Horizontal Translation: Modifying Function Inputs
The graph of f(x) = m(x - h) + b shifts the graph of f(x) = mx + b horizontally by h units — a transformation concept in enVision Algebra 1 Chapter 3 for Grade 11. For f(x) = 2(x - 3) + 1, the graph of f(x) = 2x + 1 shifts right 3 units. For f(x) = -½(x + 4) + 3, rewrite as -½(x - (-4)) + 3 — a shift left 4 units. Horizontal translations change every x-value on the graph by h, effectively moving the entire line while preserving its slope. The y-intercept changes but the slope remains identical to the original function.
Key Concepts
The graph of $f(x) = m(x h) + b$ shifts the graph of $f(x) = mx + b$ horizontally $h$ units. If $h 0$, shift the line horizontally right $h$ units. If $h < 0$, shift the line horizontally left $|h|$ units.
Common Questions
How does f(x) = 2(x - 3) + 1 relate to f(x) = 2x + 1?
f(x) = 2(x - 3) + 1 shifts the graph of f(x) = 2x + 1 three units to the right. The slope remains 2, but every point moves 3 units horizontally.
How do you graph f(x) = -½(x + 4) + 3?
Rewrite as -½(x - (-4)) + 3. This shifts the graph of f(x) = -½x + 3 four units to the left.
Does a horizontal translation change the slope of a linear function?
No. The slope m is determined by the coefficient in front of (x - h), not by h itself. Only the horizontal position changes.
Why does (x - h) shift the graph right when h > 0?
A positive h means the function reaches the same y-value at a larger x. The entire graph slides h units to the right to compensate for subtracting h from x.
What is the new y-intercept after shifting f(x) = 2x + 1 right by 3?
Set x = 0 in f(x) = 2(x-3) + 1: f(0) = 2(-3) + 1 = -5. The new y-intercept is (0, -5), different from the original (0, 1).