Describing motion along a line

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# 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.

Figure 22: A straight-line graph of z against y

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

1. It applies only to straight-line graphs.

2. The units of the gradient are the units of z divided by the units of y.

3. 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 P1 and P2 in Figure 22) are used to determine it.

4. 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.

Figure 23: Two graphs constructed from the same data but plotted with different scales. The gradients are the same, 4 ×103 kg m−3
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