Absolute Value Function Calculator | Math Function Evaluation

The V-shape of the absolute value function comes from its piecewise definition: for negative inputs, it returns the negative of that value (making it positive), and for positive inputs, it returns the value unchanged. This creates two linear pieces: a line with negative slope (-1) for x < 0 that reflects across the y-axis to create a line with positive slope (1) for x > 0. These two lines meet at the origin (0,0), forming the characteristic "V" shape. The function y = |x| is not differentiable at x = 0 (the vertex of the V) because the slope changes abruptly from -1 to 1, but it is continuous at this point.

To graph y = a|x - h| + k, apply these transformations sequentially: (1) Horizontal shift: Replace x with x - h, shifting the V-shape h units right if h > 0 or |h| units left if h < 0. (2) Vertical stretch/compression: Multiply by |a|, making the V steeper if |a| > 1 or flatter if 0 < |a| < 1. (3) Reflection: If a < 0, reflect the graph across the x-axis (upside-down V). (4) Vertical shift: Add k, moving the graph up if k > 0 or down if k < 0. For example, y = -2|x - 3| + 1 means: shift right 3 units, stretch vertically by factor of 2, flip upside down, then shift up 1 unit. The vertex of this transformed function is at (3, 1).

Absolute value functions appear in many real-world contexts: (1) Error measurement: The absolute difference between measured and actual values represents error magnitude. (2) Distance calculations: The absolute value of the difference between coordinates gives distance on a number line. (3) Tolerance specifications: Manufacturing parts must be within ±τ units of a target dimension, expressed as |x - target| ≤ τ. (4) Financial analysis: Price deviation from average is often measured in absolute terms. (5) Signal processing: Signal amplitude often uses absolute value to measure magnitude regardless of direction. (6) Economics: Absolute deviations are used in some economic indicators to measure volatility. Understanding absolute value helps quantify magnitude in situations where direction (positive/negative) is less important than size.