1. What is the gradient of the line?
The gradient of a line refers to the steepness or slope of that line. It is a measure of its slope.
- The bigger the gradient, the steeper the line.
- Gradient of a line can be positive or negative and does not have to be a whole number.
- When the gradient of two lines is the same, they are parallel.
- When the gradients have a product of -1, they are perpendicular.
2. Types of Gradients:
Understanding the different types of gradients is crucial when working with lines and their equations. Depending on the value of the gradient, there are four different types of gradient positive, negative, zero and undefined.
- Positive gradients: A line with a positive gradient has a slope that goes up from left to right, and the value of the gradient is greater than zero.
2. Negative gradients: A line with a negative gradient has a slope that goes down from left to right, and the value of the gradient is less than zero.
3. Zero gradients: A line with a zero gradient is a horizontal line. For a line that has a slope of zero, the value of the gradient is zero.
4. Undefined gradients: A line with an undefined gradient is a vertical line.
3. How to find the Gradient?
The gradient of a line is calculated by dividing the difference between the y- and x-coordinates.
The formula for the gradient of a line can be written as:
Gradient of a line =
where (x₁ ,y₁) and (x₂ ,y₂) are any two points on the line.
3.1 Gradient through given line:
The gradient of the given straight line on a graph can be determined as follows:
Step 1: Take any two points on the given line and write their coordinates.
Step 2: Consider the first point’s coordinates as (x₁, y₁) and the second point’s coordinates as (x₂, y₂).
Step 3: Find the change in the y-axis: (y₂-y₁)
Step 4: Find the change in the x-axis: (x₂-x₁)
Step 5: Substitute the change in y and x-axis in the gradient formula.
Step 6: Simplify it if required to get the value of the gradient of a given straight line.
Let's look at some examples.
Example 1 : Calculate the gradient of the line drawn below.
Example 2 : Calculate the gradient of the line drawn below .
3.2 Gradient through Two Points:
The gradient of the straight line that passes through the given points can be determined as follows:
Step 1: Consider the first point’s coordinates as (x₁, y₁) and the second point’s coordinates as (x₂, y₂)
Step 2: Find the change in the y-axis: (y₂-y₁)
Step 3: Find the change in the x-axis: (x₂-x₁)
Step 4: Substitute change in y and x-axis in the gradient formula.
Step 5: Simplify it to get the value of the gradient of the straight line.
Let's look at an example.
Example 1 : Find the gradient of the line that passes through (-5, 2) and (3, 4).
