Functions - Graphing Functions and Transformations

Vertical and Horizontal Translations: A vertical translation $y = f(x) + k$ shifts the graph up by $k$ units if $k > 0$ and down if $k < 0$. A horizontal translation $y = f(x - h)$ shifts the graph right by $h$ units if $h > 0$ and left if $h < 0$. Visually, these movements preserve the shape and orientation of the curve, simply sliding it across the Cartesian plane.

Vertical Stretching and Compression: The transformation $y = a f(x)$ represents a vertical stretch from the $x$-axis by a scale factor of $a$. If $a > 1$, the graph appears 'taller' or steeper as points move further from the $x$-axis. If $0 < a < 1$, the graph is compressed vertically, appearing 'flatter'. If $a$ is negative, the graph is also reflected across the $x$-axis.

Horizontal Stretching and Compression: The transformation $y = f(bx)$ represents a horizontal stretch from the $y$-axis by a scale factor of $\frac{1}{b}$. Visually, if $b > 1$, the graph is compressed toward the $y$-axis (pushed inward), and if $0 < b < 1$, the graph is stretched away from the $y$-axis (pulled outward).

Reflections: The transformation $y = -f(x)$ is a reflection in the $x$-axis, where all positive $y$-values become negative and vice versa, creating a 'mirror image' across the horizontal axis. The transformation $y = f(-x)$ is a reflection in the $y$-axis, mirroring the graph horizontally across the vertical axis.

Composite Transformations: When multiple transformations are applied, such as $y = a f(b(x - h)) + k$, the order matters. Generally, horizontal transformations (inside the function) are handled first, often starting with the shift $h$ or factor $b$, followed by vertical transformations (outside the function) like the stretch $a$ and finally the vertical shift $k$.

Inverse Functions: The graph of the inverse function $y = f^{-1}(x)$ is a reflection of the original function $y = f(x)$ across the identity line $y = x$. Visually, every point $(a, b)$ on the original graph is mapped to $(b, a)$ on the inverse graph, effectively swapping the domain and range.

Absolute Value Transformations: The transformation $y = |f(x)|$ takes any part of the graph lying below the $x$-axis ($y < 0$) and reflects it above the $x$-axis, while parts already above remain unchanged. The transformation $y = f(|x|)$ discards the portion of the graph where $x < 0$ and replaces it with a reflection of the portion where $x \geq 0$ across the $y$-axis, making the function even.