General Translation Formulas for All Functions
General translation formulas apply universally to all function types: g(x) = f(x) + k for vertical translation, g(x) = f(x − h) for horizontal translation, and g(x) = f(x − h) + k for combined translation. Grade 11 students in enVision Algebra 1 (Chapter 10: Working With Functions) learn that positive k shifts up, negative k shifts down, positive h shifts right, and negative h shifts left — regardless of whether the function is quadratic, exponential, radical, or absolute value. This unified framework allows students to translate any function.
Key Concepts
For any function $f(x)$, translations follow these universal formulas: Vertical translation: $g(x) = f(x) + k$ Horizontal translation: $g(x) = f(x h)$ Combined translation: $g(x) = f(x h) + k$.
Common Questions
What is the general formula for a vertical translation of any function?
g(x) = f(x) + k, where positive k shifts the graph up and negative k shifts it down.
What is the general formula for a horizontal translation of any function?
g(x) = f(x − h), where positive h shifts the graph right and negative h shifts it left.
How do you combine horizontal and vertical translations?
Use g(x) = f(x − h) + k, applying the horizontal shift h first and the vertical shift k second.
Do these translation formulas work for all types of functions?
Yes — these formulas apply universally to quadratic, exponential, radical, absolute value, and any other function type.
What is the translated graph of f(x) = x² if h = 3 and k = −2?
The translated function is g(x) = (x − 3)² − 2, with vertex shifted from (0, 0) to (3, −2).
Why does subtracting h inside the function shift the graph right?
The graph reaches its reference point when x − h = 0, which requires x = h. So a positive h means the reference point moves to x = h (to the right).