One pattern for every function
Every function family has a parent function (y = x², y = √x, y = 1/x, y = |x|, y = 2ˣ, y = sin x …). Transformations create the rest of the family:
y = a · f(k(x − d)) + c
| Parameter | Effect |
|---|---|
| a | vertical stretch (|a| > 1) or compression (|a| < 1); reflection in the x-axis if a < 0 |
| k | horizontal stretch by 1/|k|; reflection in the y-axis if k < 0 |
| d | horizontal translation: right if d > 0 |
| c | vertical translation: up if c > 0 |
Vertical changes (outside the function)
These act on y-values, as you’d expect.
- y = f(x) + 3: up 3
- y = 2f(x): twice as tall
- y = −f(x): flipped over the x-axis
Horizontal changes (inside the function)
These act on x-values, and they work the opposite way to what you might expect.
- y = f(x − 3): right 3
- y = f(2x): compressed horizontally by ½
- y = f(−x): flipped over the y-axis
Order of transformations
Apply in this order:
- Stretches/compressions and reflections
- Translations
Mapping points
For y = a f(k(x − d)) + c, each point (x, y) on f maps to:
(x/k + d, a·y + c)
Example: y = −2(x − 3)² + 1 from y = x²
| Parent point | Image |
|---|---|
| (0, 0) | (3, 1) |
| (1, 1) | (4, −1) |
| (−1, 1) | (2, −1) |
| (2, 4) | (5, −7) |
Examples across families
| Function | Description |
|---|---|
| y = √(x + 4) − 2 | √x shifted left 4, down 2 |
| y = 3|x| | |x| stretched vertically by 3 |
| y = 1/(x − 1) + 5 | 1/x shifted right 1, up 5 |
| y = 2^(−x) | 2ˣ reflected in the y-axis |
| y = sin(2x) | sin x compressed horizontally (period 180°) |
Effects on key features
- Vertical asymptotes move with horizontal shifts.
- Horizontal asymptotes move with vertical shifts.
- Domains change with horizontal transformations; ranges with vertical ones.
Worked examples
Example 1: Describe y = (x + 2)² − 5. Parent x²; left 2, down 5.
Example 2: Describe y = −√(3x). Reflected in the x-axis; horizontally compressed by ⅓.
Example 3: Map (4, 2) on y = √x under y = 2√(x − 1) + 3. → (5, 7).
Common mistakes
- Shifting the wrong direction horizontally.
- Applying a horizontal stretch as k instead of 1/k.
- Translating before stretching, when the order matters.
Try it yourself
- Describe y = |x − 4| + 1.
- Describe y = 0.5(x + 3)².
- Image of (1, 1) on y = x² under y = −(x + 2)² + 6?
- Write the equation: y = √x reflected in the x-axis and shifted up 3.
Answers: 1) right 4, up 1 2) vertical compression by ½, left 3 3) (−1, 5) 4) y = −√x + 3
Quick recap
- Transformations turn a parent function into a family member.
- Outside changes act vertically; inside changes act horizontally and in reverse.
- Stretch and reflect first, then translate.