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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.

Screen Shot 2015-11-19 at 6.56.50 PM.png

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 \(x – y \ge 1 \)
    Screen Shot 2015-11-19 at 8.27.20 PM
  • \(x \le 0, y \ge 0 \) implies |x| = -x. Thus in second quadrant the inequality becomes \(-x -y \ge 1\) or \(x+y \le -1 \)
    Screen Shot 2015-11-19 at 8.30.30 PM
  • \(x \le 0, y \le 0 \) implies |x| = -x, |y| = -y. Thus in third quadrant the inequality becomes \(-x +y \ge 1\) or \(x-y \le -1 \)
    Screen Shot 2015-11-19 at 8.31.44 PM
  • \(x \ge 0, y \le 0 \) implies |x| = x, |y| = -y. Thus in fourth quadrant the inequality becomes \(x + y \ge 1\)
    Screen Shot 2015-11-19 at 8.33.15 PM

Hence the final picture is:

Screen Shot 2015-11-19 at 8.34.00 PM

(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

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