Horizontal Translations of Piecewise Functions
Horizontal translations of the absolute value function f(x) = |x - h| shift the graph of f(x) = |x| left or right by h units — a piecewise function transformation in enVision Algebra 1 Chapter 5 for Grade 11. For f(x) = |x - 3|, the vertex moves from (0,0) to (3,0) — a shift right 3 units. For f(x) = |x + 4|, rewrite as |x - (-4)| and shift left 4 units, moving the vertex to (-4, 0). The key is that h in |x - h| represents a rightward shift when positive and a leftward shift when h is negative (written as |x + |h||).
Key Concepts
The graph of $f(x) = |x h|$ shifts the graph of $f(x) = |x|$ horizontally $h$ units. If $h 0$, shift the graph horizontally right $h$ units. If $h < 0$, shift the graph horizontally left $|h|$ units.
Common Questions
How does f(x) = |x - 3| differ from f(x) = |x|?
The graph of |x - 3| is the graph of |x| shifted 3 units to the right. The vertex moves from (0, 0) to (3, 0).
How do you graph f(x) = |x + 4|?
Rewrite as |x - (-4)|, so h = -4. This shifts the graph 4 units to the left, moving the vertex from (0, 0) to (-4, 0).
Why does |x - h| shift right when h > 0?
Replacing x with (x - h) shifts the input threshold to the right. The vertex, which occurs when the inside equals zero, now happens at x = h instead of x = 0.
Does a horizontal translation change the shape of the absolute value graph?
No. The V-shape remains identical — only its position changes.
What is the vertex of f(x) = |x - 7| + 2?
The horizontal shift moves the vertex to x = 7, and the +2 shifts it up to y = 2. Vertex: (7, 2).