The function f is defined by $f(x)=1+2\cos 2x$ for $0\le x\le\pi$.

1. State the amplitude of the function

The function is $f(x) = 1 + 2\cos 2x$.
The amplitude is the coefficient of the cosine term.
$\text{Amplitude} = 2$

2. Calculate the period of the function

The coefficient of $x$ is $2$.
$\text{Period} = \frac{2\pi}{2} = \pi$

1. Determine the maximum value of the function

The maximum value of $\cos 2x$ is $1$.
Maximum value of $f(x) = 1 + 2(1) = 3$.

2. Determine the minimum value of the function

The minimum value of $\cos 2x$ is $-1$.
Minimum value of $f(x) = 1 + 2(-1) = -1$.

1. Identify the key coordinates for one period of the graph

The period is $\pi$. The key points for $y = 1 + 2\cos 2x$ in the interval $0 \le x \le \pi$ are:

• $x = 0 \implies y = 1 + 2(1) = 3$
• $x = \frac{\pi}{4} \implies y = 1 + 2(0) = 1$
• $x = \frac{\pi}{2} \implies y = 1 + 2(-1) = -1$
• $x = \frac{3\pi}{4} \implies y = 1 + 2(0) = 1$
• $x = \pi \implies y = 1 + 2(1) = 3$

2. Sketch the graph

Draw a smooth, continuous curve passing through $(0, 3)$, $(\frac{\pi}{4}, 1)$, $(\frac{\pi}{2}, -1)$, $(\frac{3\pi}{4}, 1)$, and $(\pi, 3)$. Ensure the shape is a standard cosine wave.