The image you've uploaded is a detailed educational worksheet on "Transformations of the Absolute Value Function." It visually explains how the basic absolute value function, `y = |x|`, can be altered through translations, dilations, and reflections.
Let's break down the problem presented in the image by analyzing each section.
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1. Translations
Translations involve shifting the graph of the function without changing its shape.
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Vertical Shifts:
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Up: The formula is `y = f(x) + k`. If `k > 0`, the graph shifts up.
- Example: `y = |x| + 3` (shifts the graph up by 3 units).
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Down: The formula is `y = f(x) - k`. If `k > 0`, the graph shifts down.
- Example: `y = |x| - 4` (shifts the graph down by 4 units).
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Horizontal Shifts:
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Left: The formula is `y = f(x + h)`. If `h > 0`, the graph shifts left.
- Example: `y = |x + 2|` (shifts the graph left by 2 units).
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Right: The formula is `y = f(x - h)`. If `h > 0`, the graph shifts right.
- Example: `y = |x - 5|` (shifts the graph right by 5 units).
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2. Dilations
Dilations change the size (stretch or compress) of the graph.
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Vertical Stretch:
- Formula: `y = a * f(x)` where `a > 1`.
- Example: `y = 4|x|` (stretches the graph vertically by a factor of 4).
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Vertical Compression:
- Formula: `y = a * f(x)` where `0 < a < 1`.
- Example: `y = (1/2)|x|` (compresses the graph vertically by a factor of 1/2).
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3. Reflections
Reflections flip the graph over an axis.
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X-Axis Reflection:
- Formula: `y = -f(x)`.
- Example: `y = -|x|` (reflects the graph across the x-axis).
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Y-Axis Reflection:
- Formula: `y = f(-x)`.
- Example: `y = |-x|` (which simplifies to `y = |x|`, so it looks the same as the original because the absolute value function is symmetric about the y-axis).
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General Form of the Transformed Absolute Value Function
The worksheet also provides the general form:
`f(x) = a|x - h| + k`
- `(h, k)` is the vertex of the transformed graph.
- `a` determines the vertical stretch/compression and reflection:
- If `a > 1`, vertical stretch.
- If `0 < a < 1`, vertical compression.
- If `a < 0`, reflection across the x-axis.
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Summary of the Solution
The image serves as a comprehensive guide to understanding how the absolute value function can be transformed. By applying the rules for translations, dilations, and reflections, one can predict how the graph of `y = |x|` will change based on modifications to its equation.
This knowledge is essential for graphing absolute value functions accurately and understanding their behavior under different transformations.
If you have a specific question or need help with a particular transformation, feel free to ask!
Parent Tip: Review the logic above to help your child master the concept of transformations of absolute value functions worksheet.
