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 $=\frac{changeiny-axis}{changeinx-axis}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{y_2-y_1}{x_2-x_1}$

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 $=2$.