AM and EM Assessment Book (GCE O format)

To assist students in their revision and preparation for GCE 'O' Level, the Mathematics Department has made arrangement with the bookshop to order the following 2 books for the students.

The information is as follows:

• Additional Mathematics by topic $7.00
• Mathematics by topic $5.50

Both will include solution booklet

Please make arrangement with your Math teacher on the procedures for purchases.

Coordinate Geometry - about a SLOPE

The Slope of A Line source: http://math.about.com

Identify a few concepts on (i) gradient properties (ii) equation of a line (iii) trigonometry and gradient and (iv) collinearity from the article below. Post as comment.

When the slope of the line is 0, you know that the line is horizontal and you know it's a vertical line when the slope of a line is undefined. In the Figures below, the subscripts on point A, B and C indicate the fact that there are three points on the line. The change in y whether up or down is divided by the change in x going to the right, this is the 'rise over run' concept. y = mx + b is the equation that represents the line and the slope of the line with respect to the x-axis which is given by tan q = m. This is the slope-intercept form of the equation of a line. (m for slope? Seems to be the standard!) When the slope passes through a point A(x1, y1) then y1 = mx1 + b or with subtraction  y - y1 = m (x - x1) You now have the slope-point form of the equation of a line. You can also express the slope of a line with the coordinates of points on the line. For instance, in the above figure, A(x, y) and B(s, y) are on the line y= mx + b : m = tan q =  therefore, you can use the following for the equation of the line AB: The equations of lines with slope 2 through the points would be: For (-2,1) the equation would be: 2x - y + 5 = 0. For (-1, -1) the equation would be: 2x - y + 1 = 0

Coordinate Geometry - Line - Point - Ray

GCE O Elementary Mathematics syllabus for COORDINATE GEOMETRY



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LINE - POINT - RAY
source: http://www.mathsisfun.com

In geometry a line:

• is straight (no curves),
• has no thickness, and
• extends in both directions without end (infinitely).
• A line has no ends

 What is the main difference(s) between (a) a line, (b) a segment and (c) a ray ?

Point, Line, Plane and Solid A point has no dimensions, only location A line is one-dimensional A plane is two dimensional A solid is three-dimensional

Coordinate Geometry - Cartesian Plane (review lower secondary work)

By Mr Johari

 Watch the following video and answer the questions that follow:

Questions: (post your responses as comments)

.1 What is the significant of Cartesian plane?
.2 Give an example of an ordered pairs and explain all the terms used.
.3 What is the significant of the 4 quadrants in the Cartesian plane?
.4 Give an example of the application of Cartesian plane. (some examples are shown below)

16/4 MATH SCRIBE POST – Justin

~(•O•)~ E Math: Trigonometry Revision

CONTENT:
1. Trigonometric Functions
2. Trigonometric Rules
3. Special Angles!

*Do note that most, if not all, of these rules apply to only right angled triangles unless stated otherwise.

1. Trigonometric Functions
REVISION: Big Legged Lady
Toa (Big) – T for tangent, and O is before A.
Therefore: tan(x) = O/A
Cah (Leg) – C for cosine, and A is before H
Therefore: cos(x) = A/H
Soh (Lady/Auntie) – S for sine, and O is before H.
Therefore: sin(x) = O/H

sin^2(x) = (sin(x))^2  <–– THIS RULE applies to all trigonometric functions


2. Trigonometric Rules

1. tan(x) = sin(x) ÷ cos(x)

2. sin(90 - x) = cos(x)

cos(90 - x) = sin(x)

tan(90 - x) = 1/(tan(x)) 

3. sin^2(x) + cos^2(x) = 1

4. sin(180 - x) = sin(x)

cos(180 - x) = -cos(x)

tan(180 - x) = -tan(x)

5. PYTHAGORAS THEOREM 

a^2 + b^2 = c^2

6. Using the Z-rule 

angle of depression = angle of elevation 

3. Special Angles

Here is a table for your viewing pleasure: 

On another note you could use two triangles, 

1. Sides 1, 2, √3. 

2. Isosceles Triangle of 1, 2, and √2. 

Trigonometry - Review of Sec 2 work

Please complete the following questions. Post your responses as comment.

What is Absolute Value?

Part 1:

Absolute Value & The Opposite of a Number

Nomenclature

Part 2:

Solving Absolute Value Equations

Part 3:

Absolute value function / graphicals

Part 4:

Solving Absolute Value Equations Graph Calculator

Performance Task 1 AE

GRAPH - LINEAR

LINEAR GRAPH : y = ax^0 or y = ax^1

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Case 1: when n =0

y= 2x^0

y=-2x^0

The points are all linear which has a constant gradient.

The gradient is dependent on the value of a. if a >0 then gradient is positive, if a < 0 then the gradient is negative. However if a =0 gradient is zero.

Absence of turning point and line of symmetry 

The y intercept is dependent on what the value of c is.

Case 2:when n=1

y= 3/4x^1+4

y=3/4x^1

y=-3/4x^1

y=-3/4x^1-4

The line is also linear.

The gradient is dependent on the value of a. if a >0 then gradient is positive, if a < 0 then the gradient is negative. However if a =0 gradient is zero.

The y intercept is dependent on b in the y=ax+b equation.

Proudly brought to you

GRAPH - CUBIC

CUBIC GRAPH : y = ax^3 

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Characteristics:
It has varying gradients.
Turing points can range from 0 to 2.
When the coefficient of B and C is 0, there will only be 1

GRAPH - SQUARE RECIPROCAL

SQUARE RECIPROCAL GRAPH : y = ax^(-2) 

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y = ax^n , n = -2

y=3x^-2

















y= -1/x^2

General characteristics of the graphs :
They have a line of symmetry at the y-axis. They are reflections of each other. They are asymptotes that never cross the x or y axis.

- Mason, Shaun and Kai En

GRAPH - QUADRATIC

QUADRATIC GRAPH : y = ax^2

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They are all parabolas (except for y=0x^2).
The graphs have both an increasing and decreasing gradient
It has a maximum or minimum point,  the points is the turning point of the graph
There is a line of symmetry parallel to the y axis

Summary of Lesson (2-4-13) (Ryan)

LESSON 1

Addition of Case 7 and 8

Case 7

y=3e^x

y=-e^x

y=e^0 (This is not exponential)
y=3e^x + 4
y= -e^x -4
y=e^-x

y = | x |

if x>0, y=x
if x<0, y=-x
if x=0, y=0

Case 8

y= 3lgx

y= -lgx (Reflection about the y-axis)

y= lnx

y= 3lgx + 4

y= -lgx - 4

y= log2 x

Case 2a

y= |-3/4x|

or

y=abs (-3/4 x)

y=|3/4x|

or

y=abs (3/4 x)

Hint: For EOY Mr Johari can convert normal graph problems to a trigonometry problem or add in different topics

Let y = f(x)

Case 1: y = -f(x)

this would be a reflection about the x-axis

Case 2: y = f(x) + c

this would be a vertical shift (moving up and down the graph)

Case 3: y = f(-x)

this would be a reflection about the y-axis

case 4: y = absolute f(x)

when x > 0 , y = f(x)

when x < 0 , y = -f(x)


LESSON 2

Prove that

1                              1
_______     +  _________ = 1
log a ab           log b ab

Answer (Courtesy of Justin) :

log a a               log b b
_______     +  _________ =  log ab a (change to base a )+ log ab b = log ab ab = 1
log a ab           log b ab

GRAPH - LOGARITHMIC

LOGARITHMIC GRAPH : y = aLg (x)

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Description:

Y-axis asymptote

y=mlg(x), when m increases, gradient increases.

y=mlg(x)+c, c is where the graph intercepts with x=1 

Done by Justin, Jemima and Crystal

GRAPH - RECIPROCAL

RECIPROCAL GRAPH : y = ax^(-1) 

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 The graph shows two curves in positive-positive region and negative-negative region. Graph also always has two asymptotes, one vertical and one horizontal.

Adding to the constant shifts the graph up and down. It also shifts the horizontal asymptote. Ie. The constant is the horizontal asymptote.

Negative coefficient of the value of x rotates the graph 90 degrees about the y axis. This is because y values is negative when it should have been positive.

 

Subtracting from the constant causes the graph to shift downwards

Subtracting a value from the x within the fraction translates the graph to the left by negative of this value. Also shifts vertical asymptote to this value.

GRAPH - EXPONENTIAL

EXPONENTIAL GRAPH : y = ae^x

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Exponential graph:

THe equation of the graph is a (e^ (bx)) +c
If b=0, the graph would be a horizontal line and intersect the y intercept at a+c
if b<0, the graph would have a decreasing gradient.
if b>0, the graph would have a increasing gradient.

The value of a would affect the position of the graph:
if a>0, the graph would be above the x axis and y intercept at a+c
if a<0, the graph would be below the x axis and y intercept at a+c

Note: For exponential graphs, c is NOT the y intercept.