Mathematics
SAT-48: Function Transformations — Shifts and Reflections
Shift and reflect a graph by changing f(x): outside moves it vertically, inside moves it horizontally (and backwards).
SAT-48: Function Transformations — Shifts and Reflections
Description: If you know the graph of f(x), you can move it without re-graphing. Small changes to f(x) cause predictable shifts and reflections.
Vertical changes (outside the function)
- f(x) + k → shift up k units.
- f(x) − k → shift down k units.
- −f(x) → reflect over the x-axis (flip upside down).
(Oʻzbekcha: funksiyaning tashqarisidagi oʻzgarish grafikni vertikal (yuqori-past) suradi.)
Horizontal changes (inside the function)
These work opposite to what you'd expect:
- f(x − h) → shift right h units.
- f(x + h) → shift left h units.
- f(−x) → reflect over the y-axis.
Rule: inside changes are backwards. f(x − 3) moves right 3, not left. (Oʻzbekcha: ichkaridagi oʻzgarish teskari ishlaydi: x − 3 → oʻngga suriladi.)
Worked Example
Starting from f(x), describe g(x) = f(x − 2) + 5.
- Inside (x − 2): shift right 2.
- Outside (+5): shift up 5.
- So g is f moved right 2 and up 5.
(Oʻzbekcha: ichkarisi oʻngga 2, tashqarisi yuqoriga 5 suradi.)
Practice
How does h(x) = −f(x + 4) transform f(x)?
Show answer
Inside (x + 4) → shift left 4; the leading minus → reflect over the x-axis.
Key words — Kalit soʻzlar
- Transformation — almashtirish (oʻzgartirish)
- Shift — surish
- Reflection — koʻzguda aks ettirish
- Vertical — vertikal (tik)
- Horizontal — gorizontal (yotiq)
- Up / Down — yuqoriga / pastga
- Left / Right — chapga / oʻngga
- x-axis / y-axis — x oʻqi / y oʻqi
- Units — birliklar
Summary
- Outside f(x): vertical — +k up, −k down, −f(x) flips over x-axis.
- Inside f(x): horizontal and backwards — (x − h) right, (x + h) left, f(−x) flips over y-axis.