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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.

November 20, 2015