Stretching and Compressing Graphs of Functions
Big Idea: When we multiply a function by a constant or multiply the input by a constant, we can stretch or compress its graph vertically or horizontally. This is one of the most powerful tools in understanding how functions behave!
🔼 Vertical Stretching & Compressing
Starting with a parent function $f(x)$, the transformation:
$$g(x) = a \cdot f(x)$$
- If $|a| > 1$: The graph is vertically stretched (pulled away from the x-axis). Every y-value is multiplied by $a$, making the graph taller.
- If $0 < |a| < 1$: The graph is vertically compressed (pushed toward the x-axis). Every y-value is multiplied by $a$, making the graph shorter.
- If $a < 0$: The graph is also reflected over the x-axis in addition to the stretch/compress.
Key insight: Vertical transformations affect the output (y-values). Points on the x-axis ($y = 0$) don't move!
↔️ Horizontal Stretching & Compressing
Starting with a parent function $f(x)$, the transformation:
$$g(x) = f(b \cdot x)$$
- If $|b| > 1$: The graph is horizontally compressed (squeezed toward the y-axis). This is counterintuitive — bigger $b$ means narrower graph!
- If $0 < |b| < 1$: The graph is horizontally stretched (pulled away from the y-axis). Smaller $b$ means wider graph!
- If $b < 0$: The graph is also reflected over the y-axis in addition to the stretch/compress.
Key insight: Horizontal transformations affect the input (x-values). Points on the y-axis ($x = 0$) don't move! The horizontal scale factor is $\frac{1}{|b|}$.