IGCSE Maths / Graphs of straight lines
Gradient from two points
Edexcel International GCSE Maths (4MA1), chapter 6: Graphs of straight lines, specification point 3.3f. Free lesson.
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The gradient tells you how much a straight line climbs or drops for each 1 unit you move to the right. You can find it from two coordinates using change in y divided by change in x.
What gradient means
The gradient is written as m. It describes how far a line goes up or down for every 1 unit it goes across. Climbing as you read to the right gives a positive gradient; dropping gives a negative gradient. So a gradient tells you more than how steep the line is. Gradient = change in y ÷ change in x. The change in y goes on top of the fraction.
How to calculate gradient from two points
Use the coordinates of the two points in the formula. Keep the same point order for the y values and the x values. Put brackets around negative coordinates before subtracting, so the signs stay clear.
Example: a line that drops to the right
For the points (−2, 7) and (4, −5), the y value drops as x increases, so the gradient should be negative.
Keep the fraction the right way up
The change in y must be on top and the change in x underneath. Reversing them gives a different value, even though the working may look neat. For the points (−2, 7) and (4, −5), putting the x change on top gives 6/(−12) = −1/2, which is not the gradient. Do not mix the point order. If you subtract the y values in one order, subtract the x values in that same order. You can swap which point is first. Both the top and bottom then change sign, so the gradient stays the same.
Simplify and check your answer
A gradient can be a fraction, so you do not need to turn it into a decimal. Simplify the fraction where possible, and use the direction of the line to check whether its sign makes sense. For (−3, −2) and (5, 4), m = (4 − (−2))/(5 − (−3)) = 6/8 = 3/4. For (−4, 1) and (2, 5), m = (5 − 1)/(2 − (−4)) = 4/6 = 2/3.