The world is ruled by simple mathematical laws and mathematical curves occur almost everywhere. In fact, scientific inventions depend on the discovery of mathematical equations through various experiments. Measurements are made by varying the values of the different quantities and by placing these observation points in the coordinate system. A scientist experiences success if the mathematical curve fits well with the observation points. Finding a curve is a sign that a mathematical equation can be written between the studied quantities. Sometimes, the graphs act in a way that makes it necessary to use piecewise functions to find an equation that describes the behavior correctly. Usually, the type of function that fits the observation point is known in advance.

Example 1

Let us study the distance traveled by a walker as a function of time. Measure the distance traveled every second and place the observation points in the time, distance coordinate system.

The observation points seem to fit well on the line that passes through the origin. As a result, a line is drawn between the observation points so that there are an equal number of observation points on each side of the line. Measurements are always slightly erroneus, and if one measurement point differs significantly from the other points, it can be interpreted as incorrect and ignored when fitting the function. Here, however, all observation points can be considered as successful measurements.

Thus, the points are not simply connected with each other, as has been done in the INCORRECT image above. After all, a set of piecewise functions would be formed, the use of which does not make sense for this purpose. The speed of the walker naturally varies, but the resulting error in the function expression can be removed so that the line is indeed placed between the observation points.

Mark the time with [[$ x $]] and the distance with [[$ y $]]. Select two points on the line to determine the slope of the equation. Let these be the origin [[$ (0, 0) $]] and the point [[$ (3, 6) $]]. The slope is obtained by completing the following calculation:

[[$$ k = \displaystyle\frac {6 - 0} {3 - 0} = \displaystyle\frac {6} {3} = 2 $$]]​

Place the slope and the point [[$ (0, 0) $]] in the equation in order to obtain the result:

[[$ \begin{align*}y - 2 &= 2(x - 0) \\y &= 2x\end{align*} $]]​

In example 1, the equation of smooth motion [[$ distance = speed · time $]] was formed. The equation is used everywhere, without further questioning. Before this equation was generally accepted, scientists had to make a huge number of measurements and fits of mathematical functions.