Determining the way one variable changes when another changes: (3) for curvilinear functions
A straight-line equation is frequently a fairly good empirical statement of the relation between two variables even when the true relation is more complex than the straight line can portray. Yet it maybe just as important to know the exact or approximate "form" of the relationship as it is to have an empirical statement of it. For that reason it is necessary to consider other ways of expressing a relationship than the straight line.
The automobile-stopping case (Figures 4.1 and 4.2) showed that the relations could not be well expressed by a straight line, especially below 15 miles per hour, and above 35. The latter might be very important for the purpose of the whole study.
The real difficulty involved would have been the assumption that the straight-line function applied. That would have assumed that an increase of one mile in the speed of the car increased the distance required for stopping by the same number of feet, no matter how fast the car was already traveling. When we examine Figures 4.1 and 4.2 closely, we see that this is not correct; the line of averages slants up slowly at first, then tends to rise more steeply as the speed is increased, until it has the steepest slope at the highest speed. It is therefore incorrect to assume that we can express the slope of the line by determining the average increase in stopping distance for an increase of one mile in the rate of speed; for the increase in stopping distance is not the same regardless of the rate of speed, but tends to become greater as the rate of speed increases. Only if our way of stating the line can express that fact too will it sum up all our observations with sufficient accuracy.
What is needed is some general way of stating the relation between