13 Graphical Methods

A very useful method if life appears difficult algebraically is to attempt graphical thinking. This can particularly help when a student confronts a formula or equation in which there are structures such as modulus signs, fractional powers, rational functions, greatest integer functions and similar. The following examples illustrate standard low-level approaches useful in the early stages of problem solving, but could be systematised more in teaching.

Example 13.1

Find the region in which |x| + |y| ≤ 1.

Solution 13.1 and comments

In these modulus sign problems start with a symbolic approach considering the various (here 4) cases for the signs of arguments x and y. In the first quadrant one gets the formula x + y ≤ 1 or y ≤ -x + 1, giving the progress solution of Figure 1.

Figure 1. |x| + |y| ≤ 1 in the first quadrant.

Repeating this over each of the four quadrants gives a complete solution as in Figure 2.

Figure 2. Complete solution.

Example 13.2

Find all the solutions of x² - [x] - 2 = 0.

Solution 13.2 and comments

In questions like this one can go straight into a graphical approach. But one needs to be careful. A common method is to try to develop the left hand side function in stages. In this case the left hand side becomes a little messy and the author’s experience is that not all people find all the solutions to be obvious. It is easy to draw qualitatively the function x² - 2 for example, but then subtracting [x] is messy.

A better strategy in this case is to note that the equation is equivalent to x² - 2 = [x] and to draw the curves of each side, as done in Figure 3.

Figure 3. Graph of x² - 2 = [x].

This leads to a clear discovery that there are three solutions, two of which are integer, (-1,-1) and (2,2) and the one where y = 1 and it turns out using symbolic calculation that this root is at (√3, 1).