1. The Cartesian Plane and Gradient
Coordinate Geometry
The Cartesian Plane and Gradient

2. Basic Terminology
The figure on the right shows 2 perpendicular lines intersecting at the point O. This is called the Cartesian Plane.
O is also called the origin.
The horizontal line is called the x-axis and the vertical line is called the y-axis

3. Coordinates of a Point
The position of any point in the Cartesian Plane can be determined by its distance from each axes.
Example: Point A is 3 units to the right of the y-axis and 1 unit above the x-axis, its position is described by the coordinate (3, 1).
Similarly, the coordinates of Points B, C and D are determined as shown.

4. Question
What coordinate represents the origin O ?
Ans: (0, 0)

5. Summary
Any point, P, in the plane can be located by it’s coordinate (x, y).
We call x the x - coordinate of P and y the y - coordinate of P.
Hence, we say that P has coordinates (x, y).

6. Solution to Exercise 1

7. Gradient (or slope) l
The steepness of a line is called its GRADIENT (or slope).
The gradient of a line is defined as the ratio of its vertical distance to its horizontal distance.

8. Examples of Gradient What is the gradient of the driveway? Ans:
Note: Gradient has no units!

9. Examples of Gradient
An assembly line is pictured below. What is the gradient of the sloping section?
Ans:

10. Examples of Gradient Ans:
The bottom of the playground slide is 2.5 m from the foot of the ladder. The gradient of the line which represents the slide is How tall is the slide?
Ans:

11. Question
For safety considerations, wheelchair ramps are constructed under regulated specifications. One regulation requires that the maximum gradient of a ramp exceeding 1200 mm in length is to be
(a) Does a ramp 25 cm high with a horizontal length of 210 cm meet the requirements?
(b) Does a ramp with gradient meet the specifications?
(c) A 16 cm high ramp needs to be built. Find the minimum horizontal length of the ramp required to meet the specifications.
Ans: No
Ans: Yes
Ans: 224 cm

12. Horizontal and Vertical Lines
The gradient of a horizontal line is ZERO
(Horizontal line is flat – No Slope)
The gradient of a vertical line is INIFINITY
(Vertical line – gradient is maximum)

13. Finding the gradient of a straight line in a Cartesian Plane
(a) Positive Gradients
Lines that climb from left to the right are said to have positive gradient/slope:
(b) Negative Gradients
Lines that descend from left to the right are said to have negative gradient/slope:

14. Examples Write down the coordinates of the points given (16, 0)
(0, 10)
(16, 0)
(0, -8)
(-15, 0)

15. Examples
(-4, 0)
(0, 6)
(3, 0)
(0, -12)

16. Summary
Infinite

17. Gradient Formula So far, we have determined the gradient using the
idea of
Using the above, we must always remember to add a negative sign to slopes with negative gradient.
Now, let’s look at the formula to determine gradient. The formula will take into consideration the sign of the slope

18. Gradient Formula A(x1,y1) x1 x2 y1 y2 B(x2,y2) Horizontal = x2 – x1
Vertical = y2 – y1 y x

19. How to apply gradient formula
Write down the coordinates of 2 points on the line: (x1, y1) and (x2, y2)
If the coordinate is negative, include its sign
Apply the formula

20. Examples: L1: 2 points on the line are (1, 4) and (0, 1)
Tip: Choose points that are easy to read!
1 square represents 1 unit on both axes

21. Examples: L2: 2 points on the line are (1, 1) and (3, 3)
Tip: Choose points that are easy to read!
1 square represents 1 unit on both axes

22. Examples: L3: 2 points on the line are (3, 1) and (1, 0) (3, 1) (1, 0)
1 square represents 1 unit on both axes

23. Examples: L4: 2 points on the line are (3, -1) and (-3, -3) (3, -1)
1 square represents 1 unit on both axes

24. Examples: L5: 2 points on the line are (0, 1) and (1, -2) (0, 1)
1 square represents 1 unit on both axes

25. Examples: L6: 2 points on the line are (0, 0) and (-4, 4) (-4, 4)
1 square represents 1 unit on both axes

26. Examples: L7: 2 points on the line are (4, -2) and (-2, 2) (-2, 2)
1 square represents 1 unit on both axes

27. Examples: L8: 2 points on the line are (0, -2) and (-3, -1) (-3, -1)
1 square represents 1 unit on both axes

28. Question Is there a difference between Is there a difference between
Ans: No.
Is there a difference between
Ans: Yes!

29. Solution to Exercise 2
In order from smallest to largest gradient: e, b, a, d, c

34. 3
Horizontal Line: Zero
Vertical Line: Inifinity