sketch the graph of the function.\n$f(x)=4 + \\cos(x)$

Explanation:

Step1: Recall properties of cosine function

The basic cosine function $y = \cos(x)$ has an amplitude of $A = 1$, a period of $T=2\pi$, and its range is $[- 1,1]$. It has a maximum value of $1$ when $x = 2k\pi,k\in\mathbb{Z}$ and a minimum value of $-1$ when $x=(2k + 1)\pi,k\in\mathbb{Z}$.

Step2: Analyze the given function $y = 4+\cos(x)$

The function $y = 4+\cos(x)$ is a vertical - shift of the basic cosine function $y=\cos(x)$ by $4$ units upwards. The amplitude remains $A = 1$ and the period remains $T = 2\pi$. The range of $y = 4+\cos(x)$ is found by adding $4$ to the range of $y=\cos(x)$. So the range is $[4 - 1,4 + 1]=[3,5]$. The maximum value of $y = 4+\cos(x)$ is $y_{max}=4 + 1=5$ when $x = 2k\pi,k\in\mathbb{Z}$, and the minimum value is $y_{min}=4-1 = 3$ when $x=(2k + 1)\pi,k\in\mathbb{Z}$.

Step3: Plot key points

When $x = 0$, $y=4+\cos(0)=4 + 1=5$. When $x=\frac{\pi}{2}$, $y=4+\cos(\frac{\pi}{2})=4+0 = 4$. When $x=\pi$, $y=4+\cos(\pi)=4-1 = 3$. When $x=\frac{3\pi}{2}$, $y=4+\cos(\frac{3\pi}{2})=4 + 0=4$. When $x = 2\pi$, $y=4+\cos(2\pi)=4 + 1=5$. Connect these points with a smooth curve that repeats every $2\pi$ units.

Answer:

Sketch a cosine - like curve with a maximum value of $5$, a minimum value of $3$, passing through the points $(0,5),(\frac{\pi}{2},4),(\pi,3),(\frac{3\pi}{2},4),(2\pi,5)$ and repeating every $2\pi$ units.