Mathematics Advanced • Year 11 • Module 1 • Lesson 11
Dilations of Functions
Build procedural fluency in vertical (af(x)) and horizontal (f(bx)) dilations, naming, factor calculation, coordinate prediction.
1. Quick recall, the dilation rules
Answer each question in the space provided. 1 mark each
Q1.1 Complete the point mappings:
y = af(x): (x, y) ↦ ( ____, ____ ). Dilation type: ____________ by factor ____ from the ____-axis.
y = f(bx): (x, y) ↦ ( ____, ____ ). Dilation type: ____________ by factor ____ from the ____-axis.
Q1.2 The horizontal trap. For y = f(3x), the dilation factor is ____, so the graph is (stretched / compressed) toward the y-axis. For y = f(x/3), the dilation factor is ____, so the graph is (stretched / compressed). Circle one in each.
Q1.3 Effect on intercepts. Tick T or F:
(a) Vertical dilation y = af(x) leaves x-intercepts unchanged. T / F
(b) Horizontal dilation y = f(bx) leaves y-intercepts unchanged. T / F
2. Worked example, image of (2, 3) and (6, −1) under y = 2 f(x/3)
Follow each line. The reason is given on the right.
Problem. The graph of y = f(x) passes through (2, 3) and (6, −1). Find the image of each point under y = 2 f(x/3).
Step 1, Identify both transformations.
2 outside ⇒ vertical dilation by factor 2 (y → 2y).
x/3 inside ⇒ horizontal dilation by factor 3 (x → 3x).
Reason: f(x/3) means the input is divided by 3, so to get the same value of f we need to feed in 3x, meaning every x-coordinate is multiplied by 3.
Step 2, Apply the horizontal dilation (×3 on x).
(2, 3) → (6, 3) (6, −1) → (18, −1)
Step 3, Apply the vertical dilation (×2 on y).
(6, 3) → (6, 6) (18, −1) → (18, −2)
Conclusion. Images: (6, 6) and (18, −2).
3. Faded example, dilate y = x² to obtain y = (2x)²
Compare y = 2x² (a vertical dilation) with y = (2x)² (a horizontal dilation). Fill in the blanks. 4 marks
Step 1, Vertical dilation: y = 2x² is y = ____ · f(x) where f(x) = x². The dilation is ____________ by factor ____ from the ____-axis.
Image of (1, 1) on y = x²: y-coordinate × 2 ⇒ ( ____, ____ ).
Step 2, Horizontal dilation: y = (2x)² is y = f( ____ · x) where f(x) = x². The dilation is ____________ by factor ____ from the ____-axis.
Image of (1, 1) on y = x²: x-coordinate × (1/____) ⇒ ( ____, ____ ).
Step 3, Why do both look "narrower"? Expand (2x)² = ____ · x². So y = (2x)² is algebraically the same as y = ____ x², which is also a vertical dilation by factor ____. This explains why two different-looking dilations give the same parabola for f(x) = x².
Conclusion. For f(x) = x², the horizontal dilation y = f(2x) is identical to the vertical dilation y = 4 f(x). This coincidence is specific to ________ functions.
4. Graduated practice
Foundation, name the dilation (4 questions)
| Q | Equation | Type (V/H) | Factor |
|---|---|---|---|
| 4.1 1 | y = 4f(x) | ||
| 4.2 1 | y = f(3x) | ||
| 4.3 1 | y = f(x/2) | ||
| 4.4 1 | y = (1/3) f(x) |
Standard, apply to coordinates and equations (6 questions)
For Q4.5–4.7 the original point on y = f(x) is (4, 2). For Q4.8–4.10 write the equation after the named dilation, simplifying.
4.5 Image of (4, 2) on y = 3 f(x) is ( ____, ____ ). 1 mark
4.6 Image of (4, 2) on y = f(2x) is ( ____, ____ ). 1 mark
4.7 Image of (4, 2) on y = 2 f(x/4) is ( ____, ____ ). 2 marks
4.8 f(x) = x³ − x. Write the equation after vertical dilation by factor 3 (i.e. y = 3 f(x)), simplified. 2 marks
4.9 f(x) = √x. Write the equation after horizontal dilation by factor 1/2 (i.e. y = f(2x)), simplified. State the new domain. 2 marks
4.10 f(x) = sin x. Write y = sin(x/2). What is the period of this dilated graph (recall: sin x has period 2π)? 2 marks
Extension, link to features (2 questions)
4.11 The graph of y = f(x) has x-intercept at x = 6 and y-intercept at y = 4. (a) After y = 3 f(x), what are the new intercepts? (b) After y = f(x/2), what are the new intercepts? 3 marks
4.12 For f(x) = x, prove algebraically that y = 5 f(x) and y = f(5x) produce the same equation. Then explain in one line why this equivalence fails for f(x) = x² (give the two resulting equations side by side). 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, Point mappings
y = af(x): (x, y) ↦ (x, ay). Vertical, factor a, from x-axis. y = f(bx): (x, y) ↦ (x/b, y). Horizontal, factor 1/b, from y-axis.
Q1.2, Horizontal trap
f(3x): factor 1/3, compressed. f(x/3): factor 3, stretched.
Q1.3, Intercepts
(a) T y = af(x) keeps points where y = 0 fixed. (b) T y = f(bx) keeps points where x = 0 fixed.
Q3, Faded example: y = x² → y = 2x² and y = (2x)²
Step 1: y = 2x² is y = 2·f(x). Vertical, factor 2, from x-axis. (1, 1) ↦ (1, 2).
Step 2: y = (2x)² is y = f(2·x). Horizontal, factor 1/2, from y-axis. (1, 1) ↦ (1/2·1, 1) = (1/2, 1).
Step 3: (2x)² = 4·x², so y = (2x)² is the same as y = 4 x² (a vertical dilation by factor 4).
Conclusion: For f(x) = x², y = f(2x) ≡ y = 4 f(x). Coincidence specific to quadratic (homogeneous) functions.
Q4.1–4.4, Name the dilation
4.1: Vertical, factor 4. 4.2: Horizontal, factor 1/3. 4.3: Horizontal, factor 2. 4.4: Vertical, factor 1/3.
Q4.5–4.7, Image of (4, 2)
4.5 (y = 3 f(x)): y × 3: (4, 6). 4.6 (y = f(2x)): x ÷ 2: (2, 2). 4.7 (y = 2 f(x/4)): x × 4 then y × 2: (16, 4).
Q4.8, y = 3 f(x) for f(x) = x³ − x
y = 3(x³ − x) = 3x³ − 3x.
Q4.9, y = f(2x) for f(x) = √x
y = √(2x). Need 2x ≥ 0 ⇒ x ≥ 0. Domain: [0, ∞). (The domain has been compressed, but [0, ∞) is invariant under compression toward 0.)
Q4.10, y = sin(x/2)
y = sin(x/2). The factor is 1/(1/2) = 2 (horizontal stretch by 2), so the period is 2 · 2π = 4π.
Q4.11, Intercepts under dilation
(a) y = 3 f(x): x-intercept unchanged (still x = 6); y-intercept ×3 ⇒ y = 12.
(b) y = f(x/2): y-intercept unchanged (still y = 4); x-intercept ×2 ⇒ x = 12.
Q4.12, Equivalence for f(x) = x vs f(x) = x²
For f(x) = x: y = 5 f(x) = 5x. y = f(5x) = 5x. Same equation. ✓
For f(x) = x²: y = 5 f(x) = 5x², but y = f(5x) = (5x)² = 25x². Different, equal only if 5 = 25, which is false. The equivalence fails because for non-linear f the input-scale enters with a power higher than 1.