Translations of Function Graphs (Shifting Graphs)
📐 Vertical Translations (Up & Down)
Starting with a parent function $y = f(x)$, we shift it vertically by adding or subtracting a constant outside the function:
Shift UP by $k$ units:
$$y = f(x) + k \quad (k > 0)$$
Shift DOWN by $k$ units:
$$y = f(x) - k \quad (k > 0)$$
Memory Tip The constant is outside the function — it directly changes the $y$-values. Positive goes up, negative goes down. It works exactly as you'd expect!
📐 Horizontal Translations (Left & Right)
We shift a graph horizontally by adding or subtracting a constant inside the function (directly to $x$):
Shift LEFT by $h$ units:
$$y = f(x + h) \quad (h > 0)$$
Shift RIGHT by $h$ units:
$$y = f(x - h) \quad (h > 0)$$
⚠️ Watch Out! This is counterintuitive! The constant is inside the function, and the direction is opposite to the sign. Subtracting moves right, adding moves left.