Amplitude, Period and Counting Solutions

Graph questions test whether you can read a transformed sine curve without plotting it point by point. Three numbers control everything: the amplitude ($b$), the period ($360^\circ / c$), and the vertical shift ($a$). Once you have them, counting solutions is just counting how many times a horizontal line crosses the wave.

The function $f(x) = 2 + 3\sin(3x)$ is defined for $0^\circ \le x \le 360^\circ$. (i) State the amplitude and the period of $f$. [2] (ii) State the maximum and minimum values of $f(x)$. [2] (iii) State the number of solutions of the equation $f(x) = 2$ in the given range. [2]

The working

(i) For $a + b\sin(cx)$ the amplitude is $|b|$ and the period is $\dfrac{360^\circ}{c}$: $\text{amplitude} = 3 \quad \text{(B1)}, \qquad \text{period} = \frac{360^\circ}{3} = 120^\circ \quad \text{(B1)}$

(ii) The sine part swings between $-1$ and $1$, so $3\sin(3x)$ swings between $-3$ and $3$. Add the shift of $2$: $\text{maximum} = 2 + 3 = 5 \quad \text{(B1)}, \qquad \text{minimum} = 2 - 3 = -1 \quad \text{(B1)}$

(iii) Set $f(x) = 2$: $2 + 3\sin(3x) = 2 \;\Rightarrow\; \sin(3x) = 0 \quad \text{(M1)}$

$\sin(3x) = 0$ when $3x = 0^\circ, 180^\circ, 360^\circ, \dots$ As $x$ runs $0^\circ$ to $360^\circ$, the argument $3x$ runs $0^\circ$ to $1080^\circ$, and $\sin = 0$ at every multiple of $180^\circ$: $3x = 0, 180, 360, 540, 720, 900, 1080 \;\Rightarrow\; x = 0, 60, 120, 180, 240, 300, 360$ $\Rightarrow \textbf{7 solutions} \quad \text{(A1)}$

This matches the picture: the curve crosses its own centre line $y = 2$ twice per cycle, three cycles give six interior crossings, plus the endpoints land exactly on it.

Where the marks are won and lost

• The period uses $c$, not $b$. A common slip is to quote $120^\circ$ as the amplitude or $3$ as the period.
• For (iii), the argument’s range is $3x \in [0^\circ, 1080^\circ]$, three times wider than $x$. Counting solutions of $\sin(3x)=0$ over the $x$-range without scaling the argument gives the wrong count.
• Endpoints count. $x = 0^\circ$ and $x = 360^\circ$ both satisfy the equation and are inside the closed range.

Common mistakes

• Swapping amplitude and period.
• Using the $x$-range instead of the $3x$-range when counting solutions.
• Forgetting the maximum/minimum shift by $a$ (quoting $\pm 3$ instead of $5$ and $-1$).

Full method: Graphs of sin, cos & tan notes. Topic home: Trigonometry pillar.