Functions - Transformations of graphs (translations, reflections, stretches)

Vertical Translation: Represented by the function $y = f(x) + k$. Visually, this transformation slides the entire graph up the $y$-axis if $k > 0$ or down if $k < 0$. The shape of the curve remains identical, but every point $(x, y)$ is mapped to $(x, y + k)$. For example, the vertex of a parabola moving from $(0,0)$ to $(0,3)$ is a vertical translation where $k=3$.

Horizontal Translation: Represented by the function $y = f(x - h)$. Visually, this moves the graph left or right along the $x$-axis. Note the sign: $f(x - 2)$ shifts the graph $2$ units to the right, while $f(x + 2)$ shifts the graph $2$ units to the left. The mapping for points is $(x, y) \rightarrow (x + h, y)$. The graph appears to 'slide' without changing its width or height.

Reflection in the $x$-axis: Represented by $y = -f(x)$. This transformation acts as a vertical flip. Visually, the $x$-axis behaves like a mirror; points above the axis move below it, and points below move above. Mathematically, the $y$-coordinates negate, mapping $(x, y) \rightarrow (x, -y)$. A 'U' shaped parabola $x^2$ becomes an 'n' shaped parabola $-x^2$.

Reflection in the $y$-axis: Represented by $y = f(-x)$. This transformation acts as a horizontal flip. Visually, the $y$-axis behaves like a mirror; the right side of the graph moves to the left and vice versa. The mapping for points is $(x, y) \rightarrow (-x, y)$. This is particularly visible in functions like $y = \sqrt{x}$, which would flip to point toward negative $x$ values as $y = \sqrt{-x}$.

Vertical Stretch and Compression: Represented by $y = a \cdot f(x)$ where $a > 0$. Visually, if $a > 1$, the graph is 'stretched' away from the $x$-axis, appearing thinner and taller. If $0 < a < 1$, the graph is 'compressed' toward the $x$-axis, appearing flatter and wider. Every $y$-coordinate is multiplied by $a$, mapping $(x, y) \rightarrow (x, ay)$. Points on the $x$-axis (the $x$-intercepts) remain invariant (fixed).

Horizontal Stretch and Compression: Represented by $y = f(qx)$ where $q > 0$. Visually, this transformation pulls or pushes the graph toward the $y$-axis. If $q > 1$, the graph is compressed horizontally by a factor of $\frac{1}{q}$ (it looks narrower). If $0 < q < 1$, the graph is stretched horizontally (it looks wider). The mapping for points is $(x, y) \rightarrow (\frac{x}{q}, y)$. Points on the $y$-axis remain invariant.

Combined Transformations: Multiple transformations can be applied to a parent function $f(x)$ to create a transformed function $g(x) = a \cdot f(b(x - h)) + k$. Visually, the graph is distorted and moved from its original origin. The standard order of operations typically involves applying stretches and reflections first, followed by translations, though following the order of operations within the algebraic expression is key to identifying the correct final position.