Transformations of Functions
Introduced| English |
|---|
| translation/trænˈsleɪʃn/ |
| dilation/daɪˈleɪʃn/ |
| reflection/rɪˈflekʃn/ |
| domain/dəˈmeɪn/ |
| range/reɪndʒ/ |
Move it, stretch it, flip it
- Learn the shape of one "parent" function, and you get a whole family for free.
- Slide it, stretch it, or flip it, and the equation changes in a predictable way.
- Every transformed graph is just the parent wearing a small edit.
- Master four moves and you can graph thousands of functions by eye.
Translations: sliding the graph
- A translation 平移 slides the graph without changing its shape.
- Adding outside, $f(x) + k$, shifts it vertically by $k$ (up if positive).
- Replacing $x$ with $(x - h)$ inside shifts it horizontally by $h$.
- Outside changes act on the output (up/down); inside changes act on the input (left/right).
Adding a constant outside the function, $f(x) + 3$, shifts the graph…
Adding outside changes the output directly, so every point rises by 3 — a vertical shift up.
Dilations and reflections
- A dilation 伸缩 stretches or squashes the graph: $a\,f(x)$ with $a > 1$ stretches it taller.
- A reflection 反射 flips it: $-f(x)$ mirrors across the x-axis; $f(-x)$ mirrors across the y-axis.
- A fraction like $0.5\,f(x)$ squashes the graph toward the x-axis.
- These reshape the graph; translations only move it.
Stretch, flip, and shift a parabola
y = a·f(b(x − h)) + k
Start from y = x². Make a negative to flip it, make a large to stretch it, and change c to slide it up or down.
Multiplying the whole function by $-1$, giving $-f(x)$, produces a ____ across the x-axis.
A negative on the outside flips every output's sign, mirroring the graph across the x-axis.
The master rule
- Every transformation fits one template: $g(x) = a\,f\big(b(x - h)\big) + k$.
- $a$ = vertical stretch/flip, $b$ = horizontal stretch/flip, $h$ = horizontal shift, $k$ = vertical shift.
- Read the constants straight out of the rule to know exactly what happened.
- Outside the $f$ ($a$ and $k$) is vertical; inside ($b$ and $h$) is horizontal.
Replacing $x$ with $(x - 2)$ inside the function shifts the graph…
Horizontal shifts work backwards: $(x-2)$ moves the graph right by 2, not left. Inside changes feel reversed.
Select all correct statements about transformations.
A constant inside shifts horizontally, not vertically. The other three are correct.
Match each change to its effect on $y = f(x)$.
Outside = vertical (shift, reflect, stretch); inside = horizontal (and reversed).
Effect on domain and range
- Vertical moves (stretch, shift, reflect over x-axis) reshape the range 值域.
- Horizontal moves (shift, stretch, reflect over y-axis) reshape the domain 定义域.
- A vertical shift up by $3$ adds $3$ to every output — the range rises by $3$.
- Key features (vertex, intercepts, asymptotes) move with the graph.
A vertical stretch or a vertical shift can change the range of a function.
Vertical transformations act on the outputs, so they reshape the range. Horizontal ones act on inputs and affect the domain instead.
Horizontal transformations feel backwards. Inside the function, $(x - 2)$ shifts the graph right (not left), and $f(2x)$ compresses it (not stretches). Inside changes always do the opposite of what the sign suggests.
Describe $g(x) = -2(x-1)^2 + 4$ built from the parent $f(x) = x^2$.
- Inside $(x-1)$ → shift right by 1.
- The $-2$ outside → stretch by 2 and reflect across the x-axis (opens downward).
- The $+4$ outside → shift up by 4, putting the vertex at $(1, 4)$.
Every graph is a parent function transformed by $g(x) = a\,f(b(x-h)) + k$: a translation (shift), a dilation (stretch), or a reflection (flip). Outside changes are vertical and reshape the range; inside changes are horizontal, reshape the domain, and run backwards.