Lesson 4: Transformations of Piecewise-Defined Functions

Property

The graph of a piecewise-defined function $f(x) + k$ shifts the graph of $f(x)$ vertically $k$ units.

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

Examples

Property

The graph of $f(x) = |x - h|$ shifts the graph of $f(x) = |x|$ horizontally $h$ units.

• If $h > 0$, shift the graph horizontally right $h$ units.
• If $h < 0$, shift the graph horizontally left $|h|$ units.

Examples

Property

The coefficient $a$ in the transformation $g(x) = a \cdot f(x)$ affects the graph of any function $f(x)$ by stretching, compressing, or reflecting it vertically.

• If $0 < |a| < 1$, the graph is compressed vertically (appears wider for functions like $|x|$)
• If $|a| > 1$, the graph is stretched vertically (appears narrower for functions like $|x|$)
• If $a < 0$, the graph is reflected across the x-axis

Examples

Property

The general form of a transformed absolute value function is $g(x) = a|x - h| + k$, where the vertex is located at $(h, k)$. The parameter $a$ controls vertical stretch/compression and reflection, $h$ controls horizontal translation, and $k$ controls vertical translation.

Examples