Transformations of Functions
Введено| English | Значение |
|---|---|
| translation/trænˈsleɪʃn/ | перенос |
| dilation/daɪˈleɪʃn/ | растяжение |
| reflection/rɪˈflekʃn/ | отражение |
| domain/dəˈmeɪn/ | область определения |
| range/reɪndʒ/ | область значений |
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…
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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…
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Horizontal shifts work backwards: $(x-2)$ moves the graph правильно by 2, not left. Inside changes feel reversed.
Select все correct statements about transformations.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
A constant inside shifts horizontally, not vertically. The other three are correct.
Match each change to its effect on $y = f(x)$.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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 shift changes only the domain of a function, not its range.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
False. Vertical transformations act on the outputs, so they change the range. Horizontal transformations act on the inputs and affect the domain.
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.
Write from a point that lands at (3, 1)
- g of x equals f of (x minus 3) plus 1.
- The parent graph has the point (0, 0). Where does that point go?
g of x equals f of (x minus 3) plus 1, and the parent graph passes through (0, 0). Where does that point land?
Input shift then output shift.
Where does (0, 0) land?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Right 3 and up 1.