Vertical Translations of Piecewise-Defined Functions
Vertical translations shift the graph of a piecewise-defined function up or down by applying f(x) + k to the entire function, as covered in Grade 11 enVision Algebra 1 (Chapter 5: Piecewise Functions). Adding a positive k moves every point on every piece upward by k units; adding a negative k (or subtracting |k|) moves every point downward. The transformation applies to all pieces simultaneously, shifting the complete graph as one unit without changing its shape or the domain intervals.
Key Concepts
The graph of a piecewise defined function $f(x) + k$ shifts the graph of $f(x)$ vertically $k$ units. If $k 0$, shift the graph vertically up $k$ units. If $k < 0$, shift the graph vertically down $|k|$ units.
Common Questions
How does adding k to a piecewise function affect its graph?
Adding k shifts the entire graph vertically: positive k moves the graph up k units, negative k moves it down |k| units. Every piece shifts equally.
Do all pieces of a piecewise function shift together during a vertical translation?
Yes. Adding k to f(x) increases every output value by k, so all pieces of the graph shift by the same amount in the same direction.
Does a vertical translation change the domain intervals of a piecewise function?
No. The domain conditions (the x-intervals for each piece) remain exactly the same after a vertical translation.
What is the general rule for a vertical translation of any function?
For any function f(x), the graph of g(x) = f(x) + k is the graph of f(x) shifted k units up (if k > 0) or |k| units down (if k < 0).
Can a vertical translation cause a piecewise function to become continuous if it was not before?
No. A vertical translation shifts all pieces equally, so any gaps or jumps in the original remain — the discontinuities persist.
How is a vertical translation different from a vertical stretch?
A translation (adding k) shifts every output by the same constant amount. A stretch (multiplying by a constant) scales every output proportionally.