Year 10+ Coordinate Geometry
Chapters
Gradient of a Straight Line
Gradient of a Straight Line
The gradient
of a straight line is a number that tells you how steep its slope is. Another word for gradient is slope
.
Finding the Gradient
The gradient is given by the following formula:
Vertical Rise Over Horizontal Run
We often call the change in height the vertical rise, and the change in horizontal distance the horizontal run, so the formula for the gradient can be written as :
Let's calculate some gradients.
Example 1
The vertical rise is \(2\) and the horizontal run is \(4\), so the gradient is
Example 2
The vertical rise is \(4\) and the horizontal run is \(4\), so the gradient is
Did you notice that this line is steeper than the line in Example 1? Its gradient is larger, too.
Example 3
The vertical rise is \(5\) and the horizontal run is \(2\), so the gradient is
This line is even steeper than the line in Example 2. Its gradient is larger, too.
Sign of the Gradient
Gradients can be positive, negative or zero.

When the gradient is
positive
, the straight line is going uphill as we move from left to right. We say that the straight line function isincreasing
. 
When the gradient is
negative
, the straight line is going downhill as we move from left to right. We say that the straight line function isdecreasing
.  When the gradient is
zero
, the straight line is horizontal. It slopes neither uphill nor downhill.
Example: Negative Gradient
The vertical rise is \(3\) as the line is sloping downwards, and the horizontal run is \(2\), so the gradient is
This line is sloping downhill, so it has a negative gradient.
Example: Zero Gradient
The vertical rise is \(0\) as the line is horizontal (flat), and the horizontal run is \(4\), so the gradient is
This line is horizontal, so its gradient is zero.
Example: Vertical Line
The vertical rise is \(4\), and the horizontal run is \(0\) as the line is vertical, so the gradient is
That's a bit of a problem, isn't it? We say that the gradient of vertical lines is undefined
.
Summary
The gradient of a line that slopes uphill is positive.
The gradient of a line that slopes downhill is negative.
The gradient of a horizontal line is zero.
The gradient of a vertical line is undefined.
Steeper (uphill) lines have larger gradients.
Steeper downhill lines have negative gradients that have larger sizes.
Description
A coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates. In this tutorial series, you will learn about vast range of topics such as Cartesian Coordinates, Midpoint of a Line Segment etc
Audience
year 10 or higher, several chapters suitable for Year 8+ students.
Learning Objectives
Explore topics related to Coordinates Geometry
Author: Subject Coach
Added on: 27th Sep 2018
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