Absolute value transformations alter the graph of a function, typically denoted as f(x). These transformations include shifts (vertical and horizontal), reflections, and stretches/compressions. Based on the provided references, we can outline some fundamental transformation rules, which also apply to absolute value functions.
Vertical Shifts
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Upward Shift: Adding a constant k to the function, i.e., f(x) + k, shifts the graph of f(x) upward by k units.
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Downward Shift: Subtracting a constant k from the function, i.e., f(x) - k, shifts the graph of f(x) downward by k units.
Example: If you have the absolute value function |x|, then |x| + 3 shifts the entire graph up 3 units. Likewise, |x| - 2 shifts it down 2 units.
Horizontal Shifts
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Left Shift: Replacing x with (x + h), i.e., f(x + h), shifts the graph of f(x) to the left by h units.
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Right Shift: Replacing x with (x - h), i.e., f(x - h), shifts the graph of f(x) to the right by h units.
Example: The absolute value function |x + 4| shifts the graph of |x| to the left by 4 units. The absolute value function |x - 1| shifts it to the right by 1 unit.
Reflections
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Reflection about the x-axis: Multiplying the function by -1, i.e., -f(x), flips the graph of f(x) upside down, reflecting it about the x-axis.
Example: -|x| reflects the graph of |x| across the x-axis, making the 'V' shape point downwards instead of upwards.
Summary Table
| Transformation | Rule | Effect on Graph | Example |
|---|---|---|---|
| Vertical Shift Up | f(x) + k | Shifts the graph upward by k units | |
| Vertical Shift Down | f(x) - k | Shifts the graph downward by k units | |
| Horizontal Shift Left | f(x + h) | Shifts the graph left by h units | |
| Horizontal Shift Right | f(x - h) | Shifts the graph right by h units | |
| Reflection about x-axis | -f(x) | Flips the graph upside down (reflects over x-axis) | - |
Note: The effects of combining these transformations are cumulative. For example, f(x + h) + k results in a horizontal shift of h units to the left and a vertical shift of k units upward. Also, keep in mind that these are basic transformation rules; scaling (stretching or compressing) is also a type of transformation.
