Starting from y=f(x):
| Transformation | New equation | Effect |
|---|
| Translation | y=f(x−h)+k | shift RIGHT by h, UP by k |
| Vertical stretch | y=af(x) | factor a from the x-axis |
| Horizontal stretch | y=f(bx) | factor b1 from the y-axis |
| Reflection in x-axis | y=−f(x) | flip top-bottom |
| Reflection in y-axis | y=f(−x) | flip left-right |
KEY caution. Inside-the-bracket changes are 'opposite' of what they look like:
- f(x−3) → shift RIGHT by 3 (not left).
- f(2x) → COMPRESS horizontally by factor 2 (not stretch).
Worked example. Describe y=−2f(x−3)+5 as a sequence of transformations of y=f(x).
- Translate right by 3 (inside-bracket).
- Vertical stretch factor 2 (outside).
- Reflect in the x-axis (negative sign).
- Translate up by 5 (outside).
(Order: do INSIDE-bracket operations FIRST, then outside.)
Worked example. The graph of y=f(x) has a turning point at (2,−1). What is the corresponding point on y=f(x+3)−4?
Inside: x→x−3 relative to original f — point becomes (2−3,−1)=(−1,−1).
Outside: y→y−4 — point becomes (−1,−5).
Always work the point through each transformation in order.