The Four Transformation Types
Every transformation is one of these four
Vertical Translation
g(x) = f(x) + k
Shifts the graph k units up (down if k < 0). Affects y-values — adds k to every output.
g(x) = f(x) − 2 → down 2
Horizontal Translation
g(x) = f(x + h)
Shifts the graph −h units (opposite sign!). Inside the parentheses → affects x → does the opposite.
g(x) = f(x + 2) → left 2 (NOT right!)
Vertical Dilation
g(x) = a·f(x)
Stretches vertically by factor |a|. If a < 0, also reflects over the x-axis.
g(x) = −2f(x) → stretch ×2, flip over x-axis
Horizontal Dilation
g(x) = f(bx)
Stretches horizontally by factor 1/|b|. If b < 0, also reflects over the y-axis.
g(x) = f(2x) → compress by ½ (NOT stretch ×2)
The Two Golden Rules
Memorise these — they explain everything
Inside parentheses → x-values
Transformations inside f(…) affect the x-values and do the OPPOSITE of what they look like.
f(x + 3) looks like +3 but shifts left 3.
f(2x) looks like ×2 but compresses by ½.
f(x + 3) looks like +3 but shifts left 3.
f(2x) looks like ×2 but compresses by ½.
Outside parentheses → y-values
Transformations outside f(…) affect the y-values and do EXACTLY what they look like.
f(x) + 5 shifts up 5.
3·f(x) stretches vertically by 3.
f(x) + 5 shifts up 5.
3·f(x) stretches vertically by 3.
Describing Transformations
Examples 1 & 2
📌 Example 1 — Describe all transformations: g(x) = 2f(x − 3) + 1
g(x) = 2f(x − 3) + 1
1
Inside: (x − 3) → horizontal translation. Opposite of −3 → right 3 units.2
Outside: coefficient 2 → vertical dilation by factor of 2.3
Outside: +1 → vertical translation up 1 unit.Right 3 · Vertical dilation ×2 · Up 1
📌 Example 2 — Describe all transformations: n(x) = −4m(2x)
n(x) = −4m(2x)
1
Inside: coefficient 2 on x → horizontal dilation. Factor = 1/|2| = ½ (compresses horizontally).2
Outside: coefficient −4 → vertical dilation by 4, and a = −4 < 0 → reflects over the x-axis.Horizontal dilation ×½ · Vertical dilation ×4 · Reflect over x-axis
Effect on Domain & Range
Example 5 — finding new domain and range after transformation
The Method
For each boundary of the original domain/range, apply the transformation algebraically.
Domain (x-values): set the new input equal to the original boundary and solve for the new x.
Range (y-values): apply the outside transformations directly to each boundary.
📌 Example 5 — f has domain −3 ≤ x ≤ 5 and range 1 ≤ y ≤ 3. Find domain and range of g(x) = 2f(x+3) − 4.
g(x) = 2f(x + 3) − 4
1
Domain: Set x+3 equal to each boundary of f's domain and solve for x.x+3 = −3 → x = −6 x+3 = 5 → x = 2
2
Range: Apply outside transformations (×2 then −4) to each boundary of f's range.2(1) − 4 = −2 2(3) − 4 = 2
Domain of g
x+3 ∈ [−3, 5] → x ∈ [−6, 2]
−6 ≤ x ≤ 2
Range of g
2y − 4: 2(1)−4=−2, 2(3)−4=2
−2 ≤ y ≤ 2
Evaluating from Tables
Examples 6, 7, 8 — using transformations with table data
📌 Example 6 — Table of f. Let h(x) = 3f(2x) − 1. Find h(−1), h(2), h(0).
| x | −2 | 0 | 2 | 4 | 6 |
|---|---|---|---|---|---|
| f(x) | 1 | −1 | 0 | 3 | 7 |
a
h(−1) = 3f(2·(−1)) − 1 = 3f(−2) − 1 = 3(1) − 1 = 2b
h(2) = 3f(2·2) − 1 = 3f(4) − 1 = 3(3) − 1 = 8c
h(0) = 3f(2·0) − 1 = 3f(0) − 1 = 3(−1) − 1 = −4h(−1) = 2 h(2) = 8 h(0) = −4
📌 Example 7 — Build g(x) = af(bx) + c from three transformations applied in order.
1
Horizontal dilation by factor 3 → b = 1/3 (inside: f(x/3))2
Vertical dilation by factor 4 → a = 43
Vertical translation by −7 → c = −7a = 4 b = 1/3 c = −7 → g(x) = 4f(x/3) − 7
📌 Example 8 — Table of h. p(x) = 3h(2x) + 1. Find p(−6).
| x | −12 | −6 | −3 | 0 | 3 |
|---|---|---|---|---|---|
| h(x) | 1 | −1 | 0 | 3 | 7 |
1
p(−6) = 3h(2·(−6)) + 1 = 3h(−12) + 12
From table: h(−12) = 13
3(1) + 1 = 4p(−6) = 4
📌 Example 9 — f(x) = 3x − 2. g is a horizontal translation of f by +4. Find g(x) = mx + b.
1
Horizontal translation right 4 → replace x with (x − 4): g(x) = f(x − 4)2
g(x) = 3(x − 4) − 2 = 3x − 12 − 2 = 3x − 14m = 3 b = −14
Quick Reference
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🔑 All Four Formulas
g(x) = f(x) + k
Vertical translation: up k (or down if k<0)
g(x) = f(x + h)
Horizontal translation: −h units (OPPOSITE)
g(x) = a·f(x)
Vertical dilation |a|; a<0 → reflect x-axis
g(x) = f(bx)
Horizontal dilation 1/|b|; b<0 → reflect y-axis
⚠️ Common Mistakes
❌ f(x+3) shifts right
Inside → OPPOSITE. f(x+3) shifts LEFT 3. Right would be f(x−3).
❌ f(2x) stretches by factor 2
Horizontal dilation by 1/|b| = ½. f(2x) COMPRESSES horizontally.
❌ −af(x) only reflects
g(x) = −4f(x) BOTH dilates (×4) AND reflects over the x-axis.