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Mathematics

Understanding the Gradient

An intuitive guide to the concept of gradient, moving from the physical 'steepness' of a slope to the formal calculation of change in y over change in x.

RiseRunSteepness as a ratio
Steepness=RiseRunSteepness = \frac{\text{Rise}}{\text{Run}}
Think about walking up a hill. Steepness isn't just how high you go, but how quickly you gain height for every step forward you take.
Step-by-step solver
1

(a) Identify Points

State the coordinates given for points A and B.

A(1,2),B(5,10)A(1, 2), \quad B(5, 10)
2

(a) Formula

Recall that the gradient is the ratio of the vertical change to the horizontal change.

gradient=y2−y1x2−x1\text{gradient} = \dfrac{y_2 - y_1}{x_2 - x_1}
3

(a) Substitution

Plug the x and y values from A and B into the formula.

gradient=10−25−1\text{gradient} = \dfrac{10 - 2}{5 - 1}
4

(a) Simplify

Calculate the difference for numerator and denominator.

gradient=84\text{gradient} = \dfrac{8}{4}
5

(a) Final Result

Divide to find the final gradient value.

gradient=2\text{gradient} = 2

Original question

gradient

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