Lesson 2: Transformations of Linear and Absolute Value Functions

Property

For any function $f(x)$, the graph can be translated:

1. Horizontal shift: The graph of $g(x) = f(x-h)$ is the graph of $f(x)$ shifted $h$ units horizontally.
2. Vertical shift: The graph of $g(x) = f(x) + k$ is the graph of $f(x)$ shifted $k$ units vertically.

Examples

Property

To reflect a function across the y-axis, replace $x$ with $-x$ in the function: $y = f(-x)$

Examples

Property

To reflect a function across the x-axis, replace $f(x)$ with $-f(x)$:

This transformation multiplies every y-coordinate by $-1$ while keeping x-coordinates unchanged.

Property

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

• If $0 < |a| < 1$, the graph is compressed vertically (appears "wider" for parabolas).
• If $|a| > 1$, the graph is stretched vertically (appears "skinnier" for parabolas).
• If $a$ is negative, the graph is also reflected across the x-axis.

Examples