In this section we are going to explain how to represent functions graphically. We will define the most basic concepts you need to know to draw graphs of functions and in other sections of the web we will learn how to represent the most important functions.
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The Cartesian Axes
To analyze and see how a function behaves, we have to resort to its graphical representation.
To do this we need to represent the two variables of a function on coordinate axes called Cartesian axes:
- The x is represented on the horizontal axis, called the abcisas axis.
- La y (f(x)) is represented on the vertical axis, called the ordinate axis.
The axes must be graduated. It is not necessary that the two axes have the same scale.
To do this you must take into account the maximum and minimum values of each scale and gradually distribute the values on each axis.
Therefore, each point on the graph has two coordinates, its abcissa x and its ordinate y.
For example, we have the graph of this line. One of its points has the following coordinates:
- x = 2 (abcisa)
- y = 1 (ordinate)
And it is represented this way:
We know it is the point (2,1) because if we draw from the point a vertical line to the x-axis, it cuts with 2 and if we draw a horizontal line to the y-axis, it cuts with 1.
Once you have understood that each point on a graph has two values, in order to know how to represent functions graphically, we need the coordinates of a few points, of the most representative points.
How many points do we need to represent the graph?
Then it depends on each type of function. Each type of function must be given a series of points that represent the function. We will see it in the following section.
For example, the previous case is a line. To draw a line you only need 2 points. If you got the coordinates of more points, you would see that in the end you would fall into the line, so you would be wasting time. Let’s see what their table of values would look like.
We are given the following function and asked to graphically represent it:
Your table of values would be:
Your table of values would be:
This table was obtained by choosing two values of x and calculating the value of y.
As you can see, we have obtained two points: (0,0) and (2,1), which are 2 points for what the previous function goes through and are enough to draw it.
Representation of Elementary Functions
As we said before, depending on the type of function, we know that it has a series of representative points.
There are functions with a defined form, called elementary functions. That is to say, the form that this function has must be known before drawing it and that is why we know its representative points. These functions are:
- Linear function
- Affine Function
- Constant Function
- Function of second degree (parabola)
- Reverse proportionality function
- Exponential function
- Trigonometric functions
- Logarithmic function
Representation of More Complex Functions
There are other types of functions that need further study for representation, such as:
- Functions defined in pieces
- Functions higher than 2
- Rational functions
To represent these functions we must study their domain, continuity, maximums and minimums… and therefore they need more detailed explanations.