Graph Scaling

Definition

Graph scaling involves non-rigid transformations that alter the proportions of a graph by stretching or compressing it either vertically or horizontally.

Why It Matters

Scaling allows us to adjust the ‘amplitude’ and ‘frequency’ of mathematical models, ensuring that abstract functions can be calibrated to match the true magnitude and rate of physical systems.

Core Concepts

• Vertical Stretch/Compression: For $c > 1$: $y = c f(x)$ (stretch) or $y = \frac{1}{c} f(x)$ (compression).
  • How to read: “The value c times f of x, or one divided by c times f of x.”
  • Meaning: Multiply outputs by $c$ — stretches vertically when $c > 1$, compresses when $0 < c < 1$.
• Horizontal Stretch/Compression: For $c > 1$: $y = f(x/c)$ (stretch) or $y = f(cx)$ (compression).
  • How to read: “The function f of the quantity x divided by c, or f of the quantity c times x.”
  • Meaning: Dividing input by $c$ slows input growth (stretch); multiplying input by $c$ speeds it up (compression).
• Inversion of Intuition: Horizontal transformations often work “backwards” from vertical ones; multiplying $x$ by $c$ compresses the graph because it reaches input thresholds $c$ times faster.

Connected Concepts

• Graph Reflecting
• Graph Transformations
• Graph Transformations: Shifting