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The following table shows the evolution of the population in Granada (city) from the beginning of the 20th century until the last census in 2011 (population censuses are carried out every ten years).
(Source: National Statistics Institute (INE))
Represent this table graphically using an appropriate scale (x-axis: years; y-axis: population).
In which year was the population the highest? And the lowest?
If we only consider the period between 1940 and 1991, in which year was the population the highest? And the lowest? Do these answers match those from the previous question?
Write in your notebook the periods in which the population increased and the periods in which it decreased.
Of all the growth periods, which one was the fastest? Justify your answer. Do the same with the periods of population decrease.
In the year 1940, was Granada's population increasing or decreasing? And in 1960?
Imagine a person running along the following black line:
There are moments when that person goes up and others when they go down. Well, if that line were the graph of a function, we would say that the function is increasing in the sections where the person goes up —that is, the orange segments— and decreasing in the sections where the person goes down — the green segments.
We refer to the parts of a function by the portion of the X-axis they occupy. So, in the previous example, which we reproduce below with the scale, we would have:
The graph decreases from the beginning until -4; it increases from -4 to 2; then it decreases again until 6, and from that point on, it increases indefinitely.
There are points on the graph where it changes from increasing to decreasing or vice versa. These are the hills and valleys shown in the following drawing.
The point where the little tree is located (2, 2) is higher than all the points around it within a certain distance. That is a maximum of the function. The points where there is water (–4, –8) and (6, –2) are lower than all the points around them within a certain distance; these are called minimums.
There are places on the graph that are higher than the little tree (for example, where the branch is). That’s why the place where the tree is located is called a relative maximum. For similar reasons, the point (6, –2), where the light blue water is, is called a relative minimum.
However, there is no point on the graph lower than (–4, –8). That point is called an absolute minimum. Similarly, an absolute maximum is defined as the point on the graph that is higher than all others. In the example drawing, there isn’t one of those because we assume the graph continues going upward indefinitely.
For now, we will find the maximums and minimums of a function using its graph, although there are analytical methods to do so.
Exercises
Exercises
Observe the following graphs and study their increasing or decreasing behavior, and their maximums or minimums. Also indicate whether they are relative or absolute.
Gráfico de la población de Granada.
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