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Math Dictionary Notes: Transformations

Here they are ...





To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

Scribe Post: Insanely Crazy Long Question And Insanely Crazy Pre-Test

Hi! This is Danny, and I'm your scribe for today. First of all I'd like to apologize in advance for having this up so late, I had vocal jazz and work after school and only had the time to start my post now (which is 11:00 pm but will be some other time when I'm done). I'd like to make a twist to this scribe post. Instead of just writing normally what we did during our classes today, I'd like to make it into a story. So with no further ado here I present my story.....

Captain K and his crew of math genius' continued their travel through The Land Of Pre-Calculus in search of the Legendary Holy Golden Credit!! When they came across an obstacle in their way of eternal bliss.....A SEA PORT! This sea port was the only way they were able to cross the sea without being swept away by its' tides of sinusoidal functions.

"What do we do now Captain K?" said Craigmyer.
"Look there's a sign near by, let's see what it says" said Melk
"Trying to get across the sea port? It's not going to be as easy as walking across, first you must solve these problems. Then you shall see what happens next..." read the crew.

At this sea port, the depth of the water, h meter, at time, t hours, during a certain day is given by this formula:

h(t) = 1.8sin[2Π (t-4.00)/12.4] + 3.1

a) State the: (i) period (ii) amplitude (iii) phase shift
b) What is the maximum depth of the water? When does it occur?
c) Determine the depth of the water at 5:00 am and at 12:00 noon.
d) Determine one time when the water is 2.25 meters deep.

"WHAT ARE WE SUPPOSE TO DO!?!?!" yells Danny Boy frantically
"No need to worry Bertman is here! We just have to sketch a graph of this sea and we'll get our answers from there!" says Bertman speaking in a voice strangely close to that of Batman's.
"Easier said then done, but I'll give it a try" says Mr. Siwwy.


"What a minute, we don't even need the graph for the first part of the question!" states Kasiaw Cole (get it, Keyshia Cole, the singer?). She continues, "to find the period don't you have to go 2Π/B? But that fraction, that makes it so much harder then usual."
"I got it!" says Vinsanity "you just got to multiple it all out first, then this will give you the answer for the period"

Vinsanity shows his work to the rest of the crew:

2Π X (t-4)/12.4
= 2Π/1 X (t-4)/12.4
= 2Π (t-4)/12.4
= 2Π/12.4 X (t-4)

B = 2Π/period
B = 2Π/12.4
PERIOD = 12.4

"Nice! I got the amplitude and phase shift!" says Dr. Grey-M "By looking at the formula we get the amplitude to be 1.8 and the phase shift to be 4.00."
"Awesome, on to the next questions!" says Danny Boy

NOTE: This is where my story kind of just stops for a bit and I just post the answers to the remaining questions b), c), d). This is taking way longer then I expected and it's already cutting into my sleep time, forget about my studying time =S. Continuing......

b) The maximum depth of the water is 4.9 meters. You get this by taking your sinusoidal axis and adding the amplitude to it:

3.1 + 1.8 = 4.9 meters

To find when it occurs you plug in 4.9 into the formula:

4.9 = 1.8sin[2Π (t-4.00)/12.4] + 3.1
1.8 = 1.8sin[2Π (t-4.00)/12.4]
1 = sin[2Π (t-4.00)/12.4]
Π/2 = [2Π (t-4.00)/12.4]
(12.4)(Π/2) = 2Π (t-4.00)
(12.4)(Π/2) / (2Π) = t-4.00
3.1 + 4.00 = t
7.1 = t
7:06 am = t

c) For each time, you just plug them into the formula to find your depth at that time:

h(5 am) = 1.8sin[2Π (5-4)/12.4] + 3.1
h(5 am) = 3.9735 m

h(12 noon) = 1.8sin[2Π (12-4)/12.4] + 3.1
h(12 noon) = 1.6766 m

NOTE: If you were given a time value of 5:00 pm this would be equal to 17 hours, and that would be the value you plug into the formula.

d) To determine one time when the depth of the water is 2.25 meters deep, we just plug 2.25 meters into the formula:

2.25 = 1.8sin[2Π (t-4)/12.4] + 3.1
Let Θ = [2Π (t-4)/12.4]
2.25 = 1.8sinΘ + 3.1
-0.85 = 1.8sinΘ
-0.4722 = sinΘ
-0.4918 = Θ
-0.4918 = [2Π (t-4)/12.4]
(12.4)(-0.4918) = 2Π (t-4)
(12.4)(-0.4918) / 2Π = t-4
-0.9706 + 4 = t
3.0294 = t
3:02 am

BACK TO THE STORY:

"Wow, that took a long time but we finally figured it out" says Richard S.
Just then they saw a magical bridge appear out of no where allowing them to cross the sea safely and on towards their next challenge while in search of....the Legendary Holy Golden Credit!!
"Hey how did you get here before we did Samus?" says Dino
"Didn't you guys see that perfectly safe bridge right down over there?" Samus points towards the bridge.
"WHY DIDN'T YOU TELL US!!??!!??" the crew screams at Samus.
"You never asked? DUH!" says Samus

PART TWO OF THE STORY: Transformator And His Evil Pre-Test..... NOTE: I'm going to use past colours again, there's too many people in our class and not enough colours =S

So after getting past the sea port and it's puzzling questions, our adventurers ventured onward along the "yellow brick road" not knowing that they were about to encounter the EVIL TRANSFORMATOR DUN DUN DUN!

"Why are we walking on this yellow road anyways?" says KaDeeM AbDul Ali JaBBar confused.
"Captain K told us to, he said this road would lead us to......the Legendary Holy Golden Credit!!" says Tim_MATH_y
"But if it lead us to....the Legendary Holy Golden Credit!! I really wish that'd stop happening everytime we mention....the Legendar Holy Golden Credit!! Sigh, anyways, if it lead us to that, then wouldn't the road be gold instead of yellow? Guys? Guys?" says KaDeeM AbDul Ali JaBBar
He turns around only to see the whole crew staring at the biggest transformation ever seen on the face of The Land Of Pre-Calculus, TRANSFORMATOR!!!

"MUAHAHAHAHA!!!! I'm Transformator and I'm here to phase and vertically shift your functions to doom! Then when I'm done with that, I'll stretch and squash your amplitude until you scream for your MAMA!! MUAHAHAHA!!!" says the hideous Transformator
"Ewww, this thing is so ugly it makes me sick. What are we going to do? Stupid Transformator we just want to get the Golden Credit. Oh my gosh leave us alone." says Jeng-Lo
"Never! You're trespassing on my territory there's nothing you can do now to escape from my parameters A, B, C, and D!" says Transformator
"Alright then, I purpose a challenge to you Transformator. I bet our whole crew together can solve any transformation problem you throw at us. If we win then you let us go." says Sandy softly but bravely.
"And if you lose?" says Transformator
"Then we have to stay here forever allowing you to do whatever you want to us" says Sandy
"Alright then, you have a deal! Here's a Pre-test chalk full of challenging problems, lets see if you can get out of this one Captain K! MUAHAHAHA!" says Transformator in delight.

The crew recieves the test and starts to begin solving the questions:

1. f(x) = 2x² - 3, where x is greater than or equal to 0, then a function g that will have domain and range that are both different from those of function f is:
a) g(x) = f(-x)
b) g(x) = -f(x)
c) g(x) = f-¹(x)
d) g(x) = kf(x), k greater than 0

"Oh! The answer is C)!" says Aichelle the Incredible
"The answer wouldn't be A) because f(-x) will only result in a different domain, and it wouldn't be B) because -f(x) only results in a different range. For C), f inverse of (x) means your x coordinates would be your y coordinates, and your y coordinates would be your x coordinates therefore giving you a different domain and range!" continues Aichelle the Incredible
"Argh, that's correct, next question" says Transformator angered

2. The graph of a function f is a parabola opening upward, with its vertex on the x-axis. The graph of a new function g, where g(x) = 2f(x), will have:
a) the same domain and the same range as f
b) the same domain but a different range than f
c) a different domain but the same range as f
d) a different domain and a different range than f

"Jojo Rocks!" yells Jojo
"The answer has to be A)! Since it's a parabola, going 2f(x) wouldnt affect the range because the parabola would be going up both sides to infinity so you could trash B) and D) already. It can't be C) either because the vertex is on the x-axis, therefore the domain will be left unchanged so the answer has to be A)!" continues Jojo
"ARGH! Right again! Next one, you won't get this!" grumbles Transformator

3. Given the graph of f(x) below, sketch 1/f(x) in the space provided:

"This one was easy no challenge to it." says Johnny Johnson
"First things first find your invariant points, on this graph there's two. One at (-1,-1) and the other at (1,1). Then find your asymtotes, it just so happens that the asymtotes are the x and y-axis. After that just remember Dr.Suesus' version of math when drawing the actually reciprocal graph in (smallering and biggering). Also remember to have arrows where neccesary and dots that shows an end on the graph where necessary. In this case the graph ends at (-5,-1) but continues on going in the other direction." states Johnny Johnson in what seems like an eternity to explain.
"NO!! RIGHT AGAIN!! THERE'S NO WAY YOU'LL GET THESE LAST TWO!" screams Transformator

4. Given f(x) = cube√3x² -4, write the equation for its inverse f-¹(x).

"They just keep getting easier and easier! I'll show you the work, no sweat." says Bond, Robert Bond



















"THIS CAN'T BE HAPPENING!! THIS LAST QUESTION IS IMPOSSIBLE, NOT EVEN I WAS ABLE TO SOLVE IT!!!" yells Transformator, now shaking the ground with his mighty power.

And this is where I'm going to stop for tonight. Sorry guys I put out more then I could chew, but this final question will be up by tomorrow! Good luck to all on the test in the AFTERNOON. Scribe as you all know already is Mark. =D

Blogging on BOBbing

Well, this unit about transformations was relatively easy for me. I dont know but I think that im good at imagining things so graphing stuff was just natural to me. I had no trouble with the other materials except this last material about modelling. I hope that i will get at least get a decent mark for this test. Good luck everyone and study hard!!REMEMBER: trust your instincts..

Billy-Joe BOB

Well Howdy there y'all. This is my BOB for the unit of Transformations. Luckily Mr. K has become a whiz at using the smart board so classes are almost back at the normal pace (save for the few uncontrollable technical difficulties). Concerning Transformations my two main obstacles to get over were to always watch the sign of A as if the number given/found is negative, it will be written as a postive within the equation, and if the number given/found is positive, it will be written as a negative within the equation. I always caught myself rushing through that part without thinking it through carefully and drawing an incorrect graph due to my carelessness. The second obstacle deals with the problem solving. I kept trying to do too many things at once instead of keeping it simple and going through each of the steps thourghly. This caused massive confusion and many minor errors throughout my work that I should have easily spotted and fixed. For the next unit I have to remind myself to slow it down and that everyone learns at their own speed, so not to rush it because I'm not helping myself. I'm definetly nervous for tomorrow's test because I missed the first introduction to this unit but I feel confident enough that with the help my classmates provided me I should make it through. So, GOODLUCK EVERYONE! :D

BOB

Hi again to everyone! This unit overall was pretty easy and I enjoyed it. I liked every part but one. The last part of the unit was a tough concept to grasp. It delt with problem solving and I really don't find it all that easy. If we only had more time to work on these types of problems, then maybe it would be crystal clear. For now I will try my best on the test to get this type of question right and I hope everyone else does too. Good luck to everyone and study hard.

BOBby Valentino

Hey fellas, well it seems like this unit on Transformations went by quicker than the last one. It was a hard one for me, harder than the last unit I guess. I really had trouble catching on to the graphing, modeling and whatever else we did in class. lawl ('lol' pronounced) Maybe next unit I should ask for help because this is really confusing. All in all, I think this a fun class to be in, even if I'm not doing so good in it. Fun times with the SmartBoard and fun times with my fellow peers. You guys are a smart bunch, I am still overwhelmed by the knowledge in the classroom. I wish I could be smrt. One last advice to everyone..........STRETCHES before TRANSLATIONS and don't take steroids.

BOB II

Hello again, what can I say, this unit on Transformations was some what tough for me, especially Trigonometric Modeling toward the end of the unit. Trigonometric Modeling gave me a little trouble, because I was over concentrating and trying to do most of the work required in my head and not down on paper, where I made most of my mistakes. I also had a little trouble in the beginning when graphing functions, although that was quickly corrected when we were told to do stretches before translations ( remember). All in all, the unit was fun, especially the graphing of absolute and inverse functions, because they were not as hard as some of the other graphs we had done. Another interesting subject in the Transformations unit was the even and odd functions which were fun to do. As I was writing this BOB so much has been refreshed in my mind, time to go study!

P.S.: Remember to study and revise your notes, and good luck on tomorrow’s test.

Today's Slides: March 7

Here they are ...





Unfortunately, the student generated text from the first two slides was lost. I must have deleted it accidentally ... oops, sorry about that. ;-)

The last slide has an extra problem similar to the one we did in class today; just a wee bit extra review for ya. ;-)

To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

BOBby Brown

So we are wrapping up our second unit on TRANSFORMATIONS. Overall it was bad because it was all pretty much graphing, and as some of yous might know i despise anything thats has to do with graphing. Doing alot of this graphing though is helping furthure my understanding of the whole concept.

The first half of the unit was pretty straight forward learning such things as ƒ(x) = aƒ(bx) controls the vertical stretch and compressions of a function. Also learning how to graph reflections and inverse functions: (x,y) (y,x). And even and odd functions. The only thing i had a alittle trouble with was the whole STRETCHES BEFORE TRANSLATIONS &or MULTIPLYING BEFORE ADDING/SUBTRACTING, but i think i got it now :).

Now the reciprocal functions was a tab bit more difficult, and learning some new things like invariant points and absolute value functions.
And everything was fine and fanastic, until....we got to Trigonometric Modeling which is pretty much putting everything we learned all together into to one big, long question that takes like 30 minutes to complete, having some trouble but i will eventually get the hang of it. Could really use another extra day, but overall this unit had its ups and downs.

!!!Good Luck Everyone!!!

Dodge BOB Mania

Wow.. This unit was a very interesting unit. It was thorough in the sense that I feel that I already learnt most of this unit from previous years of mathematics. So anyways.. These classes were really indifferent. The class was progressing at an astounding rate of mathematical learning and contributions. All because of the smart board. This has shown that our class is getting smarter because of the smart board every day. I feel that I have achieved a different level of understanding due to all of the technologies involved within the class.

The transformations unit for me was just another understanding of another unit. The unit in general is jammed pack with operations that I understand and can actually remember in this class. The unit in general is split up into a few different sections of graphing mathematically and periodically. From graphing equations to finding equations. Mr. K has jam packed this unit like a review for me and the class has contributed to my learning in general.

The classes’ scribes have attributed to my learning in this unit very thoroughly. The scribes have shown that there has been a steady progression of new technologies from flickr to slideshare. This has shown that our class has given a thought into how we learn in the class and there seems a wanting on us students teaching the class ourselves online. This is really cool, because not only do we learn but can also teach other generations. For example EDDIE who is an awesome grade 5 student reading at a grade 10 level.

What's confusing me the most in the unit is graphing, however graphing is more or less a nutshell of review of knowledge that has to be un hurdled but can not due to the absence of our less prominent technologies like the Math Dictionary which some classmates (including me) have been waiting forever to figure out what comes next into the math dictionary in order to install new terms and attributes to our mathematical lives.

That's all I got to say about that. GOOD LUCK ON THE TEST GUYS!!! AND HAVE FUN!!!

BOB

Hi everyone! Our unit on Transformations is almost done. We are now getting ready for the 2nd test. This unit has been really interesting. We learned a lot of things on how to shift, stretch and/or compress a function graph. We learned about the Standard Form, f(x) = Af(Bx-C) + D. Its a bit similar to the standard form of Trig. Functions, but the values work in a different way.

A affects the graph by stretching (A > 0), or compressing (0< A < 1) its y coordinate.
B affects the graph by stretching (0< B <1), or compressing (0 < B) its x coordinate.
C affects the graph by shifting it to the left (0>C), or shifting it to the right (C>0). Watch for the sign of C because the standard form changes the sign of C.
D affects the graph by shifting it to the left (0>C), or shifting it to the right (C>0).

We also looked at the Reciprocal Functions. We learned that the root of the original function, f(x), is the vertical asymptote of the reciprocal function. The invariant points, -1 & 1, are both in the function and the reciprocal function since the reciprocal of 1 & -1 are themselves. We learned how to graph the reciprocal function by doing the "Smallering & Biggering" Game.

We looked at The Absolute Functions. The first step is to graph the original function. Then any part of the graph below the x-axis must be reflected along the positive side of the y-axis.

Finally, we looked at Trigonometric Modeling. We're not quite familiar with the material yet, that's why we are continuing with our discussion.

I hope everyone will do well on the next test. ^_^

Simple Bob

Most units are the same to me; they usually include a hill that you must climb over to achieve your goal. An example of this is like the trig problem solving. It's hard to think about all at once, especially by yourself, but quite easy to understand when the class or Mr. K is going over the question or working as a group. All of this narrows down to the solution: using your new found knowledge to practice independently. I think that is one of my weak points.. not practicing enough.

Another thing that I think would help a lot are math dictionary notes. They are concentrated knowledge all in a notebook. It's easy to access and would aid any misunderstandings greatly. I'm personally an independent learner, with the occasional team efforts. My strong suit is usually by myself.

I am not confident enough to say that I'm ready for the test. However, since we still have time to practice and we have a pretest before the actual test, I'm sure it's enough to get me over that hill.

Good luck to you all on the test and pre-test! Stretches before Translations!

Transformations Review Homework

As promised, here it is ... remember, the Pre-Test is tomorrow afternoon ...



And here are the answers ...

Bob v2

My second blogging on blogging for this class. I've done many in the past yet it never gets old to me. Hmm, what interesting tales shall I write to entertain you this time? Just kidding, it's all about our current unit that we've been studying for the past week and a half. Took me about 10 minutes looking through my notes looking for the name of the unit and its topics and I finally found it. It's TRANSFORMATIONS!!! This unit flew by pretty fast. So many funny moments on the smart board too. Anyways overall I think I get everything that's been going on. I'm still slow on these new graphing problems that we've been doing for the past few days but hopefully I'll catch on quick.

REMEMBER : Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations, Stretches before Translations

I doubt I can be more explicit than that... STRETCHES BEFORE TRANSLATIONS! Peoooooooow

BOB 2

This unit I've had some trouble with. I understand a majority of what needs to be done but these problem solving questions are pulling me down. I'm sure a lot of people feel the same about them. It's not that hard, i just happen to get confused along the way most of the time. But working on it in class is helping me understand how i do these things. I don't have much else to say, but good luck everyone on the pre test and test! ..

>Bob<

Well I must say that this unit has been a smashing good time. That smart board is just a riot. Though I must say that with Mr. K's learning curve on that smart accompanied with the interruptions it seemed to me that we did have some wasted classes. But now that the curve has smoothed out a bit our learning is faster than ever as put our answers up, check and correct faster than ever and more visually than ever. So that leads me to my feelings about the unit and I feel pretty secure about it going into the test. One thing that I drilled into my mind after that quiz we had though is STRETCHES BEFORE TRANSLATIONS, STRETCHES BEFORE TRANSLATIONS, STRETCHES BEFORE TRANSLATIONS and if you are trying to find the original graph from some transformed version do it in the reverse order, translations then stretches. Now after doing a couple of the trigonometry problem solving questions I feel fairly confident about those as well. I wish everyone luck on the upcoming test!

BOB - Transformations

"The time has come," the Walrus said, "to speak of many things..." ...but mainly our most recent unit, TRANSFORMATIONS. It seems like it was only two weeks ago that I was writing the B.O.B. for Circular Functions... ...Oh. Anyway, the unit just seemed to fly by, considering we missed a grand total of 4.5 (four and a half) classes in the short two week span.

I did not find this unit particularly challenging until Friday when Mr. K. introduced PROBLEM SOLVING. This is because the unit was mainly a combination of last year's Functions unit and our previous Circular Functions unit. However, once we reached the problem solving section of the unit I was surprised by how difficult it actually was. It only uses the concepts that we just learned, but it incorporates them in a way where we must logically make a "plan" or order as to how we will solve the problem. I think that is the hardest part, figuring out what concepts to use where and how to fit them into the question. However I think with some help from the review sheet, exercises, pretest, and a little more class time, we all have a chance to do excellent on the upcoming unit test. Good Luck to all, and remember...

Transformations

Good afternoon everyone, I'm Sandy and I'm scribe for today. During today's class we started off by completing the question Mr. K asked us to do yesterday. Just in case you didn't write it down or you're too lazy to scroll down and look for the slide, this was the question:

The pedals on a bicycle have a maximum height of 30cm above the ground and a minimum distance of 8cm above the ground. Jeng pedals at a rate of 20 cycles per minute.
a) What is the period, in seconds for this function?
b) At t = 0, Jeng's right foot is closest to the ground.
i) Write 2 equations that represent the height of her right foot above the ground; 1 sine; 1 cosine.
ii) For how long per cycle is Jeng's right foot 20cm, or higher, above the ground?


To start of, part A of the question was fairly straight forward. To find the period, we take 1 minute and convert it to seconds which is 60 seconds. And to find out how long each cycle took, we divided the time, 60 seconds, by the number of cycles per 60 seconds, which is 20 cycles.
60 seconds
20 cycles
And this results in,
3 seconds per cycle.

For part B, i, we're asked to make up two equations, one for sine, and one for cosine. For some people it might be easy to make the equations, but sometimes it's easier to determine the equations for the function by making a graph of it first.

NOTE: This graph has some errors. For one, when drawing a graph don't forget to label your x and y axis and draw the arrows. For two, don't forget to add arrow(s) or dot(s) to the end of the function depending on the context of the original questions. An example can be time, because time does not go backwards there would be a dot at the point where the function touches the y-axis and an arrow on the other end indicating that the graph continues with time. For three, you should always number at least the 4 points (for your max, min, and where it touches the sinusoidal axis). Usually it's labeled as π/2, π, 3π/2 and 2π. In this question we should label it 0.75, 1.5, 2.25 and 3.

From the graph, it's a lot easier to determine the equations that go with it.
Making a list also helps come up with the equations.
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It's usually easier to do it in the DABC order. So starting with D, we can easily find it by finding the middle between the max and min value. In this case it's 19. Since the sign is always positive for D, we can put 19 beside D for both sine and cosine. For A, we simply find the distance from either the max or min, depending on what you prefer, to the sinusoidal axis. For this question A is 11. The sign for A we don't know yet, but we'll come back to that afterwards. The period, which we calculated earlier (3), is the same for both sine and cosine so we can place that where B is. For C, we're looking for a phase shift. For cosine we know that the max or min value starts at the y-axis.. therefore C for cosine is 0. Knowing this, we can figure out the sign of A, which in this case is negative (-) because cosine is starting at it's min. For sine, we know that there has been a phase shift of 0.75 and it's moved to the right so C for sine is +0.75. We can tell from the graph that A for sine is positive so parameter A is +11.

Part B, ii, we can do on our calculator. First we take the equation we made and put it into Y₁. Then for Y₂ we put in a 20.

Now that we've done that, we find the intersection of Y₂ and the function -11cos((2π/3)x)+19.

The first intersection was 0.7935 and the second intersection was 2.2065. To find the amount of time that Jeng's foot is 20cm or higher above the ground we subtract 0.7935 from 2.2065 and we end up with 1.4131. 1.4131 is the amount of time that Jeng's foot is 20cm or higher above the ground. Wasn't that easy?

Well we started another question in class today, but I'll leave that for Danny to do tomorrow since we're not even close to being finished it. Homework will be posted on the blog if it hasn't already and during tomorrow's morning class we'll be completing the question we started today. Hope you enjoy the rest of early dismissal!. (: Tootles.

Today's Slides: March 6

Here they are ...





To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.