~ THE END ~

I guess that's all for chapter5! Im finally done with my blog! HOOORAYY=)) but remember guys, keep on practising to be excellent in this chapter.. afterall, practise makes permanent!



5.5 Making Connections and Instantaneous Rate of Change

The instantaneous rates of change of a sinusoidal function follow a sinusoidal pattern, and usually undrgoes transformation when modelling real-world processes.

Here's a simple video illustrating this subtopic :) Enjoy!=D

5.4 Solve Trigonometric Equations

1.Trigonometric equations can be solved in two ways:

• algebraically by hand
• graphically with technology

2. There are often multiple solutions. So, be sure to find all solutions that lie in the domain of interest.
3. Quadratic trigonometric ewuations can usually be solved by factoring.

Here are some interesting videos on solving trigonometric equations:) It's very beneficial..hope you guys gain the best of it! Have fun watching!:)

Solving Algebraically

Solving Graphically

LOVEEE MATHS! hehe:)

General Idea of 5.3

• The transformation of a sine/ cosine function f(x) to g(x) has the general form g(x) = a f [k(x-d)] + c, where |a| is the amplitude, d is the phase shift, and c is the vertical translation.
• The period of the transfored function is given by

5.3 Sinusoidal Functions of the form f(x) = a sin [ k(x-d) ] + c & f(x) = a cos [ k(x-d) ] + c

  f(x) = a sin [ k(x-d) ] + c 
  f(x) = a cos [ k(x-d) ] + c

a = amplitude
k-value = 2 / Period
d = phase shift to left / right
c = vertical translation upwards / downwards

EG:

y = sin x

y = 5 sin x ( where An =5)

y = sin x + 1

Maths Jokes (hahaha!! :P)

If only maths was as simple as this:

5.2 Graphs of Reciprocal Trigonometric Functions

Domain = {x Є R | x ≠ kx } , whereby k is an integer.

Range = { y > 1 or y < -1 }

Period = 2π

Vertical Asymptotes = x = kx , whereby k is an integer

Domain = {x Є R | x ≠ (2kx+1)(π/2) } , whereby k is an integer.

Range = { y > 1 or y < -1 }

Period = 2π

Vertical Asymptotes = x = (2kx+1)(π/2) ‘ whereby k is an integer

Domain = {x Є R | x ≠ kπ } , whereby k is an integer.

Range = { -1< y < 1 }

Period = π

Vertical Asymptotes = x = kx , whereby k is an integer

5.1 Graphs of Sine, Cosine, and Tangent Functions

here's an interesting song! haha:) it's quite entertaining! Well, that is the primary trigonometry identities: [SOH,CAH,TOA]
Sin x = Opposite / Hypotenuse
Cos x = Adjacent / Hypotenuse
Tan x = Opposite / Adjacent

• Basic graphs of trigonometric functions are:

Sine Graph

Cosine Graph

Tangent Graph

• Graphs y = sin x, y = cos x, y = tan x are periodic.

 

Original Graph	y = sin x	y = cos x	y = tan x
+ vertical translation v  Moves upward/downwards by c units	y = sin x + c	y = cos x + c
+ amplitude v  Amplitude increases/decreases by a factor	y = a sin x	y = a cos x	No amplitude because there is no max/min value
+ phase shift v  Shifts by d units to the left/right	y = sin (x-d)	y = cos (x-d)
+ period (k-value) v  Number of cycles in one period	y = sin kx	y = cos kx	π

Chapter 5 : TRIGONOMETRIC FUNCTIONS

Okay, now its time to update my blog again!:) This time, i'll be focussing on Chap5, trigonometric functions, one of the most interesting topics in advanced functions. I'm not being sarcastic here, really, trigo is indeed very interesting and mind-boggling. The satisfaction of solving a complex trigo question with a full-page answer is indescribable! Trust me, the feeling of accomplishment is the same as drinking a cup of hot chocolate with tiger biscuits, on a rainy day!:) it feels so g-o-o-d :) that is why I enjoy this chapter very much! Before I start off with the syllabus, let me enlighten you guys with some interesting maths facts, just as a warm up!=D

Did you know that??

1)      π=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 ...

2)      A sphere has two sides. However, there are one-sided surfaces.

3)      In a group of 23 people, at least two have the same birthday with the probability greater than 1/2

4)      Among all shapes with the same perimeter a circle has the largest area.

5)      Among all shapes with the same area circle has the shortest perimeter.

6)      As in philosophy, there are transcendental numbers

7)      As in the art, there are imaginary and surreal numbers

8)      You are wrong if you think Mathematics is not fun

9)      Trigonometry aside, Mathematics comprises fields like Game Theory, Braids Theory, Knot Theory and more

10)   The next sentence is true but you must not believe it.

11)   The previous sentence was false.

12)   One can cut a pie into 8 pieces with three movements.

13)   The only triangle with rational sides and angles is equilateral.

14)   0!=1

15)   At any given time in New York there live at least two people with the same number of hairs.

Life is All about Maths!

If people don't believe that mathematics is simple, it is only because they do not realize how complicated life is. ~ =)

~symmetry~ even?odd? odd even? even odd? @@!

So, now we're gonna look at how is symmetry represented in the equation of a polynomial function. There are so many things around us that are in symmetry. Here's few of them:

 The symmetry for a polynomial function can either be a line symmetry or a point symmetry. But the question is how do we determine which one is for which?

Even Function = LINE SYMMETRY about x=0 Odd Function = POINT SYMMETRY about the origin

Remember that not every even degree functions are even functions!! An even degree polynomial function may be an odd function!

Now, let us look at the table below :

How is symmetry represented in the equation of a polynomial function?
EVEN             y  =  x⁴- 8x² An even function always has line symmetric about the y-axis / at x = 0 HOW TO KNOW WHETHER y = x⁴-8x²  IS AN EVEN FUNCTION?? Let f(x) =  x⁴-8x² F(-x) =   (-x)⁴-8(-x)²          =  x⁴-8x² SINCE f(x) = f(-x), f(x) IS AN EVEN FUNCTION!	ODD 1)      y = x³-4x An odd function always has a point symmetry about the origin (0,0) HOW TO KNOW WHETHER y = x³-4x  IS AN ODD FUNCTION?? Let f(x) = x³-4x f(-x) = ( -x)³-4(-x)          = -x³+ 4x          = -(x³-4x) SINCE -f(x) = f(-x), f(x) IS AN ODD FUNCTION!

1.3 Equations And Graphs of Polynomial Functions

Okay now..you're walking down the road, and someone stops you and asks "how do you sketch a graph of a polynomial function?" Most probably you would think he's out of his mind, but try thinking of the question he asked... what are the factors you need to take into consideration when sketching a graph?
A polynomial function graph can be sketched using:

• x-intercepts
• the degree of the function
• sign of leading coefficient

Let's look at this example:

Given y = (x-1)(x+1)(x+3)

Degree	Leading Coefficient	End Behaviour	Zeros and x-intercept	y-intercepts
Each factor has one x. Their product is x³. The function is cubic (degree 3)	The product of all the x-coefficients is 1.	A cubic with a positive leading coefficient extends from Quadrant 1 to Quadrant 3.	The zeros are 1,-1 and -3. These are the x-intercepts.	The y-intercept is (0-1)(0+1)(0+3) = -3

  

Mark the intercepts. Since the order of each zero is 1, the graph changes sign at each x-intercepts. Beginning in quadrant 3, sketch the graph so that it passes up through x=-3 to the positive side of the x-axis, back down, through x=-1 to the negative side of the x-axis, through the y-intercept at y=-3, up through x=1, and upward in quadrant 1.

AND...that's how you sketch a graph!=D easy as ABC,isnt it? he



Graphic Calculator, the killer??

Later we'll be having a short quiz testing us on our calculator knowledge~oh-oh!!:O we'll people,i know we are all having the oh-shit-I'm-screwed feeling...but no worries! Using the calculator can be quite simple once you master the right techniques..just read up on page 512-519 from our textbook and you'll be all good=D trust me,i tried finding some video clips on tutorial for graphic calculator,and they were all boring and even more confusing! So yeah,I thought better not upload it and confuse you guys..hehe ohh,and remember how to find finite difference using calculator! That's like the very basic,and fun thing to do using calculator:) And do remember that,if you found out your graphs are missing or too small or too big, do not freak out (like I did)..its probably just your windows setting..just adjust it,and then you'll be able to get the right graphs=D

One,Two,Three... :)

Its 4am in the morning...as I was working on this blog, I found myself ending up thinking deep..about advanced functions,of course... Now things are so different,aren't they? Now,we are learning new terms like finite differences, leading coefficient,factorial,exponential,secant,instantaneous rate of change and more.. well,to some these are familiar terms, but for many,including myself, these are alien words,which takes time for us to understand:) i know we all will get the hang of it eventually.. but just think back, perhaps 15years back................................................5years...............3,2,1...when we were young.I  was 3years old back then..and numbers were just from 1-10, and nothing more than that..how foolish were we! Take a look at this video and enjoy :) haha release some stress :P

Quizzz....

Oh well,last Friday's quiz was not so bad after all,was it? At first i was freaking out cos i did NOT prepare well for it and started emo-ing in fb:( but then,to my surprise,it was not as tough as i expected..but it was not easy as well..haha it was an "okay okay lah" for me..the first page of the quiz was quite easy,and i started smiling but then..when I turned to the next page, *tadaaa* the bomb! The last question was kinda confusing:( the one about order 1 and order 2,with the graph required to be drawn...i think i screwed that particular questions:( i think most of us did..hmmm,can't say much about the quiz for now..we shall see when we get back our results! ohhh...i almost forgot..there's a continuation of the quiz this tuesday people! Remember Ms.Joanne said she'll test us on our calculator knowledge? so yeah,good luck for that! Its Sunday today,let us all play around with our calculator :P

Power Functions @#$%&*@??

Okayy...i know its been quite a while since we studied this...hmm,around two weeks ago,maybe? yeahh,so let's recap!:) Power functions is bacially the SIMPLEST type of polynomial functions,not so powerful after all :P So, we are given this long freaky continuous polynomial expression like a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ You guys dont have to memorize that, so chill!:) just do a simple self-check by asking these questions to yourself:
1)what's a and n in a function?
2)what are the characteristics of a graph when a function is even or odd :
-domain
-range
- x-intercept
- y-intercept
-symmetry = line/point
-end behaviour
3)the quadrants (this is important! always get them mixed up:/)

IF you know all these,then you are safe,for now at least:)

Advanced Functions,Love It ♥

Well, I've just started my blog :) so,let's see...hmmm,advanced functions~ well,so far the classes are great and Ms.Joanne is awesome! seriously,I have advanced functions like every single morning at 8am, and I look forward to it..it keeps me excited,in a way i guess..cos i LOVE maths!♥ hehe oh,but the part where there is homework everyday is not-so-good :/ hehe trust me,life can get pretty frustrated when you are awake at 2am,trying to solve a polynomial functions question and cant manage to get the answer  :( and the only reliable source you have (the answers at the back,of course) is sooo unreliable.. especially when you see things like "2d) answers may vary. " But otherwise..if you have the patience and passion towards this subject, you'll get through it and do just fine:)