The Cartesian Plane and Gradient Coordinate Geometry The Cartesian Plane and Gradient

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

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.

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

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).

Solution to Exercise 1

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.

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

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

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 0.68. How tall is the slide? Ans:

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

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)

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:

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

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

Summary Infinite

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

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

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

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

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

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

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

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

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

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

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

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

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

3 Horizontal Line: Zero Vertical Line: Inifinity