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2.1.4 Transformations of Absolute Value Functions

Applying translations, reflections, and stretches to absolute value functions and analyzing their effects on the graph.

定义

Transformations of absolute value functions refer to the systematic modifications applied to the parent function \(f(x) = |x|\) through translations, reflections, and stretches/compressions. A general transformed absolute value function can be written as \(f(x) = a|b(x - h)| + k\), where: - \(a\) represents a vertical stretch (if \(|a| > 1\)) or compression (if \(0 < |a| < 1\)), and a reflection across the x-axis if \(a < 0\) - \(b\) represents a horizontal compression (if \(|b| > 1\)) or stretch (if \(0 < |b| < 1\)), and a reflection across the y-axis if \(b < 0\) - \(h\) represents a horizontal translation (right if \(h > 0\), left if \(h < 0\)) - \(k\) represents a vertical translation (up if \(k > 0\), down if \(k < 0\)) \nThe vertex of the transformed function is located at the point \((h, k)\), and the graph maintains the characteristic V-shape of absolute value functions unless reflected.

核心公式

  • \(f(x) = a|b(x - h)| + k\)
  • \(f(x) = a|x - h| + k\) (when \(b = 1\))
  • \(|a| > 1 \text{ causes vertical stretch; } 0 < |a| < 1 \text{ causes vertical compression}\)
  • \(|b| > 1 \text{ causes horizontal compression; } 0 < |b| < 1 \text{ causes horizontal stretch}\)
  • \(\text{Vertex: } (h, k); \text{ Axis of symmetry: } x = h\)

易错点

  • ⚠️ Confusing the direction of horizontal transformations: students often think \(|x - h|\) shifts left when \(h > 0\), when it actually shifts right. The transformation \(|x - h|\) moves the graph RIGHT by \(h\) units, not left.
  • ⚠️ Incorrectly applying the coefficient \(a\): students may forget that \(a\) affects both the direction (reflection if negative) and the steepness. A negative \(a\) reflects across the x-axis, and \(|a| \neq 1\) changes the slope of the V-shape.
  • ⚠️ Misidentifying the vertex: when the function is written as \(f(x) = a|b(x - h)| + k\), students sometimes incorrectly identify the vertex as \((bh, k)\) instead of \((h, k)\), or fail to account for the vertical shift \(k\).
  • ⚠️ Neglecting the effect of \(b\) on the domain and range: students often ignore the horizontal compression/stretch factor \(b\) when analyzing transformations, focusing only on \(a\), \(h\), and \(k\). The factor \(b\) changes how quickly the function increases on either side of the vertex.