. I'm not entirely sure, because at 1:00am I kinda start to get a bit loopy and my thoughts aren't so coherent (thats probably evident based on how my writing progressed.) If you guys have any questions or whatever lemme know. I'll try to answer them, and post em up. Or if you find any errors, or know something I dont/forgot to add, lemme know so I can add it.

Well I'm off, but of course, not without letting you know who the next scribe poster is :)


http://i24.photobucket.com/albums/c21/skyL193/JamieSlide.jpg

Remember, vote for Zeph for VP.

K? Kthxbai

Justus- FINALLY out.

Bob(ing for Combinatorics...instead of apples)

Haha I just realized I make up my bob titles as I decide to write the bob itself. Thus the title is basically just a direct brain to keyboard link of the first thing that pops into my head :]

Anyways to the point, the unit combinatorics (also known as counting, for those who missed that little piece of unit defining info.)

For starters I'd like to say that I thoroughly enjoyed this unit, a first for any of our units to date. Now that isn't to say that Mr.K didn't teach the other units adequately, I just had to really work at those ones to even get semi decent at them. This unit on the other hand, came kinda easy to me (some of it anyways) and thus, I enjoyed it!

Some of the things I liked about the unit were...
- The practicality of it. This unit gave me alot of knowledge that I can *gasp* actually use in REAL LIFE!!!!11!111!!!11!1!
- The fact that it wasn't really concept, or practice, or anything extensive, but seemed to have just the right amount of everything in it (unlike some other units *cough*logarithmsandUNFUNexponents*cough*)
- All the clever little things we learned along the way, that were totally unrelated to the unit itself. (We should have more of these)
- All the clever little things we learned along the way, that were actually related to the unit itself (and these also)
- The whole unit? Almost

On the other hand (or the second option here, which would be 1 choose 1, because I already went over the first topic.)

I didn't so much enjoy
- The formulas. Overall I found them bulky, and annoying, and hindering to my understanding of the unit as a whole. They just made everything more complicated I think.
- The algebra involving the whatever choose something else, and or the whatever pick something else. The whole factorial deal kinda gets me sometimes (errors with my algebra cause of the ! in the middle of it. Makes no sense I know.)
-The fact I left this bob till so late (I was horseback riding for my sisters birthday party. There goes half my day D:)

So ja, overall this unit gets 8 out of 10 justus league points, not 10, because I didn't much enjoy some of the algebra involved in it but meh, its pre-cal, OBVIOUSLY theres going to be algebra, so I'll just get over it then.

I think thats everything, so I'm going to study for awhile, then sleep, then hopefully mega super power up overshield bxrrxyy final boss strongside ogre 2 wavedash fox shine infinite aleph not pwnzor the test tomorrow. I wish you all the same amount of goodness on your tests :]

Goodnight! and Ciao!

Justus - out, and studying now.

Edit: ohmg! I just remembered I'm missing like a bajillion delicious box links. I'm super sorry guys I forgot all about that thing until I scrolled down and saw it D: I promise to go and find a bunch of super kewl links tomorrow, like lots of them, cause I owe you guys -_-; Hopefully you can find it in your textbooks (or TI-83's) to forgive me. :]

Okay, now I'm really gone :p

Intro to Combinatorics

So, the long awaited blog post on combinatorics from last Friday. At least its not 12 at night right? Haha. Anyways, here we gooo!

During the first period class on Friday we had our Logarithms and Exponents test. It wasn't fun. If you werent there for the test It would probably be a good idea to talk to Mr. K about a time in which you can re-write the test. They are worth marks you know :P

The afternoon class introduced us to this wonderful thing called combinatorics. Now Mr.K told us that this is basically a branch of math that involves counting. When prompted as to why it was called combinatorics rather then something like, countinatorics or something he gave us a sample problem.

Given 5 students and 5 chairs, how many different ways can those students be seated in those chairs? Now it is important to note that the question is HOW MANY different ways, and now WHAT ARE the ways. When asked what are the ways, we are prompted to list all possible combinations, which is a long and tedious (although not necessarily difficult) task. In the following image, we used a tree diagram, to go through all the possible combinations or students (named a,b,c,d and e).

(*note* this image can also be found in the slides, in case this is too small to read.)

Although we found the answer to our question of how many but listing out what all the combinations were, there is an easier way to do this, thanks to that handy dandy thing called a calculator. By hitting 5x4x3x2x1 (five times four times three times two times one) on our calculators we can find the total number of calculations without drawing out every single one. We did this by looking at the possibilities we had left. So if theres 5 open seats, and 5 students, we have a 5 choices. When we pick one, we're left with 4 students to choose for the 4 seats. When we pick another student theres 3 spots left, and so on. Thus you multiply 5 x 4 x 3 x 2 x 1 (1 because once you've used 4 of the 5, there aren't really any choices left.)

So Lets recap.

So Far...
-Unit is Called Combinatorics
-Tree Diagrams help
-Looking at your options and choices allows you to multiply to find HOW MANY different ways.
-HOW MANY, and WHAT ARE are two totally different things.
-Theres a difference between Long and Tedious, and Hard/Difficult.

Now Moving onward!

the next problem presented a twist, what happens when there are more then 1 option for each choice? An example of this was in the next problem.

HOW MANY different ways can a nickel, dime, and quarter land on a table?

So to solve this we have to look at the options available to us (or use a tree diagram again :] ).


To solve this we looked at our options. For any 1 coin there are 2 possible outcomes, either heads or tails. now because order doesn't matter in this case, we set up our tree like this. you read the tree, by following its branches. For every flip of the nickel that lands heads or tails, the dime can land either heads or tails, and for every flip of the dime, the quarter can land either heads or tails. This the total number of combinations is, 2 x 2 x 2 = 8 (as shown in the red there.)

Now the next problem changed one of the coins to a die, thus adding a bit of a wrench to the system. BUT since we're smarter then the average bears, we figured out quite quickly that the total number would be 2 x 2 x 6 = 24 (because the die has 6 sides...normally.)

After having gallivanted through a few problems like this Mr. K unveiled the pattern behind this specific branch of math.

Fundamental Principal of Counting
-If you have "M" number of ways to do one thing, and "N" number of things to do another thing, then there are M times N number of ways to do both things.

Above is the simplified version of the Principal of Counting. Basically if you can do one thing x number of ways and another y number of ways, then you can do both of them at the same time, x times y number of ways. (haha I basically just repeated it :P)

Shortly after we were introduced to this we were introduced to Factorial Notation

Factorial Notation
n! = n · (n-1)(n-2)(n-3)...3·2·1


Basically what that means is that (correct me if I'm wrong here ^_^;) is n factorial equals n times, n minus 1, times n minus 2, times n minus 3, etc etc until you get down to, 3 times 2 times 1. Now because not everyone wants to input that into their calculator, they have an EVEN EASIER WAY!

By hitting, [n][math][<][4] you can get the factorial of whatever you want. This input reads in normal terms, means, hit your n value (whatever it is, 4, 7, 234872398, 0.0000000000002) followed by math, the arrow left key, and then 4. You should end up with your n value followed by a ! symbol (which means factorial, not I really mean the letter n)

Now we come to the final little bit of this blog, using the slot method, and irregular combinations.

In the first problem illustrated on this slide, the question reads, "How Many "words" of 4 different letters can be made from the letters A,E,I,O,R,S,T?" On the slide you can see that we set up our slots. Because there is a 4 letter limit, we only used 4. With 7 letters, we simply filled in the slots as we went down the line, multiplying as we went. The result was 7 x 6 x 5 x 4 = 840 words.

The next question gives even more restrictions, asking how many of the words begin with a vowel, and end with a consonant. Yet again we set up our slots, with 4 places. This time though, we set up our restrictions first. Because there are only 4 vowels, and 3 consonants, we fill those in first (as shown in the diagram.) Then we take the remaining letters and fill in the two slots in the middles (5 and 4 respectively, because we already used two with the first and last letter).

Finally in the last question, all the stops are pulled, and we are required to use two separate "slot" mechanisms, to solve it. Using all the same methods as before we came up with 144 as our answer. Thus we completed the class and our introduction to Combinatorics and I've just about finished my scribe post.

To conclude
-Using Tree diagrams helps, but is sometimes really long
-Factorial Notation saves the day (on your calculator [n][math][<][4])
-Fundamental Principal of Counting is if there are M ways for one thing, and N ways for another, there are MN ways for both.
-Using a slot system like those seen in the slides can greatly help
-When there are restrictions on a problem, solve the restrictions FIRST, then everything else.

Alrighty, I think that about sums it up. You know the deal if somethings wrong, tell me, or edit it or whatever, or if you don't get something ask me and I'll try to put it a different way, and edit the post on here.

So the next scribe post is....

*checks scribe list*

Francis....

If francis already did it

Lawrence. Okay? so lemme make sure yous all gots it.

I chose francis, but cause the scribe list seems a bit behind maybe, IN THE EVENT THAT FRANCIS ALREADY WENT. Lawrence. So we dont have to do the whole shenanigans in class thing. k? Alrighty, I'm going to skate :]

Ciao!

Justus out.

BOB version whatever we're at -_-;

Okay well I almost forgot about this. I had to work tonight and was studying and doing crazy amounts of english and blah blah and when I finished I was just ready for bed. Then I remembered the bob for this unit of ours, logarithms and exponents. So because I'm tired, lets get this show on the road.

Pros (or other assorted things I found somewhat easy):

-I found this unit was helped along quite well with the phrase, "A Logarithm is an exponent." Many times when I found myself stuck on a question, I'd turn to this, and usually a eureka moment was soon to follow.

-Much of this unit was algebra, so as long as you had that locked down, almost half the work was done for you.

Cons (or other assorted things I found somewhat difficult):

-For me, one of the hardest parts of this unit was grasping the concepts in it. Usually when I learn something new, I take what I'm taught, and translate it in my head, and change it into a form I can remember. Now for the most part this process goes off without a hitch, but with this unit I found parts of it...trying. For example, I knew the change of base law worked, and I kinda knew where and when to use it, but I had nooo idea HOW it worked, until recently, when I got that cleared up with Mr. K

-Alot of the time I found it downright confusing. Like much of the unit made sense, but seeing the logs and ln's and such all over the place was sometimes disorienting, and a bit smothering. I had to learn to break it down, and take those problems one step at a time, or else I'd get overwhelmed.

Overall:

Not my favourite unit. Not by a long shot. It was interesting yes, and as always Mr. K's teaching methods were inventive and got the point across, and when I needed help he clarified things quite well. However I am almost looking forward to this test tomorrow, because that means they're finished, done, finito; until exam time. I struggled the most with this unit out of all of them by far.

Anyways I'm going to sleep haha. z_z

Justus out,

Ciao.

BOB the Blogger is Back! (Identities Unit)

Its that time of the unit again, where we must do our BOBs!. thus I commence with mine own BOB for Identities.

So to begin I'd like to say that this unit was easily the most up and down one for me. I'll explain what that means exactly right now. With most of the other units, it was either steadily easy, or steadily difficult. However, with the identities unit, I found that it was all over the place. Some of it I got extremely quickly, and other parts of it I struggled profoundly with.

The easiest parts I think, involved the use of the sum and difference identities. Substituting the values of sine or cosine, or substituting the identity itself seemed relatively straightforward to me.

One of the more difficult parts, involved solving identities in general, as I'd usually get stuck in the paralyzed frozen mode Mr. K was talking about because I couldnt see the end of the problem, and thus, didnt want to advance into an uncertainty. I am working on forcing myself to continue and try new things though to get rid of that bad habit :P

One of the things I found most helpfull was the sine dance. Expect to see me doing it during the test :D

Well I think thats all, so Ciao!

Proofs; The Sum and Difference Identities.

Alright, lets get this show on the road as I'm now the replacement scribe for today. First things first.

PAUL IS THE SCRIBE FOR TOMORROW (March 25th, 2008)

Anyways now that thats dealt with lets begin.

We started class off with a quiz, which for many of us (or at least I think it was many of us) had a bit of trouble getting into. One of the main points to remember in this little intro portion from the quiz was to look at difference of squares, and look out for one of the factors of them. Many of the questions were solved by algebraically massaging the expression using a difference of squares (where applicable.) Anyways the answers to the quiz(and in most cases a step by step guide of how to get to those answers) can be found;

Here

Things to Remember
-Look for a difference of squares, or a factor of a difference of squares
-Remember that you dont need to memorize the corollaries for the trig identities, but rather, derive them yourself when you need them from the Pythagorean theorem. Sine^2 + Cosine ^ 2 = 1.

*Note* In case the images are to small, they're all on the slides, so simply jet on over to the link above and hunt em down :)

So with that little opening bit of the lesson out of the way, lets continue with the main content of this scribe post, The proofs of the Sum and Difference Identities.

If you remember from the previous class on Thursday, we were right on the precipice of uncovering the, "clever idea" behind these identities. If you recall, we worked with the angles ROP, and QOP. Now assuming the angle was in a unit circle, we were able to find the coordinates of points Q, P, and R in terms of Sine and Cosine. After that we implemented the first portion of our clever idea, a rigid transformation of the overall angle, which put it on the x and y axis perfectly. The finding of the angles all lie in the fact that angle QOP, is equivalent to angle alpha subtract angle Beta. Thus when the whole thing is rotated, the angle QOP on the x axis may be described as Alpha - Beta.


Now that our recap is completed lets get into the new stuff! By looking at the above you may see an orange line. This represents the distance between Q and P. This here, is the bread and butter of the clever idea for today. So with the rigid transformation from the first to second diagram, the distance from Q to P must have remained the same (hence rigid transformation.) It is from this point the real magic can be worked. Next we used the Distance formula √[(X2-X1)^2 + (Y2-Y1)^2] (which is the Pythagorean theorem by the way.) and plugged in the values we found before. Thus we got everything seen on the following slide.


Once the Sine and Cosine values have been plugged in you begin to simplify it all down until you have the cosine difference identity. Nice eh? Now because that only covered the difference identity, we have to find the sum identity. To advance here however, we must go back, to Odd and Even functions. If your memory serves you correctly you should remember that Cosine is an Even function and Tangent, and Sine are both Even functions, that is to say that Cosine (x) = Cosine (-x), Tan(-x) = -Tangent(x), and Sine(-x) = -Sine(x). So keeping all that in mind, and remembering we have already proven one identity, we can now prove the cosine sum identity As follows

Cos(α+β) = Cos (α-[-β]) <---------- Clever Idea!
|Cos (α) Cos (-β) + Sin (α) Sin (-β)
|Cos (α) Cos (β) + Sin (α) Sin (β) *
Cos(α+β) = | Cos (α) Cos(β) + Sin (α) Sin (β)


By substituting in the even or odd identity for Sine and Cosine we were able to come up with the proof for the Sum identities.

Now I think that wraps up just about everything we did in this class. Today.
Remember that homework for today is to (using the above) come up with the sum and difference proofs for Sine. To do this use what we learned today, on R'Q' = RQ rather then Q'P' = QP. Also we have Exercise the Next, which is known more specifically as Exercise 17.

So ja, thats it. Hopefully it made sense. In case you missed it up top, paul is scribe for tomorrow. Now I must go build a catapult for physics! -_-;

Nothing Less Than The Best....Group

SLIDE 18

On a typical day at an ocean port, the water has a maximum depth of 20 m at 8:00 a.m. The minimum depth of 8 m occurs 6.2 hours later. Assume that the relation between the depth of the water and time is a sinusoidal function.

Let's draw a graph!


a) What is the period of the function?

From the information we have been given...
* We can set the 8am as t = 0 hours.
* A maximum value is when t = 0 hours and when d = 20.
* A minimum value is when t = 6.2 and d = 8.

We can see from the graph that the period is 12.4 hours.

b) Write an equation for the depth of the water at any time, t hours.

cosine equation's parameters...
A = 6
B = (2pi) / 12.4 = pi/6.2
C = 0
D = 14

To get A, amplitude, calculate the distance from the sinusoidal axis to a maximum value or minimum value.
sinusoidal axis = (20+8)/2 = 14
amplitude = 14-8 = 6

B = pi/period = pi/6.2

C, the phase shift, is 0.

D is the sinusoidal axis, 14.

D(t) = 6cos [(pi/6.2)t] + 14


c) Determine the depth of the water at 10:00 a.m.

10:00 am = 2 hrs from when t = 0 or 8:00 am. Plug in the 2 as t into the equation to get the answer.

D(2) = 6 cos [(pi/6.4)2] + 14 = 17.1738 metres

d) Determine one time when the water is 10 m deep.

The wave is 10 metres deep, so the qestion is asking for what the time is when D = 10. Plug in 10 as D, then solve for t.

10 = 6cos[(pi/6.2)t] +14
-4 = 6cos[(pi/6.2)t]
-4/6 = cos[(pi/6.2)t]
arc cos(-4/6) = (pi/6.2)t
2.3005 = (pi/6.2)t
2.3005/(pi/6.2) = t
2.3005 x 6.2/pi = t
14.2632/pi = t
t = 4.5401

Convert the 4.5401 into "actual time" because we use hours:minutes:seconds to show time, so...

4.5401 hrs + 8am = 12.5401 hrs

Obviously, its not efficient to say .5401 hrs so we convert that to minutes.

0.5401 x 60 = 32.406 min.

12:32:24pm

We can round that to 12:30pm.

BOB Loves Transform(ers)ations!

Well I'm back, this time for the periodic posting of the BOB's (who is that guy anyways?)

SO, transformations. For the most part I found this unit fairly straightforward and easy, mostly due to the fact that I remembered a lot of the steps to solving functions that had been transformed from last year. I thought many of the methods Mr. K taught worked well, and were thus, effective.

One of the things I did have a bit of a problem with initially was dealing with with graphing reciprocal graphs but I figured a lot of it out when I had to do the scribe post for that specific lesson :P

The only thing I'm actually still trying to get a better grasp of is, solving algebraically, inverse functions (eg. given f(x) = x+9 what is the arc f(x)?) I haven't quite got the whole process involved in changing a function to its inverse in an algebraic form. Graphically I don't have any issues but when its just the equation involved I need some help (so if anyone has any tips they use that they could tell me that'd be great :D)

Well I think that about sums it up. Transformations seemed to be kinda a down shift for me, from our previous unit, which I struggled with grasping in the beginning. I'm doing well now that I've finally gotten used to having math again, and am feeling pretty confident going into the pre-test tomorrow :)

Anyways I've got English homework now, so I'm off!
Ciao!

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.
http://www.merriam-webster.com/dictionary/infinitesimally

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

BOB.1 [Circular Functions]

Well now that I'm back from my Kenora basketball trip, and once again have computer access I thought it would be an oppourtune moment to do my BOB post for this unit. Alas, I find myself here doing exactly that. BOBbing!

Well I'll start with the bad, and work my way towards the good, seeing as how it also goes like that in terms of chronological order from the beginning of the unit to the end :)

The hardest part of this unit was definitely the very beginning. Now this had alot to do with the fact that I missed the first two days of class, having been sick with the flu (>_<). So by the time I had returned to the class on Wednesday (that being the third day of class) I had basically no idea of what was going on. However, with some help from my classmates, and a couple nights spent awake much later then I should have been, I managed to figure my way around the first couple days of work, and catch up to everyone else.

On the flip side, the easiest part of the unit, has probably been the memorization, of the patterns in the unit circle (which eventually lead to the memorization of the unit circles basic values for sine, cosine, tangent, etc.) I found this rather easy, because all I had to do, was draw the unit circle over the weekend, without referring to my notes. How simple is that? So after drawing the unit circle, probably 7 or 8 times over the weekend, taking about 3-5 mins each time (with that time frame getting smaller as I went along,) I managed to get a fairly good grasp of basic angles locations on the unit circle, and the trigonometric function values of those said angles.

Overall this unit wasn't very difficult to understand or grasp (at least for me.) I'd say the hardest parts of it all were related to the memorization of certain things (like what A,B,C,D mean in graphing f(x) = AsinB(x-C) + D). I also found it somewhat difficult at times to remember what to do when adding radicals or similar things that we learned in previous grades, and I haven't had to use in awhile. Besides that I found much of what we did within my abilities to understand and put to use in the exercises we've been assigned.

Well I think that about sums it up for this BOB. I hope everyone enjoyed their long weekend and I also hope everyone enjoys the rest of today :) I shall see everyone on Monday!

Ciao!
~Justus

Circular Functions (February 13, 2008)

Well guys it's me, Justus here, and as the scribe for today I'm writing this blog post :)

To begin I'd like to say that we had a substitute today, a very nice lady by the name of, Ms. Cheekie. Anyways after introducing herself, she informed us that we had a quiz on circular functions, which we had the whole class to do, but must be handed in at the end of the class. If you missed the class I'd sugges talking to Mr. Kuropatwa and seeing if you'd be able to write the quiz, as I assume its for marks.

After that, we were handed out a worksheet on circular functions, entitled, "Working With the Unit Circle Exercises." Which was to be worked on once completed the quiz. If we didnt finish the worksheet during regular class time, Ms. Cheekie said that it would be due during TOMORROWS CLASS (as in February 14th, 2008, at 9:00am.)

That basically sums up what happened during todays class, and thus, sums up my scribe post. However, because this scribe was significantly shorter then most posts, and I'm not entirely sure if this material counted as a legitimate scribe post (regardless of the fact that we do need to know of the hand in sheet), I decided to just post anyways. Should it be unsatisfactory as a scribe post, I will help Richard (whom I've selected as the next scribe) out with tomorrows scribe (since it is a two period day.)

That is all, until next time.