Key points
In order to work with gradients and straight lines successfully, a good understanding of coordinates and linear graphs is needed.
The A measure of the slope of a line. The steeper the line, the greater the gradient. The gradient is represented by 𝒎 in the equation 𝒚 = 𝒎𝒙 + 𝒄 is a measure of slope. The greater the gradient, the steeper the slope.
When the gradient of two lines are the same, they are Always equidistant (at equal distances). Parallel lines, curves and planes never meet however far they extend. . When the gradients have a The result of multiplying two or more terms. of -1, they are Perpendicular lines are at 90° (right angles) to each other. .
The gradient of a line is calculated by dividing the difference in the \(y\)-coordinates by the difference in the \(x\)-coordinates. This may be referred to as the change in \(y\) divided by the change in \(x\), or the vertical divided by the horizontal.
Understanding the gradient of a straight line
The gradient is the amount of The up-down direction on a graph or map. movement for each unit of The right-left direction on a graph or map. Parallel to the horizon. movement to the right. The greater the gradient, the steeper the slope.
A positive gradient slopes up from left to right. A negative gradient slopes down from left to right.
A gradient of 2 and a gradient of -2 have the same steepness. A gradient of 2 slopes up from left to right, and a gradient of -2 slopes down from left to right.
Parallel lines have the same gradient.
Perpendicular lines are sloped in opposite directions. One has a positive gradient and the other has a negative gradient. The product of their gradients is -1
Examples
Image caption, The gradient is a measure of the slope of a line. It is the amount of vertical movement for each unit of horizontal movement to the right. The greater the gradient, the steeper the slope. The gradient of 3 is steeper than the gradient of 1 and the gradient of 2
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Question
Which of these lines have the same gradient?
A and C have the same gradient. For each unit of horizontal movement to the right the vertical movement is -2 (2 down). They are parallel.
B and D have the same gradient. For each unit of horizontal movement to the right the vertical movement is 1 (1 up). They are parallel.
Working out the gradient of a straight line on a graph
To work out the gradient of a straight line:
Choose two points on the line.
- Any two points will work.
- Whole numbers make the working easier.
Draw a triangle showing the horizontal movement to the right and the vertical movement (up or down).
Label the triangle with the change in the \(x\)-coordinate and the change in the \(y\)-coordinate.
Work out the value of the change in the \(y\)-coordinate divided by the change in the \(x\)-coordinate.
Examples
Image caption, The gradient is the change in the 𝒚-coordinate divided by the change in the 𝒙-coordinate.
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Question
Work out the gradient of the line.
There are three points to choose two from, (0, -1), (1, 3) and (2, 7).
Using (0, -1) and (2, 7), draw a triangle showing the horizontal movement to the right and the vertical movement down.
The change in the \(y\)-coordinates (8) is divided by the change in the \(x\)-coordinate (2), 8 ÷ 2 = 4
The gradient of the line is 4
For a different pair of coordinates, (0, -1) and (1, 3) or (1, 3) and (2, 7), the change in the \(y\)-coordinates (4) is divided by the change in the \(x\)-coordinate (1), 4 ÷ 1 = 4
The calculation gives the same answer.
Finding the gradient and intercept
The A mathematical statement showing that two expressions are equal. The expressions are linked with the symbol = of a straight line can be written as \(y = mx + c\)
- \(m\) is the gradient.
- \(c\) is the \(y\)The point at which the line crosses the 𝒚-澱. Commonly referred to as 'the intercept'..
To write the equation of a straight line:
- Work out the gradient.
- Find the \(y\)-intercept, the value at which the line crosses the \(y\)The line on a graph that runs vertically (up-down) through the origin. It is used as a reference to measure from..
- Write the equation in the form \(y = mx + c\)
Example
Image caption, Find the equation of the line.
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Question
Work out the equation of the line.
Choose two coordinates on the line. Here (1, 4) and (3, 10) are used.
Draw a triangle showing the horizontal movement to the right and the vertical movement up.
Label the triangle with the change in the \(x\)-coordinate (from 1 to 3 is 2) and the \(y\)-coordinate (from 4 to 10 is 6).
Work out the value of the change in the \(y\)-coordinate divided by the change in the \(x\)-coordinate. This gives 6 ÷ 2 which is 3. The gradient of the line is 3
Find the \(y\)-intercept of the line. The value where the line crosses the \(y\)-axis is 1
Write the equation in the form \(y = mx +c\) where \(m\) is the gradient (3), and \(c\) is the \(y\)-intercept (1). The equation is \(y = 3x + 1\)
Practise working out the gradient of a straight line
Quiz
Practise reading and plotting quadratic graphs with this quiz. You may need a pen and paper to help you with your answers.
Real-life maths
The gradient of a line gives the rate of change.
Although graphs in real life may not always be straight lines, the principle of using gradients as a rate of change is useful.
Scientists, such as physicists, might carry out experiments where they need to track the distance a All substances are made of particles. Particles could be atoms, molecules or ions. travels over time. They could use a graph to show this and the gradient of the graph would show the speed of the particle.
Understanding gradient is important for A person who designs and draws plans for buildings. when calculating the slope of a roof, otherwise known as the roof ‘pitch’.
There are rules and regulations for how steep a roof can be.
For example, if the steepest incline allowed for a house extension is 15° (this is a gradient of approximately 0۰268), this will have an impact on the distance that the extension can be built out to the original walls of the house.
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