Translations of Functions
Simple Explanation
A translation slides a graph to a new position without changing its shape at all. y = f(x−h) shifts the graph h units horizontally; y = f(x)+k shifts it k units vertically. Combined, y = f(x−h)+k moves the whole graph to a new location.
Why Do We Need It?
Once you know one function's shape, translation lets you instantly picture (and write the equation for) the same shape moved anywhere else on the plane — without re-deriving anything.
See It
A parabola y equals x squared, and a second identical parabola shifted 2 units right and 3 units up
Formula
Translating a Function
y = f(x − h) + k
Shifts the entire graph of y=f(x) horizontally by h units and vertically by k units, without changing its shape at all.
- h
- — the horizontal shift (right if h>0, left if h<0 — note the subtraction)
- k
- — the vertical shift (up if k>0, down if k<0)
When to use it: Whenever a graph needs to be moved to a new location without changing its shape.
Worked Example
Write the equation of a translated function
The graph of y = x² is shifted 4 units left and 5 units down. Write the new equation.
Why Does This Work?
Replacing x with (x−h) means every output that used to occur at some x-value now occurs at x+h instead — shifting the whole graph right by h (and left if h is negative). Adding k separately shifts every output value up by k, with no effect on which x-values are used — the two shifts are independent.
Real-Life Example
Adjusting a schedule's start time
A factory's production graph over the day is shifted 2 hours later when the shift schedule changes, with no change to the shape of the day's pattern.
This is exactly a horizontal translation — the same production pattern, just occurring at a shifted time.
Practice
The graph of y = √x is shifted 3 units right and 2 units up. What is the new equation?
MediumCommon mistake
Getting the horizontal shift direction backwards — y=f(x−3) shifts RIGHT (even though it looks like subtraction), while y=f(x+3) shifts LEFT.
Quick Review
- y = f(x−h) + k: horizontal shift h, vertical shift k.
- Horizontal shifts feel "backwards": (x−h) shifts right, (x+h) shifts left.
- Vertical shifts are intuitive: +k shifts up, −k shifts down.