Transformations of Functions
Function Transformations
Transformations describe how a parent graph such as $y=f(x)$ becomes $y=a,f(b(x-h))+k$. Every symbol has a predictable geometric effect, and the exam tests whether you can read both directions: from equation to graph, and from graph to equation.
Concepts to master. Vertical stretches and reflections are governed by $a$: $|a|>1$ stretches, $0<|a|<1$ compresses, and $a<0$ reflects across the $x$-axis. Horizontal stretches and reflections are governed by $b$: the factor is $\tfrac{1}{|b|}$, so $b=2$ compresses horizontally by $\tfrac12$, and $b<0$ reflects across the $y$-axis. Translations are $h$ (horizontal) and $k$ (vertical). Know the mapping rule: a point $(x,y)$ on the parent moves to $\left(\tfrac{x}{b}+h,; ay+k\right)$.
Traps. The horizontal shift has the opposite sign of what students expect: $f(x-3)$ shifts right 3, not left. Horizontal changes operate inside the argument and therefore act in the reverse order and reverse sense from vertical ones. When several transformations combine, order matters for reading but the mapping rule handles all of them at once. Also watch domain and range: a reflection or translation changes them, and a horizontal stretch changes the period of trigonometric graphs.
Rule of thumb. Rewrite any expression in the standard form $a,f(b(x-h))+k$ before naming transformations. If the coefficient of $x$ is not 1, factor it out first.
Worked example. Start with $y=\sqrt{x}$ and describe $y=-2\sqrt{-x+4}+1$. Factor inside: $\sqrt{-(x-4)}$. Then $b=-1$ (reflect in $y$-axis), $h=4$ (shift right 4), $a=-2$ (reflect in $x$-axis, stretch vertically by 2), $k=1$ (shift up 1). The parent point $(0,0)$ maps to $(4,1)$, and $(1,1)$ maps to $(3,-1)$.