# 3.7 A note on straight-line graphs and their gradients

We end this section by reviewing some of the important features of straight-line graphs, though we do so in terms of two general variables *z* and *y*, rather than *x* and *t*, in order to emphasise their generality. If the graph of *z* against *y* is a straight line of the kind shown in Figure 22, then *z* and *y* are related by an equation of the form

where *m* and *c* are constants. Here *c* represents the intercept of the graph and is equal to the value of *z* at which the plotted line crosses the *z*-axis (provided the *z*-axis passes through *y* = 0). The constant *m* represents the gradient of the graph and is obtained by dividing the change in *z* by the corresponding change in *y*, taking full account of the sign of each change.

You may find it useful to remember that the gradient of a graph is given by its 'rise' over its 'run'.

There are several points to notice about this definition.

It applies only to straight-line graphs.

The units of the gradient are the units of

*z*divided by the units of*y*.The gradient of a straight-line graph is a constant (a number, multiplied by an appropriate unit). The same constant is obtained, no matter which two points (see P

_{1}and P_{2}in Figure 22) are used to determine it.The gradient of a straight-line graph can be positive, negative or zero. Equation 10 assigns a positive gradient to a graph sloping from bottom left to top right, as in Figure 22 for example, and a negative gradient to a graph sloping from top left to bottom right, as in Figure 15. A horizontal line has zero gradient.

Although the appearance of a graph can be changed by plotting the points on different scales (compare Figures 23a and b for example) the gradient, defined by Equation 10, is independent of the shape or size of the graph paper or display screen.