Question 1 [2 marks]
Sine wave through (0,0), (90°,1), (180°,0), (270°,−1), (360°,0); y-axis labelled −1, 0, 1.
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Sine wave through (0,0), (90°,1), (180°,0), (270°,−1), (360°,0); y-axis labelled −1, 0, 1.
Cosine wave through (0°,1), (90°,0), (180°,−1), (270°,0), (360°,1); y-axis −1 to 1.
Branches through (0,0), (180°,0), (360°,0); vertical asymptotes x = 90°, 270°; increasing branches between asymptotes (curve leaves the window near the asymptotes). y-axis −1 to 1.
Increasing exponential; crosses y-axis at (0, 1); passes through (1, 2) etc.; x-axis labelled.
Hyperbola in 1st and 3rd quadrants; asymptotes x = 0 and y = 0; passes (1,1) and (−1,−1) in window.
(i) 12 (ii) −12
(i) −12 (ii) 12
(i) k (ii) −k
(i) x ≈ 17° and x ≈ 163° (ii) sin x° = 0.7 ⇒ x ≈ 44° and x ≈ 136° (accept reasonable reads from graph).
(i) x ≈ 114° and x ≈ 246° (ii) cos x° = 0.75 ⇒ x ≈ 41° and x ≈ 319° (graph reads).
(i) x ≈ 50°, 230° (ii) x ≈ 149°, 329° (one solution per branch in range; accept reasonable estimates).
(i) x ≈ 2.6 (ii) x ≈ 3.8 (accept 2.55–2.65 and 3.75–3.85 from graph).
When x = 0, y = p = 8. Then 8q = 18 ⇒ q = 94 = 2.25. k = 8 × (94)¹.5 = 8 × (278) = 27.
a = xy = 2 × 3 = 6. Check: 64 = 1.5. k = 66 = 1.
(a) Cosine oscillation between 4 and 8 with period 12 h (30t = 360° ⇒ T = 12). (b) Minimum when cos(30t) = −1 ⇒ 30t = 180° + 360°n ⇒ t = 6 + 12n; t = 6 and t = 18.
Sketch y = sin x° and y = cos x°; intersections where tan x° = 1 ⇒ x = −135°, 45°.
y = sin x°: usual wave; y = 12 sin x°: same period, amplitude 12; both pass through (0,0), (±180°,0).
(a) Horizontal compression period 180°. (b) Translated down 1. (c) Reflection in x-axis.
(a) Period halved. (b) Period doubled. (c) Phase shift 90° left (same as cos x°).
(a) Vertical compression amplitude 12. (b) Translation +1 in y (oscillates between 0 and 2). (c) Vertical stretch amplitude 2 (oscillates between −2 and 2).