What do the formulae for linear functions look like in general? The easiest way to answer this question is to work backwards - start with a straight line graph and work out what its equation must be. Suppose then that we have a straight line graph:
Let (x,y) be a typical point on the graph; label it T. We want to find the rule relating y to x.
Now, the line must meet the y-axis somewhere (we will not consider vertical lines in our work). Suppose it meets the y-axis at the point (0,c); remember that the x-coordinate of a point on the y-axis is zero. We denote this point P, and draw a horizontal line from P to some other point (chosen arbitrarily). Then we draw a vertical line from Q, meeting the straight-line graph at the point R, as shown. Denote the ratio below by m:
Next we extend the line PQ to the point SA on the same vertical line as T. Then the triangle is just a magnification of the triangle PQR, so their sides are in the same proportion, that is:
Thus ST = m(PS). But as can be seen from the diagram, ST = y - c and PS = x, so y - c = mx. Rearranging we obtain
This is therefore the equation of the straight line. We call c the intercept and m the slope of the line. All linear functions have equations which can be written in this form, by suitable rearrangement if necessary. Note that if c = 0 then the line passes through the origin.