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# Graphs of absolute value functions (Tomato Subjective 126)

Problem: Sketch, on plain paper, the regions represented, on the plane by the following:

(i) $$|y| = \sin x$$

(ii) $$|x| – |y| \ge 1$$

Discussion: First we need to understand what |y| signifies. It is the absolute value of y, that is it is +y when y is positive and -y when y is negative.

Lets test with $$x = \frac{\pi}{6}$$. Clearly then sin x = 1/2. This implies $$|y| = 1/2$$ or $$y = 1/2, -1/2$$.

Again let us test with $$x = \pi + \frac{\pi}{6}$$. Then $$\sin (\pi + \frac{\pi}{6}) = – \frac{1}{2}$$ implying $$|y| = – \frac{1}{2}$$. But this is impossible as absolute value cannot be negative.

Using these observations we get a clear idea about what is happening.

• The values of x where sin (x) is positive (that is when $$\displaystyle {\frac {4k \pi}{2} \le x \le \frac {(4k+2) \pi }{2} }$$, where k is any integer), we draw the graph of y = sin x and reflect it about x axis (as y = sin x and y = – sin x both satisfies the equation).
• The values of x where sin (x) is negative (that is when $$\displaystyle {\frac {(4k+2) \pi}{2} \le x \le \frac {(4k+4) \pi }{2} }$$, where k is any integer), the given relation is not defined as absolute value of y cannot be negative.

For Part (ii)

Notice that |x| gives distance of a point from y axis and |y| gives distance of a point from x axis.

Let us split the problem into cases:

• x, y both non negative (first quadrant). Then |x| = x, |y| = y, implying $$|x| – |y| \ge 1$$ is same as $$x – y \ge 1$$
• $$x \le 0, y \ge 0$$ implies |x| = -x. Thus in second quadrant the inequality becomes $$-x -y \ge 1$$ or $$x+y \le -1$$
• $$x \le 0, y \le 0$$ implies |x| = -x, |y| = -y. Thus in third quadrant the inequality becomes $$-x +y \ge 1$$ or $$x-y \le -1$$
• $$x \ge 0, y \le 0$$ implies |x| = x, |y| = -y. Thus in fourth quadrant the inequality becomes $$x + y \ge 1$$

Hence the final picture is:

(only the shaded zones, not the lines).

## Chatuspathi:

• What is this topic: Graphing of functions, inequalities
• What are some of the associated concept: Absolute value functions, inequalities
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, discusses these topics in the ‘Calculus’ module.
• Book Suggestions: Play with Graphs (Arihant Publication)
November 20, 2015