Hint
Angles are radians and A,B are positive. Reflect the negative half-waves upwards.
Worked solution
(a)
Every negative sine half-wave is reflected before the whole graph is shifted up by D.
Answer: \(\left[9, 11\right]\); \(\displaystyle \frac{\pi}{3}\)
(b)
The original sine zeros give the minima.
\(9\)
Answer: \(9\)
(c)
Use \(\displaystyle \frac{\pi }{B}\) after taking the modulus.
\(\displaystyle \frac{1}{3}\)
Answer: \(\displaystyle \frac{1}{3}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Reflect the negative sine portions, determine attainable heights and account for the halved period.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The modulus changes the period of this sine graph.
(b) · 2 marks
- 1 mark: The original sine zeros give the minima.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The reflected parts do not move horizontally.
(c) · 2 marks
- 1 mark: Use \(\displaystyle \frac{\pi }{B}\) after taking the modulus.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The modulus changes the period of this sine graph.
Total: 6 marks.