Coordinate Geometry & Graphs

Coordinate Geometry & Graphs

​By : kasskassmaths

​General Mathematics Syllabus

Section #7 , Objectives ​6 - 15, 20 - 26

TABLE OF CONTENTS

​Graph Points to Note :

(a) the x axis is the horizontal axis & the y axis is the ​vertical axis.

(b) always draw the axes with a pencil and ruler & label them clearly.

(c) the both axes can have the same scale or different scales.​ (such as

1 cm = 1 , 2cm = 2 etc or 2 cm = 1, 4 cm = 2 etc)

(d) graph sheet : 1 small block = 1 cm x 1 cm and contains 25 very small blocks, 1 big block = 2 cm x 2cm and contains 100 very small blocks.

(e) ​when plotting points use x's.

(f) draw line graphs with a ruler, draw quadratic graphs free-handed​

(g)​

Lines

​Method 2 :

This method need ​the gradient and 1 point on the line in order to find c / the y intercept. They are subbed into y = mx + c, then c is made the subject through simplifying. (Note : this is done before stating final answer/ equation of the line.

Determine the equation of the straight line that contains the points A (1,8) and B(2,11).

​Plot the following and determine the equation of the line in form y = mx + c where m and c are values to be determined.​

​

​x	-2​	-1​	0​	1​	2​	3​
y​	​-11	-7​	-3	​1	5​	9​

​Method / Type 3

This method involves either

1. Points given to plot the line and then find the equation using the graph and / or table with points. OR

2. Given the graph of the line and then finding the equation using observation (y intercept) & points on the graph (gradient).

Method / Type 4​. This involves the relationship between 2 lines.

Vertical Lines

​

There are s​ome cases where the lines are vertical.

​Vertical lines are said to have an "undefined slope," as their slope / gradient appears to be some infinitely large, undefined value. That is, m = undefined. A vertical line has no y intercepts, therefore the c = 0.

The equation of a vertical line takes the form ​

x = k, where k is a real number.

E.g. for the line x = 3.

The points (3,2), (3,0), (3,-1) etc all lie on the line.​

​

Horizontal Lines

​There are s​ome cases where the lines

are horizontal.

Horizontal lines have no slope, therefore

the gradient is 0. That is, m = 0.

A horizontal line​, (in most cases)

has a y intercept and that y intercept

is the equation of the line.

The equation of a horizontal line​ has the

form y = ​c.

E.g. for the line y = 3

The points (1,3), (-1,3) etc all lie on the line.​

Intercepts

​The x intercept(s) is/are the values of x for which a graph cuts or intersects the x axis. At this point(s), the corresponding value(s) of y is/are 0. That is, at an x intercept, the coordinate is (k,0),where k is a real number. Graphically, this can be spotted easily. Algebraically, it is found by subbing y = 0 into the equation of the line. The y intercept(s) is/are the opposite. It has the coordinate (0,c), where c is real number. Algebraically it is found by subbing x = 0 into the equation of the line.

​

Magnitude / Length

​Midpoint

​E.g.Given points A (-2,-5) and B (4,1) on a line segment, where m is the midpoint of A & B, calculate a) the length of AB to 1 decimal place and b) the coordinates of m.

Quadratic Graphs

From Graph : y intercept ​= 3

Algebra: when x = 0, y = (0)² +4(0) + 3 = 3​

​From Graph: roots of the equation: x = -1 and x = -3

Algebra: Solve x²+ 4x + 3 = 0 through
factorizing or completing the square [1(x+2)² -1 = 0]​

​Drawing the Quadratic Graph

​x	-5​	-4​	-3​	-2​	-1​	0​	1​	​2	​3
x	25​	​16	9​	​4	​1	0​	​1	4​	9​
​+2x	-10​	-8​	-6​	-4​	-2​	0​	2​	8​	6​
-8​	-8​	-8​	-8​	-8​	-8​	-8​	-8​	-8​	-8​
y​	​7	0​	​-5	-8​	-9​	​-8	-5​	​0	7​

​Table 1

​Table 2

​x	​y
​-5	7​
-4​	0​
-3	-5​
-2​	-8​
-1​	-9​
0​	-8​
1​	-5​
2​	0​
3​	7​

​y intercept : -8

axis of symmetry : -1

roots : x = -4 & x = ​2

​

​

​Sketch of Graph showing

Minimum point (-1,-4)

Axis of symmetry : x = -1 ​

Roots : x = -3 & x = 1

Y intercept ​: -3

Using a Graph to solve

simultaneous equations (linear)

​x	y​
​-5	​-1
-4​	​0
-3​	1​
-2​	2​
-1​	3​
0​	4​
1​	5​
2​	6​
3​	7​
4​	8​
5​	9​

​x	y​
​-1	​-5
0​	-2​
1​	1​
2​	4​
3​	7​

​x	y​
​-1	​-5
0​	-2​
1​	1​
2​	4​
3​	7​

​x	y​
​-5	​-1
-4​	​0
-3​	1​
-2​	2​
-1​	3​
0​	4​
1​	5​
2​	6​
3​	7​
4​	8​
5​	9​

​Solution is

x = 3 and y = 7​

Using a Graph to solve simultaneous

equations (linear and non-linear)

​x	​-4	​-3	-2	-1​	0	​1	2​
y​	-1​	0​	​1	​2	​3	​4	​5

​x	-4​	-3​	-2​	-1​	0​	1​	​2
y​	​5	​0	​-3	​-4	​-3	​0	​5

Linear Inequalities (2 variables)

E.gs

​A shop sells x guitars and y violins.

Write and inequality to represent

(i) the no. of violins must be less than or equal to the number of guitars

(ii)If the cost of 1 guitar is 150, the cost of 1 violin is 300 and the maximum amount that can be spent is 4500

(iii) the buyer must buy at least 5 violins

​

E.gs

​The amount of apples bought y, must always be more than the amount of bananas x, after disposing of two bad ones found in each heap.

​Her money spent on juice y, must be less than or equal to twice the amount of money for water plus 2 extra dollars from savings.

​For some questions, the solution is a region formed when 2 or more inequalities intersect.

July 2021 P2

​

​Maria buys 2 types of mobile phone, B-Flo and

C-Flex, from a company retail.​ One B-Flo mobile phone costs $60 while one C-Flex costs $80. She buys x number of B-Flo phones and y number of

C-Flex phones.

(a) i) Marla must not spend more than $1200. Write an inequality to represent this information.

ii) The number of ​B-Flo phones must be greater than or equal to the number of C-Flow phones. Write down an inequality in x and y to show this information.

(iii) Represent the two inequalities on the grid provided and Label as R the region which satisfied both inequalities.

​

​when y = 0

3x = 60 (divide 3)

x = 20

(20,0)​

​using x = 20 as ending point : ​

​when x = 20​, y = 20

(20,20)​

(since the grid given reaches 20 on both axes)

​Grid

Provided​

CXC

​

Past Paper Questions To Practice

May/June 2024 P2 No. 8(b)
May /June​ 2023 P2 No. 8
May / June 2022 P2 Nos. 4(b), 8 (a) - (d)

​July 2021 P2 No. 4(a), 8 (b)

June 2019 P2 No. 4(c)