State the amplitude and period of $2\sin 3x + 1$. Sketch the graph of $y = 2\sin 3x + 1$ for $0 \le x \le \pi$. By drawing a suitable straight line on your sketch, determine the number of solutions of the equation $\pi\sin 3x - x = 0$.

1. Identify the amplitude from the coefficient of the sine term

The function is in the form $y = a\sin(bx) + c$. The amplitude is given by $|a|$.
Amplitude = $|2| = 2$.

2. Identify the period from the coefficient of $x$

The period of $\sin(bx)$ is given by $\frac{2\pi}{|b|}$.
Period = $\frac{2\pi}{3}$.

1. Determine the key features of the graph (maximum, minimum, and starting points)

The graph of $y = 2\sin 3x + 1$ has an amplitude of $2$ and is shifted upwards by $1$.
Maximum value $= 1 + 2 = 3$.
Minimum value $= 1 - 2 = -1$.
At $x = 0$, $y = 2\sin(0) + 1 = 1$.

2. Determine the period and the number of cycles in the given domain

The period is $\frac{2\pi}{3}$.
The domain is $0 \le x \le \pi$.
Since $\pi = 1.5 \times \frac{2\pi}{3}$, the graph will complete $1.5$ cycles.

3. Sketch the curve and label key coordinates

Key points to plot:
$(0, 1)$
$(\frac{\pi}{6}, 3)$ (Maximum)
$(\frac{\pi}{3}, 1)$
$(\frac{\pi}{2}, -1)$ (Minimum)
$(\frac{2\pi}{3}, 1)$
$(\frac{5\pi}{6}, 3)$ (Maximum)
$(\pi, 1)$
Draw a smooth curve through these points.

1. Rearrange the given equation to match the function sketched in part (b).

$\pi\sin 3x - x = 0$
$\pi\sin 3x = x$
$\sin 3x = \frac{x}{\pi}$
Multiply by 2 and add 1 to both sides:
$2\sin 3x + 1 = \frac{2x}{\pi} + 1$

2. Identify the straight line to be drawn.

The suitable straight line to draw is $y = \frac{2x}{\pi} + 1$.

3. Determine the number of intersections between the curve and the line in the given domain.

The line passes through $(0, 1)$ and $(\pi, 3)$.
Drawing this line on the sketch of $y = 2\sin 3x + 1$ for $0 \le x \le \pi$:

• At $x = 0$, both the curve and the line are at $y = 1$ (1 intersection).
• In the first cycle, the curve rises to $y = 3$ at $x = \frac{\pi}{6}$ while the line is at $y = 1\frac{1}{3}$, so they intersect once as the curve descends (1 intersection).
• In the third cycle, the curve rises to $y = 3$ at $x = \frac{5\pi}{6}$ while the line is at $y = 2\frac{2}{3}$. The curve crosses the line on the way up and on the way down (2 intersections).
  Total number of intersections = $1 + 1 + 2 = 4$.