I'll see you in 3 months Bob.

Is it just me, or does anyone else see the countdown on our blog state that the exam is in 220 days?! OH man, wish we had that much time, but it's just my computer's clock playing tricks on me, haha. Well anyways, on to my final bob for the course.
I can't even begin to comprehend how fast our year passed by, but all I can really remember is all of the times that we shared in this class. I really believe that this was a momentous semester for me, and never before have I resided in such a healthily and welcome classroom. There were so many occasions that I simply cannot forget, such as Eddie and the event of conversing with his class in Georgia, or how we intuitively utilized the smartboard for everyday class. Well, enough about that, I believe there's still an overall course review BOB or something similar at the end of the year. So on to talking about our final unit; sequences.
When we first started sequences, I couldn't believe how simple it was. The questions themselves were relatively straightforward, and even the thought-provoking questions just require the implementation of given formulas. All we really have to do is stop and analyze what is being asked, the sixth term, the distance the ball travelled, what is S100, etc. I believe that this test shouldn't prove too difficult for anyone in the class, and hopefully everyone will perform incredibly on our final test.
GOOD LUCK TO EVERYONE ON THE PRE-TEST, TEST AND NEXT WEEK'S FINAL EXAM!!!

Research Assistant

Here I am again with another research assistant! For our final unit in the entire course, Sequences, I have compiled a list of links to help anyone review or practice for the upcoming test. I found these sites earlier today, hopefully they can help anyone who needs help trying to understand or anyone in search of extra practice for this unit's test.

REVIEWS/TUTORIALS

http://home.alltel.net/okrebs/page131.html

http://www.sparknotes.com/testprep/books/sat2/math2c/chapter12section2.rhtml

(The following four are from the same site, just broken into separate portions.)

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54a_seq.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54b_series.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54c_arith.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54d_geom.htm

PRACTICE QUIZZES/TESTS

http://rutherglen.ics.mq.edu.au/math106S206/exercises.html
(scroll all the way down, select the second choice then click "get the quiz")

questions: http://www.math.umn.edu/~stanton/3113/exam1.html

solutions: http://www.math.umn.edu/~stanton/3113/pracexam1soln.html

I hope that these links can be utilized to help anyone review for the upcoming test, of which I wish everyone good luck! Have a great weekend everyone!

Boron-Oxygen-Boron

I almost forgot to bob for the upcoming probability test until I checked the blog and saw Sandy's. And to iterate Sandy's statement, the provincial exam is only 2 weeks away! I can't believe that we've blown through the year in almost no time at all (well at least it felt like that) and what a blast it's been. Though I'm feeling really down because of all the cruddy marks I've been getting recently, but I've really enjoyed this course so far. This was the first class which presented a functional lesson solely integrated upon the smartboard. But enough about reflecting over the entire year, this bob's supposed to be about just this unit xD. This unit of course was probability. This unit wasn't terribly difficult and didn't present a lot of tribulations, especially seeing as the final portion of probability incorporated one of our previous lessons: combinatorics. Just remembering that a probability can be found by dividing the amount of favorable outcomes by total amount of outcomes is how you arrive at the wanted probability. Well this unit was very relaxed and somewhat straight forward, but as Mr. K left us (made me sad )= ) we were responsible for our own learning, and showing by my quiz marks, I wasn't exactly a very reliable worker. But as for the rest of the unit, there wasn't really anthing new that was unique to other units except for the exciting colloquy we had with Eddie's class in Georgia! It was quite a surreal experience, seeing as it's quite intriguing to converse with people from other regions of the world. I think we were all excited to chat with the class over in Georgia, and when the day came, we really enjoyed it. I think it was quite an extravagant episode in our bevy of pre-calculus memories. But with that said, I can't wait for further conversations from the class over in Georgia. This was my bob, good night everyone. As for the test:

I WISH EVERYONE GOOD LUCK ON THE PRE-TEST AND TEST!!!

Exactly why are doctors so cagey?!

Hey it's MrSiwWy here to present to you the scribe post for today! I'm the scribe for today's rather intriguing but yet solemn class. I say solemn as I think that the main question our class was focused around pertained to very serious matter. But anyways, onto the scribe post for
Tuesday, May 22, 2007.

Our class began with Mr. K outright stating to the class that we're on the verge of the end of probability, and the beginning of our final unit: sequences. He then told us that he won't be present the two following days, but that we only have two things left to learn in probability, one of which incorporates one of our earlier units: counting and combinatorics. He presented us with an example of such a question. This question was regarding the arrangement of students, like we have been presented with in the past.
If students are to be seated in 7 desks, what is the probability that Mark and Robert are sitting together?
Basically, we take amount of favorable outcomes, which would be the amount of ways Robert and Mark can be seated together, then divide that by the total amount of seating possibilities.

We quickly moved onto the class for today, and very soon, we transitioned into the question that Mr. K left us off with on Thursday; "Why do doctors have do so many tests? Why are they so cagey?" Once Mr. K put up the question we were to dissect, he set us up in small groups of four. This smoothly yet appropriately forwarded the momentum of our class into the actual lesson. This question can be seen on the following slide:

If we think about each of the four possibilities, we have to realize that the test is to check for cancer, so if it tests positive when the person really does have cancer, then the test succeeded, meaning that it has a probability of 98%, or 0.98. This means the remaining probability (that the test failed by showing negative, when they really have cancer) is 2%, or 0.02.
But as for people who do not have cancer, if the test shows positive, this means that the test failed, and this probability is in fact 2%, or 0.02, and the remaining probability (to test negative for those who do not have cancer) is 98%, or 0.98.
And as I stated at the end of the slide, a tree diagram can really help. So here is a tree diagram that was used to calculate the probabilities of each of the questions on the following slide.

Mr. K later asked us to construct this tree diagram to help us understand the question's calculations, but really, to create one beforehand is more of a help.

Here we see that Mr. K coded certain important aspects for various reasons. These reasons are very vital when contemplating a method to approach each question.
The blue underlined portions indicate that we're dealing with people who have cancer.
The green underlined portions indicate that we're dealing with people who do not have cancer.
The small circular flower symbols indicate total amounts of people that have cancer.
The small stars indicate total amounts of people that have been tested positive for cancer.
The last two are particularly important because the main question that we were originally asked was what the probability of a patient that has cancer is also tested positive for cancer. In order to arrive at this, we must dignify how many total people have cancer, and divide that by the amount of people that were tested positive. We've already learned that in order to find the probability of an event, we take the amount of favorable outcomes and divide that by the total amount of outcomes. So in this case, we take the amount of subjects who have cancer, and divide that by the total amount of times that the test would show a positive result. That is why Mr. K signified each of the above portions. This was taken care of in part 4 of the above questions, but before that, I'd like to outline how to achieve each of the answers in the entire slide.
In order to determine each of the answers shown on this slide, we used the tree diagram above in conjunction with the sample group of 1,000,000 people.

1. This part was simple. Since we know that there are 1,000,000 people, and 0.5% of these people have cancer, that means that 5000 people have cancer, and the rest don't. The rest can be found by subtracting how many that have cancer from the total of people, this gives us 995000.
2. In this part we use the tree diagram to determine what percent of people that have cancer will test positive. To do this we multiplied the amount of people that have cancer, then multiplied that by the probability of testing positive, which is 0.98. This gave us 4900, the answer presented in the slide. For b, since we know 4900 tested positive, the rest of the people who have cancer must have tested negative, and that number is 100.
3. Here we repeated the same process used in part 2, exact that we used different values since we're looking at only the people who do not have cancer. We know that there are 995000 people that do not have cancer, and as I outlined earlier, since the test would fail if it said positive, we would have to find 2% of 995000. This, as shown, is 19900, and to find how many tested negative, we just subtract this from the total. 995000-19900 gives us an answer of 975100.
4. The final part was compounding what we determined in the previous steps, and in essence, determining the answer to our original question. First we take how many people altogether tested positive, we get this simply by adding those that have cancer that tested positive together with those that don't have cancer but tested positive. For b, we know that of the people that tested positive, 4900 of them actually had cancer. So to find the original question at hand, this was the work shown. I already described how this was found, here is the work:
   

As he showed us that vertical line separation between the C and the P, the class began wondering what that represented. This introduced the class to conditional probability. The way Mr. K described such a probability was "when we have inside information" beforehand. Usually when we have dealt with probability we knew that either one thing or another can happen, then another event can happen in one way or another. But in conditional probability, we know that one of these events is going to happen in such a way. In this question, we knew that our answer only concerns when we know that the test is going to be positive. He gave another example of conditional probability using three jars containing marbles. He says that if jar one has 3 blue and 3 red marbles, jar 2 has 2 blue and 3 red, jar 3 has 2 blue and 2 red, then he doesn't look at any of the jars and pulls out a blue marble out of one of them. In this case, we know that the marble that he pulled out is blue. Given the above information, what is the probability that it is from jar 2 of pulling a blue marble from jar 2? This isn't word for word, but I believe that was the basis of the example Mr. K conveyed to us. The basic idea is that we know that the marble is going to be blue, similar to how we knew that the test was to be positive in the first question today. So to sum up conditional probability:
Conditional probability is when we have some inside information on the events at hand, basically when we know some of the information before we even start to calculate.
When we are dealing with conditional probability we write P(P1P2), where P1 is the probability that we're looking for, the vertical line says "given that we already know", and P2 is the event that we already know.

Once we were finished with that question, we moved onto a very similar question, in which he just changed "cancer" to "industrial condition," 98 % to 99 % and 0.5% to 1%. Here is the question along with the work, of which I will of course explain.

Here we see both a tree diagram and the work to the solution for the question.

• I represents those with industrial disease.


• H represents those without industrial disease.


• P represents those that tested positive for the disease.


• N represents those that tested negative for the disease.

Here we see that we found the probability that someone has the disease and also tests positive for the disease by dividing the favourable outcome (probability of having the disease) by the total amount of testing positive. This is exactly the same case as finding the probability of having cancer and testing positive for cancer, just with different probabilities.

So in this class, we fully determined the answer to the question that Mr. K had left us with on Thursday afternoon.

Doctors aren't necessarily cagey, but run the tests more than once to make sure. If we look at the results that we achieved by both questions, we arrive at the fact that if someone really has the disease and tests positive, there's not a good chance that it's for sure. Thus, doctors run tests more than once not for fun or anything of the sort, but in both cases it's more likely or equally likely to be tested positive and not have the disease, so it's good to be sure before breaking such news to a person.

And that concluded our class for today, with the lesson of conditional probability that was presented to us today, as Mr. K said, we don't have a lot left in this unit. This was my scribe post, and I hope it was clear enough for anyone to understand. I still need to fix some content up, really don't like how my Computer is so resilient, especially seeing as I can't get to any pod casts. =/ But as soon as my computer begins to cooperate with me I'll get everything fixed up and tidy. Please just contact me if there are any concerns, questions, complaints, errors, etc. Have a good night everybody!
oh yes, thank you John for reminding me. The scribe for tomorrow will be: Jann!

Research Assistant

I remember when Mr. K told us about this, I was ecstatic. I want to help =) This research assistant post will be concerning the Conics unit , and I hope I do it correctly. Well, I went on the net and found some resources to help anyone who doesn't fully understand the unit yet, and to help give people some extra review for the test tomorrow.

REVIEW/TUTORIALS
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/emat6690/Insunit/conicsunit.html

http://www.scit.wlv.ac.uk/university/scit/maths/calculus/modules/topics/precalc/conics/learn.htm

PRACTICE TESTS/QUIZZES
http://www.absolutevaluepublications.com/studentworkbooks/principlesmath12/PM12%20Conics%20Lesson%207.pdf

http://mathforum.org/mathtools/cell/a2,12.15.4,ALL,ALL/

I hope I did this right, and even more so, I hope it helps whoever see's this and follows the links. I already used some of the practice quizzes to help me review, hope they help you as well. Good night!

Pre-BOBulus

Wow, I hate my internet. Took me hours just to get this fully functional BOB up, it's a miracle I arrived at the "create post" successfully. Well here goes my BOB.

I think the unit of Conics was a breeze. Extremely easy, nothing truly difficult was presented to us, and the very little homework. But, I do have a rant regarding what Mr. K said concerning people going up to the board. Throughout the course, we've seen the utilization of the smartboard optimize and manifest itself as the base learning mechanic of the class. But we always seem to see the same people go up, while the rest of us just watch. I guess it's good to experience learning through visual means, but kinesthetic learning is a good change of pace to help us fully engulf ourselves in the information being presented. I don't think it's a big deal, however. Many of us are probably more visually-oriented learners now. Well, that was just a rant, all this free time due to this incredibly easy unit got me to thinking. I don't suggest anything though, I'm just sharing my view. Back to the unit, Conics brought back the well known parabola and circle back from last year's precalculus class, and also introduced some new funky sections for us, such as the ellipse and the hyperbola. As long as you know the properties and definitions of each conic section, you'll be fine. Reiterating my previous description, this unit was extremely easy. I hope (and am strongly believing) that everyone will do great on this test.

GOOD LUCK ON THE TEST TOMORROW EVERYONE!!!

BOB The gaurdian

I accidentally posted twice, I hate my Internet! =/ So I'm just going to edit the double post (this post actually) so that it's not so bad. The only real thought I had of how I could edit this post appropriately is... well... a BOB. Well here I go:
This unit, though not quite finished yet, was so far quite easy actually. I really enjoy this unit, and if you do your homework, it is extremely easy. Though before we started, I bet logarithms seemed quite intimidating, but really, they're softer on us than a kitty cat. (= Well not entirely so much, we still have a little bit to learn, and what we learned yesterday wasn't completely a piece of cake, but the test on this unit should produce a good mark boost for us (I hope!). This unit was full of repetition, and I'll pretty much end this BOB with some of that repitition, because people, remember...

Logarithms are exponents

Logarithms are exponents

Logarithms are exponents

Logarithms are exponents

Logarithms are exponents

Well that's my BOB, but if you didn't read it at least read this..

DON'T WORRY PEOPLE THERE'S NO TEST QUITE YET!!! I JUST DID THIS TO COVER UP MY DOUBLE POST =/

The test is probably going to be around next week friday? Or maybe the week after next (hopefully xD) Have a great weekend everyone!

Flickr Assignment!

Hey, this is my flickr assignment! Just click to be linked there =)

OH yeah and before I go, HAPPY ADVANCED BIRTHDAY TO GRAEME!!!

X-ponent men 3: The last stand

Hey everybody! I'm MrSiwWy and I'll be your scribe today! I don't have much time to complete my scribe post for today, as I won't be home until much later and I still have to practice for the talent show on Friday! Well, not much time so I'll head onto it.

Tuesday:
As we all know, yesterday was the day when we wrote the terrifying identities test, and I bet that test left everyone with anything but a smile on their face. As Mr. K said the previous day, it was obvious that there was some anxiety concerning the formidable identities test. But despite all worries and the incredible tension in the room when we were attacking the paper with our minds, I hope everyone managed to perform adequately on the test. Now on for some actual knowledge that was conveyed our way.

Wednesday Morning:
Now that the identities test was clearly out of our way, it was inevitable that the following day would bring forth a new unit to our growing intellect. This seemingly excruciating unit is of course exponents and logarithms. But I must add, I do applaud Mr. K's method to introduce this unit, which will be explained in the very next paragraph.

This method of course, all begin with Mr. K up at the Smartboard. He waited there until class had essentially began, then showed us the introductory slide. This slide of course, consisting of numbers to help us segway smoothly into the unit. The slide consisted of these numbers:
2
3
4/9
1/4

Then Mr. K told exactly what we were to do, write each of these numbers in different ways. We were told that we must write each of the above numbers in different forms, but each must consist of an exponent. He started the ball rolling by giving examples of such for the first number, 2. The examples he gave were 4 ^1/2 and (1/2)^-1 . These examples fit as 4 to the exponent half equals 2 since if you have an exponent 1/2, the number is first calculated to the power of 1 then the square of that number. In this case, 4 to the exponent 1 is 4, then the square root of 4 is 2. For 1/2 to the exponent -1, negative exponents, as we have learned in the past, mean we must take the reciprocal of the base, which in the case of 1/2 is 2. After he gave us these examples, he gave us time to determine more for each of the number. He said to either come up with four different ways or as many as we can for each. Then he gave us a while to work on them (although it felt like more than a while =p).

Once Mr. K stopped us, we began to convey all of our thoughts onto the Smartboard. We collectively assembled many different methods for writing each of these, and through writing these we noticed some very obvious patterns. Here are the slides of which our answers are integrated upon:

Just some of the possibilities of writing 2 exponentially. From writing these possibilities, we began to uncover a pattern to determine how to write a number in different ways using exponents. This pattern can be seen through the 8^1/3 which is equivalent to (1/8)^-1/3. Using such examples can show that when we write any number to an exponent x, we can also write the reciprocal of that number to the exponent negative x.

From writing these values we further elaborated upon the aforementioned pattern. Then we quickly continued writing all of the possibilities for the next number, 4/9.

At this point, Mr. K asked if there were any ways we could come up with numbers with exponents that do not include the number 1. For exponents which do not contain 1, such as the exponent 2/3, this simply means that we must square both the numerator and denominator (if the base is a fraction) then take the cube root of both. For the exponent -2/3, we simply take the reciprocal of the base then apply the exponent 2/3.


At last, we wrote down some possibilities for the number 1/4. Another pattern emerged, concerning the number 1/4, this pattern was shown by the expressions (1/8)^2/3, (1/32)^2/5. This pattern seems to always have 2 as a numerator in the exponent, but increases by odd intervals for the denominator. As for the denominator of the base, it increases by a certain exponent, then is divided by 2. This pattern along with the other can help us when trying to write two numbers with the same base, which will be described in a little bit.

After we were done with all of those, and now that we knew there are an infinite number of ways to write any number using exponents, we moved on. But all of this was a great preparation for our real lesson for today. Solving the following equations:

2^x = 64
27^2x-1 = 3
3^x = 1/27

He taught us that we must solve these by making the bases equal to each other. If you set them equal to each other, since the bases are equal, they must equal the same number in the end, meaning that the exponents must also be equal. So if you set the bases equal, you can determine an unknown exponent since they must be equivalent. This is what we must know for this unit, as this is how we will perform and carry out logarithmic expressions.
Such as for the equation 2^x = 64, we can solve this numerous ways, but as Mr. K said, and as we have learned before, we must set the bases equal to each other. The two most obvious ways to do so are as follows:

The above also states what to do in each step, although there are bound to be more complex problems in the future, leading to more steps. But each of the above show the basic structure for solving these equations; rewrite the equation with equal bases then you can get rid of the bases and set the exponents equal to each other. But remember, such as in the second example, make sure you keep the x when you alter the exponent if the x is in that exponent. Such as in 64^(1/6)x, when the 2^x was changed to a power to the base of 64, the x must still remain. Don't just change it to 64^1/6.
As for the other equations, the same mechanism was used for each solution, so just remember that if the bases are the same, then the exponents must also equal each other.
The work for each equation can be seen here.

Check them out if you want some extra reference. And that was that for the morning class, as these equations concluded the class.

Wednesday Afternoon:
In the afternoon class, we didn't learn anything as the afternoon was marked by a workshop concerning racism. In class we just talked about certain events, and Mr. K read out a story of which he was given to conduct the workshop. We began talking about incidents, and certain stories which pertained to the subject, and most of what we talked about in class can probably be found on the podcast. And that's about all I can say for now, as I'm running out of time.

Well that's my scribe everybody! Sorry if it came up late, I've already went back and forth with practicing for talent show and now my parents are angry with me so I have to cut this short. I won't have much time to edit this and even entirely complete it, so I'll be adding more to this post maybe later tonight or tomorrow, sorry for the inconvenience. If there's any mistakes, complaints, or anything whatsoever, just tell me, I still have to edit much of this post. I hope my scribe is useful to anyone that didn't attend class and it's at least sufficient considering the high standards on our blog. See you all tomorrow! OH and one last thing... the scribe for tomorrow will be...

Craig

Have a good night everyone! Don't forget to do your homework!

The BOBinator

Hey everybody! I hope Mr. K doesn't consider this a late bob, but this was kinda the only time I could do it, seeing as I've been pretty busy over the weekend. Thank god I remembered to blog or else I would've lost one whole mark!!! Well, enough about that...on with the bob...
So we're now done our third unit, the unit of identities. I think this unit began quite quickly, and lots of information was just thrown at us (especially for me since I wasn't there for the first class of Identities). But as the unit progressed, I think we kind of moderated our learning pace and things began to become more clear and visible in my mind. Mr. K was right, and throughout my bob I'll probably stress it repeatedly, practice is the only way to get good at identities. Homework in this unit is especially beneficial to understanding how to manipulate identities and mathematically "massage" them accordingly. If you payed attention and continued with your homework regularly, I don't think this unit would be particularly difficult. Don't forget, practice is the only way!!! So, overall, this unit went by extremely quickly, but for the amount of material we learned I think we went at well-regulated pace. In essence this unit wasn't too difficult, particularly if you kept up with your homework. There isn't much more to say, so I'll end this bob with a list of the common identities you should know for the test. They are as follows:

• sin²θ + cos²θ = 1
  (note: you can derive other equations by dividing every term by either sin²θ or by cos²θ, which will give:
  tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ)
• tanθ = sinθ / cosθ
• cscθ = 1/sinθ
  secθ = 1/cosθ
  cotθ = 1/tanθ or cosθ/sinθ
• remember the sine dance :
  sin(α+β) = sinαcosβ + cosαsinβ (change sign for α-β)
  cos(α+β) = cosαcosβ - sinαsinβ (change sign for α-β)
• don't forget to watch for a difference of squares!

GOOD LUCK ON THE TEST AND PRETEST EVERYONE!! DON'T FORGET TO PRACTICE THOSE IDENTITIES!

BOB BARKER

Speaking of Bob Barker, why don't we play a game of The Price is Right on the smartboard? Or even better, Wheel of Fortune! Wouldn't that be the coolest thing? And above all else, Mr. K could be Vanna White! Haha. We all know he'd want to get to use the smartboard as much as he can. Ha, Well enough about that, it's BOB time.

So, our second unit is coming to a close and now another gruesome test approaches our cowering souls. But despite the oncoming test, I feel as though this unit wasn't even a full unit, but yet we learned quite a bit in this unit, the unit of Transformations. That might be due to the fact that I have always found graphing to be very simple and straight forward, and this unit didn't present any exception to this. Some review questions did require deep contemplation, although these questions weren't particularly concerning graphing so there isn't much worry there. I hope everyone else, as I successfully accomplished, finished all of the homework that was given to us. It's very important, and in graphing I really think it's helps with our conceptual understanding as we become more habitual to graphing each type of function transformation. But overall, the unit of Transformations didn'y present any particular tribulations to me, and was a really easy unit. Only thing I didn't like was the quiz. Why do I never take time to read the questions? Now I'll know, good timing as well, as our pretest is tomorrow, and I'll try not to fall victim to one of Mr. K's tricks. But now we know all about that now, since we are past our first marker of the year, the end of our first unit.

But our second unit not only marked the triumph over our first obstacle, the circular functions test, but it was also a marker for another triumphant feat. This feat of course is the utilization of our smartboard throughout the unit of Transformations. Since the time we first received the smartboard, Mr. K has been practicing his smartboarding skills and now it's more than just incorporated into our daily lessons, but is the basis of where we conduct all of our lessons. We now all come up to share our answers with the class on the smartboard, and I think it's ingenious the way Mr. K set up this peer evaluation process mixed with some competitive vigor to help our overall education. With the smartboard, now we're all willing to go up and put up our answer, and whether we're correct or incorrect, we inevitably end up learning. I wonder what the year would have been like without the almighty smartboard.

So I hope everyone is ready for the pretest tomorrow (maybe it's tomorrow), and I hope you've all completed all of your exercises. As for me, I know I've completed them all, so I'll end my BOB with a quick review of important reminders for the pretest, then subsequently the great Transformations test:

• Inverse functions are different from recipricol functions. f^-1(x) is not f to the exponent -1, it's an inverse where you switch y and x coordinates, or flip it over both the y and x-axis'. Simple.
• Always stretch the graph before translating it.
• To graph reciprocal functions, first find the invariant points. These are points where both graphs go through, and these points are always -1 and 1. Then find the root(s), these are where the vertical asymptote(s) will be for the reciprocal. Then, draw the reciprocal by drawing it increasing in value positively when the function is decreasing positively, the same for the negative direction (or vice versa if drawing the original function).
• Even functions are different from odd functions. Even functions are symmetrical over the y-axis, while odd functions are symmetrical over the origin, meaning odd functions are the same if you turn the graph 180 degrees.
• Watch for the sign of a when considering translations in the general form: y = f(x-a)+b

And that's my BOB folks, again, sorry it's long, but I tend to ramble on... (=

GOOD LUCK TO EVERYONE ON TOMORROW'S PRETEST, AS WELL AS OUR TEST ON WEDNESDAY. (Dates may not be exact, as Mr. K said)

Lost in translation...and stretching.

Hey everybody! I'm going to be scribing instead of Kadeem today since he could not do so. And since he told me he was going to choose me tomorrow, I offered to switch with him so that things would work out efficiently. So I'm saying it now, and I will say it again later, tomorrow's scribe is Kadeem.

Friday Morning:
I was not in class on Friday morning, as I had to go to "Doors Open," an extracurricular program for history class. But despite my absence, I asked around and determined that we didn't learn anything in the morning. All we were told was that exercise nine was assigned for homework over the weekend, but we weren't taught about stretches yet so we could not complete the exercise. So basically, the entire class was spent working on other subject work or "working" on the smart board. Nothing new was learned in this class.

Friday Afternoon:
I was present for the afternoon class, but then again there really wasn't any point to be there. We again didn't learn anything or accomplish any work, so we were either completing work concerning other subjects or (as the majority of the class did) played hangman about "math" on the board. Yeah...that was the main topic...math. Well, that was what the afternoon pretty much consisted of. But now on to where the most recent information that he taught us.

Weekend:

I know it's odd that we actually learned something on the weekend, but it's ultimately true. Late Saturday night, Mr. K posted on the blog a post which contains links to helpful sites, a java tutorial applet and a video lesson he created for us. If you did not get to see it, here's a direct link to his post here, or just scroll down (I think the latter requires more effort =p).

A summary of the movie:

In the video he taught all about compressions, stretches and reflections (all of which were elaborated upon in today's class). He began by stating that we've went over that any function, y=f(x) is where "the output is something done to the input." But if we alter the output, such as by multiplying y=2f(x), we alter the y-coordinates of the function. He later elaborated upon this in the video, and in class, and I too will later elaborate upon this as well. After he showed how this works he went more into detail about how vertical stretch will occur if the value of that number is greater than 1, and will vertically compress the graph is less than 1. Then we moved onto altering the input of the function, such as y=f(2x). This alters the x-coordinates of the function, but not by the value given, but by the reciprocal of that. In the function y=f(2x), the graph is not stretched horizontally, but is compressed horizontally, as it makes things speed up. This was later explained in today's class. Once he was done explaining vertical and horizontal stretches/compressions, he went into reflections:

y=-f(x) <-- This function will be reflected over the x-axis, as the negative value is changing the y-values of the function. That negative sign is -1, so if you multiply all y-values by -1 the graph will appear to be flipped over the x-axis.

y=f(-x) <-- The same fits for this function, but it will be reflected over the y-axis. The negative 1 is multiplied by all the x-values of the function, causing them to be flipped horizontally over the y-axis.

He then ended it by saying that inverse functions will be explained in class, which was later proved to be true.

Monday's Class:
Today's class, we learned a lot useful and very interesting information about our new unit. I came into class and I saw that Mr. K had the aforementioned video up on the smart board, and I naturally assumed that we were going to go over it. This was true. So we began the class by watching the video, as much of the class did not get to watch it over the weekend. He paused the video at certain points and elaborated upon and fixed what he said in the video. The first time he paused it was to explain stretches and compressions.
When the function y=f(x) is altered by adding a value to multiply the output, in the case he showed us, y=2f(x), we can find the new y-values of the function easily. The 2 in front of the function alters y-coordinates, so all we must do is multiply the y values in the original graph by 2 to find the new functions y-coordinates. This causes a vertical stretch, and he showed this stretch by using the following graph:

In this graph, the black line represents y=sin(x).
The red line represents y=2sin(x).
The graph shows how once the graph was transformed by multiplying all y-values by 2, the graph was stretched vertically. But if the value is decreased, say to less than 1, such as 1/2, how will the graph look?

In the graph, the transformation is compressed vertically, which shows what happens to the y-values of the original function when the output is multiplied by a number less than 1. The graph becomes compressed, and the y-values are multiplied by, in this case, 1/2, which causes the new function to have lower y-values, causing it's compression. But this is not the only type of stretch that he showed us, as he also showed us horizontal stretch.

The graph now shows y=sin(x) compared to y=sin(2x). The graph is compressed, and Mr. K went in depth into his explanation of why this is. He explained that:

• In this case, you're doubling the input before it was calculated by the sin function. Before you do this, you multiply the x value by 2, doubling the original value before you find the sin.
• This makes the function happen faster, as you're multiplying the value before it is input into the function.
• To determine the x-values of the transformed function, you multiply all x-values by the recipricol of the value. In this case, to find any x-coordinate in the new function, you must multiply the original functions value by 1/2.

This graph shows what happens to the function y=sin(x) when you transform it to y=sin((1/2)x). In this graph, everything is slowed down, and if you take a look at the x-values of the original function, they appear to have been doubled. The graph is stretched horizontally, and as detailed before, to find the new x-coordinates of the transformed function you must multiply the original values by the recipricol of the value.

Each of the above graphs help portray how functions are stretched and compressed and then once we were shown all of these graphs, we were given a general equation to represent these parameters. In general we see:

y=af(bx)

The value of a represents the vertical stretch of the functions transformation. The value of a is used to multiply the y-values of the original function to find the new function. A alters the y-values of a function.

The value of b represents the horizontal stretch of the functions transformation. Though, the value of b is not the value used to determine the new function's coordinates, we must use the reciprocal of b. We must do this since b alters the values being input into the equation before it is actually calculated. For example, in the function y=sin(x), the transformation of y=sin(2x) causes the values of x to be doubled before it is calculated into the function of sine. In order to counteract this, we must use the reciprocal so that the values will end up equivalent before going into the sine function. In this case, if x=Π/2.

y=sin(x)

y=sin(Π/2)

y=1

But when the graph is transformed to y=sin(2x), our y value will end up being 0 instead of 1. This is because the value of b multiplied the x value before it went into the sine function. So if we take the reciprocal of 2, which is 1/2, and multiply x by the reciprocal, then our answer will balance out before going into the sine function.

y=sin(2x)

y=sin(2(1/2(Π/2)))

y=sin(2(Π/4))

y=sin(2Π/4)

y=sin(Π/2)

y=1

So remember, the value of b is not what we're multiplying our x-values of the original function by, we're multiplying by the reciprocal.

Then we partially reviewed translations from previous classes:

y=f(x-a)+b

The value of a represents the horizontal slide of the function.

• If a is greater than 0 then the function is shifted to the right a units. In this case, the x-coordinates increase.
• If a is less than 0 then the function is shifted to the left a units. In this case, the x-coordinates decrease.

[Don't forget to pay attention to the sign of (x-a)]

The value of b represents the vertical slide of the function.

• If b is greater than 0 then the function is shifted up b units. In this case, the y-coordinates increase.
• If b is less than 0 then the function is shifted down b units. In this case, the y-coordinates decrease.

After we finished reviewing translations, we then worked on graphing some of these newly taught stretches and compressions, as well as translations. He showed us a graph on the smart board, and then we had to alter it accordingly. These questions can be found on slide 2 here. The last question was composed of both stretches and translations, and from this question he taught us an important rule:

[Stretches come before translations]

Always work out stretches before translations. Well, now that all that was done, he quickly taught us about inverse functions.

He began by outright telling us that an inverse undoes whatever has been done to something. He then contrasted this to the "inverse" relationship between his daughter and his wife and himself. Every morning, his daughter goes into her clean room, then she transforms the room into a messy room. At night, the super inverse parents come into the messy room to clean up the mess, and trasnform the room back into a clean room. He showed a diagram of this:

Then we went more into the work involving the inverse function. The -1 above the inverse function's F does not mean it's to the exponent -1, that would be written as [f(x)]^-1 . An inverse function "undoes" the transformations of a function through a routine, and the function of a regular function switches y and x values.

f --> f^-1

(a,b) --> (b,a)

If you consider Mr. K's example, the messy room is the domain of f(x), and the clean room is the range. In other words, in the function f(x), the messy room represents all the x-coordinates, while the clean room represents all the y-coordinates. In the inverse of the function, however, we switch the messy and clean rooms around. By doing this, we're switching the domain and range around, which is exactly what happens in an inverse function.

(a, b) = (messy room, clean room)

(b, a) = (clean room, messy room)

He then showed us the inverse function of f^-1(x) through different methods.
Algebraically
The inverse of a function can be determined algebraically. In the equation, switch the x and y variables, then solve for y to receive the naturalform of a function. We must due this so that we can graph it. Consider the function f(x).
f(x) --> y=√2x³-4

f^-1(x) --> x=√2y³-4

x²=2y³-4

x²+4=2y³

x²+4=y³

y=√((x²+4)/2) <-- we are now able to graph this.

Conceptually
We can also display the inverse of a function by undoing whatever has been done to the function. Whatever order is taken to transform the function, the inverse will undo these transformations by reversing the order and reversing the operations. An example is shown in here on slide 3.

Graphically
In order to graph the inverse of a function, you must graph the function then exchange all x values with all y values. You can also check here on the third slide to see an example of a graph, or for any of the inverse information.

And that concludes my scribe post for today. To reiterate my earlier statement, tomorrow's scribe is:

KADEEM

I know Craig chose him last time, but me and Kadeem "switched" our scribe posts so thing can work out better, since he was not able to today. Good night everyone, hope everyone understands our unit so far. Good luck in the rest of the unit and see all of you guys tomorrow!

Some helpful sites.

Hey everybody, I know that yesterday we didn't really find out anything new and didn't learn anything, so I went out looking for some sites to help me learn it. I found some helpful sites which explain transformations (look for stretches to make up what we were supposed to learn yesterday) in depth and hopefully make sense of the unit so far. You don't have to use these sites if you don't want to, and I hope Mr. K doesn't get mad at me for posting these sites for any reason =/ Well hope they help guys!

http://www.bmlc.ca/Principles%20of%20Math%2012%20%20-%20Transformations%20Lesson%204.pdf <-- This site contains steps for how to determine each parameter in a standard transformed function, such as stretch, shift, etc.

http://www.mathsnet.net/transform/index.html <-- Here's some interactive tools to show how all the parameters work.

http://archives.math.utk.edu/visual.calculus/0/functions.12/index.html <-- Good summary of the principles of translation.

I hope these sites can help, and I hope this kind of makes up for those classes we missed.

The test of Doom

Hey folks! Craig, today's scribe, is at work today so I'm going to scribe for him today (doesn't really count as a scribe post on my behalf since we accomplished so little today). Craig will still be scribe for tomorrow, so there's no need to cry about his absense in the blog today. =D Well, today we had a morning and afternoon class, but still didn't really do anything each class so this post is bound to be short.

Morning Class:
In the morning , we were given, as told, the greatly anticipated circular functions test. It was not what I had expected, and to be honest, I expected a lot more from the test. But still, the morning class was an intense one, as an ambient tension spread across the class once the test was given out. Complete silence overtook the room, with the exception of pencil scratching and calculator button mashing. Class progressed with this seemingly ominous silence, only to be interrupted by the substitute teacher telling us to place our tests on the desk at the front on our way out of the class. Period two was imminent, so some scrambled to check over their test or just to finish their test, while others blissfully laid their papers on the desk of fortitude. Once the bell rang, everyone handed in their tests, if they did not already do so, and that concluded the morning class.

Afternoon Class:
There's nothing much to say concerning the afternoon class, as we really didn't do much. We were "assigned" to read pages 10 to 16 I believe, and to complete the assignment it stated on the board. When did we ever receive this assignment? It's odd because I could have sworn he said something about "applied 40S." I really didn't know what he was talking about, but class turned out okay. The afternoon was a big slack class, everyone just sat in their desks (or moved around) and completed homework and did practice problems and such. I myself finished homework in the class, and talked about the test from the morning class. Many of us were still conversing and arguing about questions that were on the test, and in many cases one side of the chattering parties ended up feeling regret for realizing their mistakes on the test. For the rest of the class, not much else happened except for the aforementioned activities, and we didn't learn anything, so I don't think I can say anymore.

Now I'm just wondering whether we're still going to have to complete the rest of the test, which is the graphing part of it. I was under the impression that this was going to be given to us in the afternoon, but it wasn't, so I'm assuming that it will be given to us tomorrow morning. Am I correct? Well either way, I think I, as the rest of you should, am going to practice some trigonometric graphing problems to prepare for tomorrow.
Good night everybody! Study for tomorrow's graphing portion of the circular functions test (at least I think it's still on for tomorrow) and I wish everyone good luck!

BOBMANIA

Hey everybody! This is my first ever post on this blog ever, or any blog for that matter, so I don't exactly know how to start. I guess I could start by saying how much I love this class so far, and then progress into what we learned in the class so far, which is the almighty unit circle.

Throughout our unit of the unit circle, I haven't quite encountered any insurmountable roadblocks in terms of difficulty comprehending anything. Overall I understood everything, except for those classes which I missed (for worthy reasons of course), but I did manage to catch up thanks to the oh so helpful podcast. Homework was another big thing, and I really am glad that for a change, I am fully completing it and am not procrastinating in doing so. I really hope the rest of you have been doing your homework too. I think it really helps with understanding each aspect of the concept further and helps with subtlizing each problem so that new problems of a similar nature won't present as much tribulation as it could have. And seeing as our test is coming up tomorrow, I think the homework Mr. K has given us can provide us with much practice solving problems that will probably be on the test. I think that I, as I advise the rest of the class to do, am going to utilize these questions to us as help to prepare for the gruesome fear-inducing test tomorrow.

But try not to forget all the mnemonics and tips Mr. K has given us to help us remember certain segments of the unit, such as:

• The exact values of the functions sin and cos found on the unit circle follow the pattern of 1, √2, √3 (all of which are over 2). Visualize where the angle is found on the unit circle, then use this to help imagine what the exact co-ordinates of that point are.
• The exact values of the tan function (sinx/cosx) follows a pattern of 1/√3 ,1, √3. It ascends along this pattern as it travels away from the x-axis, and reaches undefined at the y-axis, then descends back down this pattern until it reaches the x-axis again. The same goes for underneath the x-axis, but remember that tan is negative in quadrants 2 and 4.
• F(x)=AsinB(x-C)+D F(x)=AcosB(x-C)+D
  Remember DABC to remember the order when graphing a trigonometric function.
  D -> A - > B -> C
  Start by visualizing or drawing in(using a dotted line) the D parameter, then A, subsequently determine B by using:
  B= 2Π /Period
  Use this value to scale the x-axis. Once all of those are done, C (the phase shift) will be used to show draw how much the graph is shifted horizontally.
• Think proportionally! It's a great way to think and a logical method to solving problems, and is accurate and simple as well.

Well, sorry if I made this BOB too long, I just got carried away. =/ Well, I guess I'll end this off here by saying that I really like this class, and I'm glad I'm understanding everything perfectly so far (despite what the recent quiz might show). So, I wish everyone good luck on their test tomorrow, and don't forget to study!