Graph Transformations

Definition

Graph transformations are algebraic modifications to a parent function that result in predictable geometric changes to its graph. These include shifts (translations), compressions, stretches, and reflections.

Why It Matters

Transformations allow us to reuse one irreducible truth for infinite contexts; by understanding how a parent function can be shifted, scaled, and flipped, we gain the power to quickly model complex real-world behaviors without starting from scratch every time.

Core Concepts

Let $c > 0$ and $a > 0$ :

Vertical Shifts: $y = f(x) \pm c$ (Moves graph up or down).
- How to read: “The function f of x plus c, or f of x minus c.”
- Meaning: Output shifts change height without altering shape.
Horizontal Shifts: $y = f(x \pm c)$ (Moves graph left or right; note that $f(x-c)$ moves right).
- How to read: “The function f of the quantity x minus c, or f of the quantity x plus c.”
- Meaning: Input shifts slide the graph horizontally — opposite sign intuition.
Reflections:
- About $x$ -axis: $y = -f(x)$ $y = - f (x)$ (Flips vertically).
  - How to read: “The value negative f of x.”
  - Meaning: Negate every $y$ -value; mirror across the $x$ -axis.
- About $y$ -axis: $y = f(-x)$ $y = f (- x)$ (Flips horizontally).
  - How to read: “The function f of negative x.”
  - Meaning: Negate every $x$ -input; mirror across the $y$ -axis.
Stretches & Compressions:
- Vertical: $y = af(x)$ $y = a f (x)$ . Stretch if $a > 1$ $a > 1$ , compression if $0 < a < 1$ $0 < a < 1$ .
  - How to read: “The value a times f of x.”
  - Meaning: Scale all outputs by $a$ — taller when $a > 1$ , shorter when $0 < a < 1$ .
- Horizontal: $y = f(ax)$ $y = f (a x)$ . Compression if $a > 1$ $a > 1$ , stretch if $0 < a < 1$ $0 < a < 1$ (Inverse effect).
  - How to read: “The function f of the quantity a times x.”
  - Meaning: Speed up inputs by $a$ — graph squeezes horizontally when $a > 1$ .

Graph Transformations

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes