There are a large number of functions that we can graph; Depending on the analytical form, we can choose which technique to use to represent them graphically. From the data of points taken, substituting values in the function, to a set of algebraic characteristics that express its graphic form.

One method of graphing is as follows:

- Determine the domain.
- Determine the cut points with the axes.
- Know the graphical form of the basic function to which the function belongs.
- Apply the rules of transformation of functions, where translations, reflections and stretching or narrowing occur.
- Check the domain and the cut points previously found.
- Indicate the cut points with the axes, ends and important points of the graph.

**Example**: Graph the following function **f (x) = √ (2 - x) - 1**.

**Step 1**: determine the domain.

Remember that the roots cannot be negative therefore:

2 - x ≥ 0

2 ≥ x

It means that the domain will be all the numbers from two to minus infinity: Dom: (-∞, 2]

**Step 2**: determine the cut points with the axes.

For the y axis, we must substitute the value of x = 0:

f (x) = y (x) = √ (2 - x) - 1

y = √ (2 - 0) - 1

y = √2 - 1 = 0.41

So, (0, √2 - 1) is the cut point with the y axis.

For the x-axis, we must equal the function to zero and clear the variable:

0 = √ (2 - x) - 1

√ (2 - x) = 1

[√ (2 - x)] ² = 1²

2 - x = 1

- x = 1 - 2

- x = - 1

x = 1

The cut point with the x axis is (1, 0).

**Step 3**: The graphical form of the basic function to which this function belongs.

That base function is:

**Step 4**: apply the rules of function transformations.

First we add + 2 to the basic function and look at the translation that occurs:

Second, we change the sign to the variable x, which will make the function reflect with respect to the vertical axis:

Third, we subtract 1 to have the graph that they are asking us for, this subtraction will make the function have a vertical translation:

**Step 5**: it is verified in the graph that the domain is the same one that was calculated algebraically Dom: (- ∞, 2] and its cutpoints are (0, 0,41) in the y axis; (1, 0) in the axis x.**Step 6**: we indicate in the graph the important points.