If only maths was as simple as this:
Tuesday, 11 October 2011
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)(
Range = { y > 1 or y < -1 }
Period = 2π
Vertical Asymptotes = x = (2kx+1)(π/2
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
Sunday, 9 October 2011
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. ~ =)
Sunday, 21 August 2011
~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
| 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)
| ||
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:
AND...that's how you sketch a graph!=D easy as ABC,isnt it? he
A polynomial function graph can be sketched using:
- x-intercepts
- the degree of the function
- sign of leading coefficient
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
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