Transformations of Functions

Parent Functions

Know the shapes of: $x$, $x^2$, $x^3$, $\sqrt{x}$, $|x|$, $\frac{1}{x}$

Transformation Rules

Starting from $y = f(x)$:

$y = af(b(x - h)) + k$

| Parameter | Effect | |-----------|--------| | $k > 0$ | Shift up $k$ units | | $k < 0$ | Shift down $|k|$ units | | $h > 0$ | Shift right $h$ units | | $h < 0$ | Shift left $|h|$ units | | $a > 1$ | Vertical stretch by $a$ | | $0 < a < 1$ | Vertical compression by $a$ | | $a < 0$ | Reflect over x-axis | | $b > 1$ | Horizontal compression by $\frac{1}{b}$ | | $0 < b < 1$ | Horizontal stretch by $\frac{1}{b}$ | | $b < 0$ | Reflect over y-axis |

Order of Transformations

Inside → Outside (affects input first, then output):

1. Horizontal reflection ($b < 0$)
2. Horizontal stretch/compression
3. Horizontal shift
4. Vertical stretch/compression
5. Vertical reflection ($a < 0$)
6. Vertical shift

Function Composition

$(f \circ g)(x) = f(g(x))$

Apply $g$ first, then $f$ to the result.

Domain of $f \circ g$: All $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Inverse Functions

$f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x$

To find $f^{-1}$:

1. Replace $f(x)$ with $y$
2. Swap $x$ and $y$
3. Solve for $y$

Graphs of $f$ and $f^{-1}$ are reflections over $y = x$.

Even and Odd Functions

• Even: $f(-x) = f(x)$ → symmetric about y-axis
• Odd: $f(-x) = -f(x)$ → symmetric about origin

AP Tip: Horizontal transformations are "opposite" of what you'd expect. $f(x - 3)$ shifts RIGHT 3, not left.