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

• point has no dimensions, only location
• A line is one-dimensional
• plane is two dimensional
• 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:

.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.
Lesson Summary for April 17th, Trigonometry: Special Angles
The special angles are 0º, 30º, 45º, 60º and 90º which answers when using sine, cosine and tangent,  can be found and recalled through a method below.
How to recall
1. Start by writing down the angles 0º, 30º, 45º, 60º, 90º
2. Label the numbers like shown in the picture below from 0 to 4 respectively.
3. Square root all the numbers.
4. Divide all the numbers by 2 to get the answer for sine for each of the special angles.
5. Then inverse the order of the answers for the sine(special angle) to get the answer for the cosine(special angle).
6. Afterwards, divide the answers gotten for sine(special angle) by the answers gotten for cosine to get the answers for tan(special angle).

Lesson Summary 16 April 2013 Trigonometry

Well here it is~ Rules and stuff.

Part 1:

Solving Absolute Value Equations Graph Calculator

Case 7 : Exponential Graphs

Exponential graphs are constant to the power of variable. A simple model of this would be the classic lily pad covering the pond question. Assuming if the lily pad double its population every day and the students want to clean the lily pad out of the pond the day before the entire pond is covered with lily pads. In the end, the students have to clean the pond up the day half of the pond is covered. This growth of the lily pad, when plotting, forms an exponential graph.

Case 2: Linear Graphs

Linear graphs with coefficient 1 usually have a gradient, when the equation of the graph, y, is less than or more than 0. The exception is when y = 0, as the graph will be parallel to the x-axis. The gradient, when present, is always constant. There is no turning point or line of symmetry. Intersections with the x and y axis have a relationship with the gradient, where (y₁-y₂)/(x₁-x₂) = gradient.

Compiled by: Dylaine Ho (2), Esther Tiey (3) and Man Jun Jie (17)

Graph Sketch of Ax^3

Cubic function explorer : www.mathopenref.com/cubicexplorer.html

Description of the graphs:

When equation is y = Ax^3, there is a point of inflection. The point of inflection is the point where the gradient is zero, but there is no change in direction.
When the equation is y = x^3 +2x, there is no point of inflection.
When the equation is y = ax^3 + bx^2 + cx + d, there will be two turning point, maximum and minimum.

Case 1 : Linear Graphs where n = 0

Case 1 : y = ax^n , where n = 0

Linear graph where the value of a determines the gradient and c represents the y-intercept.

Since a^0 = 1 and a must be a real number, y = 1 + c .

Equation 1 : y=2x^0
=2(1)
=2

Equation 2 : y=-2x^0
=-2(1)
=-2

Equation 3 : y=0x^0
=0(1)
=0

The features of these graphs include a constant gradient and no x-intercept because the x is raised to power 0. Because there is no x-intercept, the graph is a horizontal straight line that cuts the y-axis at 2, -2 and 0 respectively for equations 1, 2 and 3.

There is also a lack of a turning point, since the gradient doesn't change.

Reciprocal Graphs - Jerome, Enoch, Kashyap

Reciprocal Graphs

y = 3/x:

The graph is split into two parts. Each part consists of 2 asymptotes, 1 towards the y-axis and 1 towards the x-axis. When y approaches +-∞, x approaches 0. When x approaches +-∞, y approaches 0. Each asymptote has a turning point, when the horizontal asymptote starts becoming a vertical asymptote and vice versa. There is a line of symmetry, which cuts through the 2 asymptotes.

y = 3/x + 4

This graph has about the same characteristics as the previous graph. Let us now overlay the two graphs to see the difference between them.

As it can be seen, the 2nd graph was "raised" by approximately 4 points. As such, we can see that any extra values added to the original equation has the effect of raising the graph.

Let us try another graph.

y = -1/x

This graph has approximately the same characteristics, though the parts of the graph lie in different regions.

y = -1/x - 4

The graph has the same characteristics, and it can be seen that it is "lowered" by 4. Now, we can also see that subtracting values also affects the graph.

Y = 0/X

y = 0/x is also y = 0, which is a straight line along the x-axis. At any point, y = 0.

y = 1/(x-2)

The graph is shifted slightly to the right, causing the bottom asymptote to intersect the y-axis. Otherwise, the graph has the same characteristics.

Let us over lay a line x = 2 on the graph.

As can be seen, the vertical asymptotes are towards the line x = 2. As y approaches +-∞, x approaches 2, but does not reach 2.

y = (- 2)(+ 3)

y = (- 2)² + 3

y = -x² - 2x - 1

y = -x² - 4

y = 3x² + 4

y = -x²

y = 3x²

They are all parabola (which a ≠ 0), except for y = 0x² (it is a linear)
If the coefficient of x² is 0, the graph will become a linear.
The graphs have both an increasing and decreasing gradient.
They have maximum or minimum turning point.
There is a line of symmetry parallel to the y axis.
The constant of the graph dictates the y-intercept.
The coefficient of x² dictates the shape of the graph.

by: Aziel, Jemaimah and Kai Chek

Case 8: Logarithmic Graphs

These graphs have asymptotes and never intersect the y-axis and have 1 intersection with the x-axis. Log graphs are functions. For every value of x, there is only one value of y. Log graphs are reflections of exponential graphs along the line y = x, and they are NOT exponential. No turning point. The line of symmetry is always diagonal to the axises.
a ≠ 0. If a > 0, curve will go upwards. If a < 0, curve will go downwards.