A reflection of an absolute value function is a transformation that flips the graph of the function across a line, typically the x-axis or the y-axis, without changing its fundamental "V" shape.
Understanding Reflections
Reflections involve creating a mirror image of the original function. In the context of absolute value functions, the most common type of reflection is across the x-axis.
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Reflection Across the x-axis: This occurs when the entire function is multiplied by -1. If the original function is y = |x|, the reflection across the x-axis is y = -|x|. This flips the graph upside down. For example, the function y = |x-1| - 1 becomes y = -(|x-1| - 1) or y = -|x-1| + 1.
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Reflection Across the y-axis: While less common because absolute value functions are often symmetrical about the y-axis (or a vertical line shifted from the y-axis), reflecting across the y-axis involves replacing x with -x. For the basic function y = |x|, a reflection across the y-axis would result in y = |-x|, which simplifies to y = |x| because the absolute value of x and -x are the same. For a function like y = |x-1|, reflecting across the y-axis would give y = |-x-1|.
Key Characteristics of Reflected Absolute Value Functions
- Shape Preservation: The basic "V" shape of the absolute value function remains unchanged. Only its orientation is altered.
- Vertex Location: The vertex (the point where the "V" changes direction) may shift vertically or horizontally depending on the other transformations applied to the function. When reflecting across the x-axis, the y-coordinate of the vertex changes sign.
- Symmetry: Reflections can affect the symmetry of the function depending on the axis of reflection and the original function's symmetry.
- Equation Modification: The algebraic equation of the function is altered by multiplying by -1 (for x-axis reflection) or substituting -x for x (for y-axis reflection).
Example
Consider the absolute value function y = |x|.
- Original function: y = |x|
- Reflection across the x-axis: y = -|x|
The graph of y = -|x| is an upside-down "V" with its vertex at the origin (0, 0). The original function opens upwards, while the reflected function opens downwards.
