This section covers how to build new functions from existing ones through transformations, combinations, and compositions.
| Skill | Description | Difficulty |
|---|---|---|
| Function Transformations | Overview of graph transformations | Beginner |
| Vertical and Horizontal Shifts | Translating graphs | Beginner |
| Reflections and Stretches | Flipping and scaling graphs | Intermediate |
| Combining Transformations | Multiple transformations in sequence | Intermediate |
| Function Arithmetic | Sum, difference, product, quotient | Intermediate |
| Function Composition | $(f \circ g)(x) = f(g(x))$ | Intermediate |
graph TD
A["Function Transformations"] --> B["Vertical/Horizontal Shifts"]
A --> C["Reflections & Stretches"]
B --> D["Combining Transformations"]
C --> D
E["Function Arithmetic"] --> F["Function Composition"]
D --> F
click A "function-transformations.html"
click B "vertical-horizontal-shifts.html"
click C "reflections-stretches.html"
click D "combining-transformations.html"
click E "function-arithmetic.html"
click F "function-composition.html"
| Transformation | Formula | Effect on Graph |
|---|---|---|
| Vertical shift up $c$ | $f(x) + c$ | Graph moves up |
| Vertical shift down $c$ | $f(x) - c$ | Graph moves down |
| Horizontal shift right $c$ | $f(x - c)$ | Graph moves right |
| Horizontal shift left $c$ | $f(x + c)$ | Graph moves left |
| Vertical stretch by $c$ | $c \cdot f(x)$ | Graph stretches vertically |
| Horizontal stretch by $c$ | $f(x/c)$ | Graph stretches horizontally |
| Reflection over $x$-axis | $-f(x)$ | Graph flips vertically |
| Reflection over $y$-axis | $f(-x)$ | Graph flips horizontally |
| ← Section 1.2 | Skills Index | Section 1.4 → |