Lesson 4: Translations of Functions

Property

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: $g(x) = f(x - h) + k$

Examples

Property

To track translations, identify key reference points on the original function $f(x)$, then apply the transformation rule: if $(a, b)$ is on $f(x)$, then $(a + h, b + k)$ is on $g(x) = f(x - h) + k$.

Examples

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

• If $k > 0$, shift the parabola vertically up $k$ units.
• If $k < 0$, shift the parabola vertically down $|k|$ units.

Examples

• To graph $f(x) = x^2 + 4$, you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$.

• To graph $f(x) = x^2 - 5$, you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$.