Gradients (AQA GCSE Further Maths)
Revision Note
Author
PaulExpertise
Maths
Gradients of Lines
What is the gradient of a line?
- The gradient is a measure of how steep a 2D line is
- A large value for the gradient means the line is steeper than for a small value of the gradient
- A gradient of 3 is steeper than a gradient of 2
- A gradient of −5 is steeper than a gradient of −4
- A positive gradient means the line goes upwards from left to right - "uphill"
- A negative gradient means the line goes downwards from left to right - "downhill"
- In the equation for a straight line, , the gradient is represented by
- The gradient of is −3
How do I find the gradient of a line?
- The gradient can be calculated using
- You may see this written as instead
- may even be used which links to the work on Calculus
- this can be read as "the difference in y divided by the difference in x"
- may even be used which links to the work on Calculus
- You need to know two coordinates a line passes through to find its gradient
- If given two coordinates and the gradient of the line joining them is
-
-
- The order of the coordinates must be consistent on the numerator and denominator
- i.e. ("Point 2" – "Point 1") or ("Point 1" – "Point 2") for both
- The order of the coordinates must be consistent on the numerator and denominator
- If given a diagram of a straight line you will need to pick two points the line passes through
- If possible, pick whole number coordinates
- positive numbers are easier to work with than negatives!
- try not to pick coordinates that are close together
- If possible, pick whole number coordinates
-
Exam Tip
- Be very careful with negative numbers when calculating the gradient; write down your working rather than trying to do it in your head to avoid mistakes
- For example,
Worked example
a)
Find the gradient of the line joining (-1, 4) and (7, 28)
Using :
Simplify:
Gradient = 3
b)
Work out the gradient of the line shown in the diagram below.
First note that this is a "downhill" line so we are expecting a negative gradient
We first need to identify two points on the line - looking for whole numbers we can see that the line passes through (-2, 0) and (2, -6)
Using
Simplify
Gradient
Parallel & Perpendicular Gradients
What are parallel lines?
- Parallel lines are equidistant meaning they never meet
- Parallel lines have equal gradients
What are perpendicular lines?
- Perpendicular lines meet at right angles
- The product of their gradients is -1
How do I tell if lines are parallel or perpendicular?
- Rearrange equations into the form y = mx + c
- m is the gradient
Exam Tip
- Exam questions are good at “hiding” parallel and perpendicular lines.
- e.g. a tangent and a radius are perpendicular
- typically this would be shown using a diagram
- e.g. a tangent and a radius are perpendicular
- Parallel lines could be implied by phrases like “… at the same rate …”
Worked example
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