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2.7.2 Reflections Across Axes

Analyzing how negating function values or inputs creates reflections across the x-axis and y-axis.

定义

A reflection across an axis is a transformation that flips a function's graph over either the x-axis or y-axis. For a function \(f(x)\):

Reflection across the x-axis: The transformation \(y = -f(x)\) reflects the graph of \(f(x)\) across the x-axis. Each point \((x, y)\) on the original graph maps to \((x, -y)\) on the reflected graph. This negates all output values while keeping inputs unchanged.

Reflection across the y-axis: The transformation \(y = f(-x)\) reflects the graph of \(f(x)\) across the y-axis. Each point \((x, y)\) on the original graph maps to \((-x, y)\) on the reflected graph. This negates all input values while keeping outputs unchanged. \nThese reflections are fundamental function transformations that preserve the shape and size of the graph but change its orientation. Reflections are isometric transformations, meaning they preserve distances and angles.

核心公式

  • \(y = -f(x)\) (reflection across the x-axis)
  • \(y = f(-x)\) (reflection across the y-axis)
  • \(-f(-x)\) (reflection across both axes, equivalent to a 180° rotation about the origin)
  • \((x, y) \rightarrow (x, -y)\) (point transformation for x-axis reflection)
  • \((x, y) \rightarrow (-x, y)\) (point transformation for y-axis reflection)

易错点

  • ⚠️ Confusing which transformation corresponds to which axis: Students often incorrectly apply \(f(-x)\) for x-axis reflection and \(-f(x)\) for y-axis reflection, when it's actually the opposite. Remember: negating the OUTPUT (\(-f(x)\)) reflects across the x-axis, while negating the INPUT (\(f(-x)\)) reflects across the y-axis.
  • ⚠️ Forgetting to apply the reflection to all parts of the function: When reflecting piecewise functions or functions with multiple terms, students sometimes only reflect part of the function, leading to incorrect graphs and domain/range errors.
  • ⚠️ Misidentifying the domain and range after reflection: A reflection across the x-axis changes the range but not the domain, while a reflection across the y-axis changes the domain but not the range. Students often incorrectly assume both change or neither changes.
  • ⚠️ Incorrectly composing reflections with other transformations: When a reflection is combined with translations or stretches, students may apply transformations in the wrong order or forget that the order of operations matters for the final result.