Function chart

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.

function chart

One method of graphing is as follows:

  1. Determine the domain.
  2. Determine the cut points with the axes.
  3. Know the graphical form of the basic function to which the function belongs.
  4. Apply the rules of transformation of functions, where translations, reflections and stretching or narrowing occur.
  5. Check the domain and the cut points previously found.
  6. 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:

function chart1

  • Step 4: apply the rules of function transformations.

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

function graph2

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

function chart3

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

function chart4

  • 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.

function chart5