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.