Function Transformations

Definition

Function transformations are operations performed on a function’s rule that result in predictable changes to its graph. These include sliding (translation), flipping (reflection), and stretching or compressing (scaling).

Why It Matters

Transformation theory allows for ‘modular’ modeling; instead of deriving every new function from first principles, we can quickly construct complex behaviors by shifting, scaling, and flipping known ‘parent’ functions, drastically accelerating the model-building process.

Core Concepts

Translations (Sliding): Moving the graph without changing its shape or orientation.
- Vertical Shift: $f(x) + C$ $f (x) + C$ moves the graph up $C$ $C$ units; $f(x) - C$ $f (x) - C$ moves it down $C$ $C$ units.
  - How to read: “The functions f of x plus C and f of x minus C.”
  - Meaning: Adding to the output moves every point vertically without changing shape.
- Horizontal Shift: $f(x + C)$ $f (x + C)$ moves the graph left $C$ $C$ units; $f(x - C)$ $f (x - C)$ moves it right $C$ $C$ units.
  - How to read: “The functions f of the quantity x plus C and f of the quantity x minus C.”
  - Meaning: Replacing $x$ with $x - h$ delays the input — the graph slides right by $h$ . Signs feel “backward” because the input must compensate.
Reflections (Flipping): Creating a mirror image of the graph.
- Reflection across x-axis: $-f(x)$ $- f (x)$ flips the graph vertically.
  - How to read: “The function negative f of x.”
  - Meaning: Negate every $y$ -value; the graph mirrors across the $x$ -axis.
- Reflection across y-axis: $f(-x)$ $f (- x)$ flips the graph horizontally.
  - How to read: “The function f of negative x.”
  - Meaning: Negate every $x$ -input; the graph mirrors across the $y$ -axis.
Scaling (Stretching and Compressing):
- Vertical Scaling: $A \cdot f(x)$ $A \cdot f (x)$ stretches the graph vertically if $|A| > 1$ $∣ A ∣ > 1$ and compresses it if $0 < |A| < 1$ $0 < ∣ A ∣ < 1$ .
  - How to read: “The function A times f of x.”
  - Meaning: Multiply all outputs by $A$ ; amplitudes scale, inputs unchanged.
- Horizontal Scaling: $f(B \cdot x)$ $f (B \cdot x)$ compresses the graph horizontally if $|B| > 1$ $∣ B ∣ > 1$ and stretches it if $0 < |B| < 1$ $0 < ∣ B ∣ < 1$ .
  - How to read: “The function f of the quantity B times x.”
  - Meaning: Inputs are sped up by factor $B$ ; the graph squeezes horizontally when $|B| > 1$ (opposite intuition from vertical scaling).

Function Transformations

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes