Cartesian Coordinates:
Cartesian coordinates are a way to specify points in a plane or space using numerical values.
They are based on a system of perpendicular axes:
- 2D Cartesian Coordinates:
- A point is represented as (x, y), where:
- x is the horizontal coordinate (left or right).
- y is the vertical coordinate (up or down).
- 3D Cartesian Coordinates:
- A point is represented as (x, y, z), where:
- x is the horizontal coordinate.
- y is the depth or second horizontal coordinate.
- z is the vertical coordinate.
Cartesian coordinates are used in geometry, physics, engineering, and computer graphics to plot points, lines, and shapes.
Functions & Linear Graphs
Learning Objectives:
1. State the coordinates of a point
2. Plot a point in a Cartesian plane
3. Draw the Graph of a linear function
4. Find the gradient of a straight line
5. State the y-intercept of a straight line
6. solve problems involving linear graphs in real-world contexts
A function is a relationship between two variables x and y such that every input x produces exactly one output y.
The cartesian plane consists of two number lines intersecting at right angles at point 0, known as the origin.
Cartesian coordinates are important because they provide a systematic way to describe and analyze positions, shapes, and movements in space.
Here’s why they matter:
1. Foundation of Geometry & Algebra
- They bridge algebra and geometry, allowing equations to represent geometric figures (e.g., lines, circles, curves).
- The coordinate plane enables solving geometric problems with algebraic methods.
2. Navigation & Mapping
- Used in GPS and navigation systems to determine locations accurately.
- Maps and blueprints rely on Cartesian coordinates for precise positioning.
3. Physics & Engineering
- Essential for mechanics, physics, and engineering to describe motion, forces, and structures.
- Used in robotics and mechanical design to program movement and positioning.
4. Computer Graphics & 3D Modeling
- Used in video games, animation, and simulations to render objects in 2D and 3D space.
- CAD (Computer-Aided Design) software relies on Cartesian coordinates for creating detailed models.
5. Data Visualization & Analysis
- Graphs and charts (bar graphs, line graphs, scatter plots) use Cartesian coordinates to display relationships between variables.
- Helps in understanding trends and making data-driven decisions.
6. Mathematical and Scientific Applications
- Used in calculus to analyze curves, slopes, and areas under curves.
- Essential for solving equations and modeling real-world scenarios.
Overall, Cartesian coordinates are a fundamental tool across various fields, providing a universal way to describe and manipulate space mathematically.
The position of any point in the plane can be determined by its distance from each of its axes.
(Draw the Graph of a Linear Function)
Ex. 1.
i) On a sheet of graph paper, using a scale of 2cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of the function y = 2x for values of x from 0 to 4.
ii) The point (3, p) lies on the graph in (i), and find the value of p.
Solution:
We first set up a table of values for x and y. These pairs of values for x and y satisfy the equation of the function y = 2x.
x 0 2 4
y = 2x 0 4 8
Using the scale of 2 cm to represent 1 unit on the x-axis and
1 cm to represent 1 unit on the y-axis, the three pairs of values are plotted as points in the Cartesian plane and a straight line is drawn to pass through these points.
Q.2. i) On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y - axis, draw the graph of the function y = 2x + 1 for values of x from 0 to 4.
ii) The point (q, 6) lies on the graph in (i). Find the value of q.
Q.3. On the sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y - axis, draw the graphs of the functions y = 3x and y = 2 - 2x for values of x from - 2 to 2.
Drawing the Grap of a Linear Function:
i) On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of the function y = 2x for values of x from 0 to 4.
ii) The point (3, p) lies on the graph in (i). Find the value of p.
iii) From the graph in (i)
when x= 3, p = y = 6
Practice 1.
1. i) On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of the function y = 2x + 1 for values of x from 0 to 4.
ii) The point (q, 6) lies on the graph in (i). find the value of q.
2. On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graphs of the function y = 3x and y = 2 - 2x for values of x from - 2 to 2.
Let O be the origin (0,0).
Vertically change (or rise) OP = 6
Horizontal change (or run) OQ = 2
Since the line slopes downwards from the left to the right, its gradient is negative.
Gradient = rise/run
= - 6/2
= -3
Distance -Time Graph:
Q.1. The travel graph shows a journey taken by a cyclist. He started his 50-km journey at 0800 hours. At 0900 hours, his bicycle tire suffered a puncture and he spent half an hour repairing it. He then continued his journey and reached his destination at 11.30 hours.
a) How far did the cyclist travel before his bicycle tire suffered a puncture?
b) Find the gradient of each of the following line segments, stating clearly what each gradient represents.
i) OA
ii) AB
iii) BC
Solution:
a) 20 km
b) i) Gradient of OA = 20/I
= 20
The cyclist traveled 20 km in 1 hour, i.e. his average speed was 20 km/h
before his bicycle tire suffered a puncture
ii) Gradient of AB = 0
He stopped cycling, i.e. his average speed was zero.
iii) Gradient of BC = 30/2
= 15
The average speed during the last part of his journey was 15 km/h.
