Hint
The midline is the average of the extreme values; the amplitude is half their difference.
Worked solution
(a)
\(\displaystyle a= \frac{7+ (- 1)}{2}= 3\) and \(\displaystyle b= \frac{7- (- 1)}{2}= 4\). From \(\displaystyle \frac{2\pi }{k}= \frac{2\pi }{3}, k= 3\).
Answer: \(a=3,b=4,k=3\).
(b)
At \(\displaystyle x=0\), sine is zero, giving (0,3). The first positive maximum has \(\displaystyle 3x= \frac{\pi }{2}\), so it is \(\displaystyle (\frac{\pi }{6}, 7)\).
Answer: \((0,3)\) and \(\displaystyle (\frac{\pi }{6}, 7)\).
(c)
Sin(3x)=0, so \(\displaystyle x= \frac{n\pi }{3}\). The half-open interval allows integers \(\displaystyle n=0,1,2,3,4,5\). There are six roots; \(\displaystyle n=6\) gives the excluded endpoint 2π.
Answer: \(\displaystyle 0, \frac{\pi }{3}, \frac{2\pi }{3}, \pi , \frac{4\pi }{3}, \frac{5\pi }{3}\); six.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Give \(\displaystyle a=3\).
- 1 mark: Give \(\displaystyle b=4\).
- 1 mark: Give \(\displaystyle k=3\).
Alternative methods: Equivalent midline/range/period equations are accepted.
Common error: Using the full output range width as the amplitude.
(b) · 2 marks
- 1 mark: Give y-intercept (0,3).
- 1 mark: Give first positive maximum \(\displaystyle (\frac{\pi }{6}, 7)\).
Alternative methods: A correct quarter-period graph argument is accepted.
Common error: Using \(\displaystyle x= \frac{\pi }{2}\) without accounting for the inner multiplier.
(c) · 3 marks
- 1 mark: Use \(\displaystyle x= \frac{n\pi }{3}\) or an equivalent zero-crossing method.
- 1 mark: List exactly the six permitted x-values.
- 1 mark: State six roots and respect the excluded endpoint 2π.
Alternative methods: A complete, correctly indexed list from the graph is accepted.
Common error: Counting 2π as a seventh root.
Total: 8 marks.