Gradient and Intercepts of a Line - SS2 Mathematics Lesson Note

The gradient or slope of a linear graph defines the movement of a point along the linear graph. It measures the change in distance along both the \(x\) and \(y -\)axes.

The gradient along a linear graph \(= \frac{change\ in\ y - axis}{change\ in\ x - axis} = \frac{\mathrm{\Delta}y}{\mathrm{\Delta}x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)

On the left, points \(A\) and \(B\) are points on a linear graph, where \(A = (2,0)\) and \(B = (8,3)\). The gradient of this linear graph \(= \frac{\mathrm{\Delta}y}{\mathrm{\Delta}x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{3 - 0}{8 - 2} = \frac{3}{6} = \frac{1}{2}\)

The gradient on a line is constant throughout that linear graph.

Intercepts of a linear graph are the points on the graph where the line graph cuts the axes. The \(\mathbf{y}\)-intercept is the point the graph cuts the \(y\)-axis that is where \(x = 0\). The \(\mathbf{x}\)-intercept is the point the graph cuts the \(x\)-axis that is where \(y = 0\). In the graph, the \(y\)-intercept \(= \ - 1\) and the \(x\)-intercept \(= 2\).