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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 11 · 11.5 Transformations of Functions

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.Translation 5 units left
2.Translation 4 units up
3.Reflection in the xx-axis
4.Vertical stretch (dilation) by factor 3
5.(0,1)(0, 1)
6.(6,−3)(6, -3)
7.(−2,5)(-2, 5)
8.(2,5)(2, 5)
9.Horizontal compression by factor 12\dfrac{1}{2}
10.Translation 3 units up
Silver
11.y=(x−3)2+2y = (x - 3)^2 + 2
12.y=−x2+4y = -x^2 + 4
13.Vertical stretch by factor 3
14.(−1,−1)(-1, -1)
15.Turning point (−1,−4)(-1, -4) — maximum; yy-intercept (0,−7)(0, -7)
16.(0,4)(0, 4) and (4,−2)(4, -2)
17.y=−(x+2)2y = -(x + 2)^2
18.(3,8)(3, 8)
19.Max at (3,8)(3, 8)
20.x=3x = 3
Gold
21.y=−(x−2)2+4y = -(x - 2)^2 + 4; vertex (2,4)(2, 4)
22.Vertical stretch ×2, then translate 3 right and 1 down.
23.x=−1x = -1 or x=3x = 3
24.(2,1)(2, 1)
25.Feature at x=13x = \dfrac{1}{3} with yy-value 6.
26.y=2(x−2)2−1y = 2(x - 2)^2 - 1
27.xx-intercepts at x=1x = 1 and x=5x = 5 (unchanged); yy-intercept (0,5)(0, 5).
28.g(x)=f(−x)−2g(x) = f(-x) - 2
29.y=2(x+3)2−5y = 2(x + 3)^2 - 5; vertical stretch ×2, translate 3 left, then 5 down.
30.(3,4)(3, 4)
Platinum
31.Maximum at (5,8)(5, 8)
32.Compress vertically by factor 12\dfrac{1}{2}, then translate 3 left.
33.72
34.Translate 3 left, then 4 down.
35.y=3sin⁡(2x)+2y = 3\sin(2x) + 2
36.Vertex (−2,−1)(-2, -1); g(x)=x2+4x+3g(x) = x^2 + 4x + 3
37.(3,7)(3, 7) and (5,13)(5, 13); average rate of change 3 (unchanged from ff).
38.h=4h = 4, k=−7k = -7
39.Corresponds to (5,2)(5, 2); reflection of ff in the line y=xy = x.
40.ff is odd. After translation, g(x)=(x−2)3−4(x−2)g(x) = (x - 2)^3 - 4(x - 2) is neither.

Pack B — Answers

Bronze
1.Translation 3 units right
2.Translation 5 units down
3.Reflection in the yy-axis
4.Vertical dilation by factor 12\dfrac{1}{2} (compression)
5.(0,6)(0, 6)
6.(1,−3)(1, -3)
7.(−3,7)(-3, 7)
8.(−1,−4)(-1, -4)
9.Horizontal stretch by factor 3
10.Translation 4 units right
Silver
11.y=(x+4)2−1y = (x + 4)^2 - 1
12.y=−(x+3)2y = -(x + 3)^2
13.Vertical compression by factor 12\frac{1}{2}
14.(−1,7)(-1, 7)
15.Turning point (−2,4)(-2, 4) — still a minimum; yy-intercept (0,7)(0, 7)
16.(0,5)(0, 5) and (4,2)(4, 2)
17.y=−(x−3)2y = -(x - 3)^2
18.(18,8)(18, 8)
19.Now a minimum at (2,−3)(2, -3)
20.x=−2x = -2
Gold
21.y=−[(x+1)2−3]=−(x+1)2+3y = -[(x + 1)^2 - 3] = -(x + 1)^2 + 3; vertex (−1,3)(-1, 3)
22.Reflect in xx-axis, then translate 2 left and 5 up.
23.x=−5x = -5 or x=1x = 1
24.(2,3)(2, 3)
25.Feature at x=2x = 2 with yy-value 32\dfrac{3}{2}.
26.y=−2(x+1)2+3y = -2(x + 1)^2 + 3
27.xx-intercepts at x=−1x = -1 and x=−5x = -5; yy-intercept (0,−5)(0, -5) unchanged.
28.g(x)=−f(x−3)g(x) = -f(x - 3)
29.y=−(x−2)2+3y = -(x - 2)^2 + 3; reflect in xx-axis, translate 2 right, then 3 up.
30.Maximum at (−1,10)(-1, 10)
Platinum
31.Minimum at (3,−2)(3, -2)
32.Translate down 4, then reflect in the xx-axis.
33.12
34.Vertical stretch ×2, reflect in xx-axis, translate 1 right, then 5 up.
35.y=12sin⁡ ⁣(x2)−1y = \dfrac{1}{2}\sin\!\left(\dfrac{x}{2}\right) - 1
36.Vertex (3,3)(3, 3) — now a maximum; g(x)=−(x−3)2+3g(x) = -(x - 3)^2 + 3
37.(1,4)(1, 4) and (3,16)(3, 16); average rate of change 6 (doubled).
38.h=−3h = -3, k=2k = 2
39.Corresponds to (3,−1)(3, -1); same reflection.
40.ff is even. g(x)=(x−3)2+1g(x) = (x - 3)^2 + 1 is neither (translation in xx breaks yy-axis symmetry).

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) Translation 5 units left. (b) Translation 3 units up. (c) Reflection in the xx-axis. (d) Vertical stretch by factor 4.
Full working
Replace xx with x+hx + h: left hh. Add constant outside: up. Negate outside: reflect in xx-axis. Multiply outside: vertical stretch.
2Problem 2
Answer
(a) (−1,−1)(-1, -1), (0,2)(0, 2). (b) (−4,4)(-4, 4), yy-intercept depends on f(3)f(3). (c) (0,−7)(0, -7), max at (−1,−4)(-1, -4). (d) (−2,4)(-2, 4), (0,7)(0, 7) unchanged.
Full working
(a) Vertical shift down 5. (b) Horizontal shift left 3: min →(−4,4)\to (-4, 4). yy-intercept becomes f(0+3)=f(3)f(0 + 3) = f(3) — value of original ff at x=3x = 3. (c) Reflect in xx-axis. (d) Horizontal stretch by 2.
3Problem 3
Answer
(a) y=(x−2)2−3y = (x - 2)^2 - 3. (b) y=−(x+1)2+4y = -(x + 1)^2 + 4. (c) y=2(x−3)2y = 2(x - 3)^2.
Full working
(a) Right 2 replaces xx with x−2x - 2; down 3 subtracts 3. (b) Reflect: y=−x2y = -x^2. Then translate: y=−(x+1)2+4y = -(x + 1)^2 + 4. (c) Stretch first: y=2x2y = 2x^2. Translate right: y=2(x−3)2y = 2(x - 3)^2.
4Problem 4
Answer
(a) Vertex (−3,5)(-3, 5); maximum. (b) x=−3x = -3. (c) (0,−13)(0, -13). (d) Vertical stretch ×2, reflect in xx-axis, translate 3 left and 5 up.
Full working
(a) Vertex (h,k)=(−3,5)(h, k) = (-3, 5); coefficient −2<0-2 < 0 → max. (b) x=−3x = -3. (c) f(0)=−2(9)+5=−13f(0) = -2(9) + 5 = -13. (d) Read off coefficient sign and shifts.
5Problem 5
Answer
(a) y=−3(x−4)2y = -3(x - 4)^2. (b) Reflect in the xx-axis, vertically compress by factor 13\frac{1}{3}, then translate 4 left.
Full working
(a) Translate first: y=(x−4)2y = (x - 4)^2. Stretch: y=3(x−4)2y = 3(x - 4)^2. Reflect: y=−3(x−4)2y = -3(x - 4)^2. (b) Inverse: reverse the order and invert each step.
6Problem 6
Answer
(a) (2,4)(2, 4), (3,7)(3, 7), (4,16)(4, 16). (b) g(3)=f(1)+3=7g(3) = f(1) + 3 = 7. (c) g(4)=f(2)+3=16g(4) = f(2) + 3 = 16.
Full working
Translate right 2 (add 2 to xx) and up 3 (add 3 to yy).
7Problem 7
Answer
(a) x=1,5x = 1, 5 unchanged; yy-intercept (0,−10)(0, -10). (b) x=12,52x = \frac{1}{2}, \frac{5}{2}; yy-intercept (0,−5)(0, -5) unchanged. (c) x=1,5x = 1, 5 unchanged; yy-intercept (0,5)(0, 5).
Full working
Vertical stretch preserves xx-intercepts (zeros remain zeros) and scales the yy-intercept. Horizontal compression halves xx-intercepts and leaves yy-intercept unchanged (it depends on f(0)f(0)). Reflection in xx-axis flips yy-intercept sign.
8Problem 8
Answer
(a) y=3sin⁡(2x)y = 3\sin(2x). (b) Amplitude 3, period π\pi. (c) y=3sin⁡(2x)+1y = 3\sin(2x) + 1.
Full working
(a) Multiply sin⁡\sin by 3 and replace xx with 2x2x. (b) Amplitude = coefficient of sin⁡\sin; period = 2πB=π\frac{2\pi}{B} = \pi. (c) Add 1.
9Problem 9
Answer
y=2f(x)y = 2f(x).
Full working
Each yy-coord has doubled; xx-coords unchanged. This is a vertical stretch by factor 2.
10Problem 10
Answer
(a) x=3x = 3 and y=2y = 2. (b) Translate 3 right and 2 up. (c) (0,53)(0, \frac{5}{3}).
Full working
(a) Asymptotes shift with the graph. (b) Inside: shift right 3; outside: shift up 2. (c) y=1−3+2=53y = \frac{1}{-3} + 2 = \frac{5}{3}.
11Problem 11
Answer
(a) Translate 1 right, vertical stretch ×3, reflect in xx-axis, translate 4 up. (b) (1,4)(1, 4), maximum. (c) (0,1)(0, 1).
Full working
(a) Inside: x−1x - 1 → right 1. Outside: −3⋅-3 \cdot → stretch ×3 and reflect. Then +4+ 4. (b) New vertex at (1,4)(1, 4); coefficient is −3<0-3 < 0 so max. (c) g(0)=−3(0−1)2+4=−3+4=1g(0) = -3(0 - 1)^2 + 4 = -3 + 4 = 1.
12Problem 12
Answer
(a) Chain 1: y=3f(x)+2y = 3f(x) + 2. Chain 2: y=3(f(x)+2)=3f(x)+6y = 3(f(x) + 2) = 3f(x) + 6. (b) Not the same — stretch is applied to the constant 2 as well in Chain 2. (c) Two translations (horizontal and vertical) commute, since each acts on a different coordinate.
Full working
(a) Apply each step to the function in order. (b) Stretch acts before the addition in Chain 1 but after in Chain 2 — so the constant gets stretched in Chain 2. (c) Horizontal and vertical translations commute because they act on xx and yy independently.