graph the function f(x) = cos(x + 3π) + 1.

Explanation:

Step1: Simplify the cosine function

Use the property $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Also, $\cos(x + 3\pi)=\cos x\cos(3\pi)-\sin x\sin(3\pi)$. Since $\cos(3\pi)= - 1$ and $\sin(3\pi)=0$, then $\cos(x + 3\pi)=-\cos x$. So the function becomes $y =-\cos x+1$.

Step2: Identify key - points of the basic cosine function

The basic cosine function $y = \cos x$ has key - points:

• When $x = 0$, $\cos(0)=1$; when $x=\frac{\pi}{2}$, $\cos(\frac{\pi}{2}) = 0$; when $x=\pi$, $\cos(\pi)=-1$; when $x=\frac{3\pi}{2}$, $\cos(\frac{3\pi}{2}) = 0$; when $x = 2\pi$, $\cos(2\pi)=1$.

Step3: Transform the key - points for $y=-\cos x + 1$

• For $x = 0$: $y=-\cos(0)+1=-1 + 1=0$.
• For $x=\frac{\pi}{2}$: $y=-\cos(\frac{\pi}{2})+1=0 + 1=1$.
• For $x=\pi$: $y=-\cos(\pi)+1=-(-1)+1=2$.
• For $x=\frac{3\pi}{2}$: $y=-\cos(\frac{3\pi}{2})+1=0 + 1=1$.
• For $x = 2\pi$: $y=-\cos(2\pi)+1=-1 + 1=0$.

Step4: Plot the points and draw the graph

Plot the points $(0,0),(\frac{\pi}{2},1),(\pi,2),(\frac{3\pi}{2},1),(2\pi,0)$ on the given coordinate grid and draw a smooth curve passing through these points. The graph of $y =-\cos x+1$ is a cosine - type wave with an amplitude of 1, reflected about the $x$ - axis and shifted up 1 unit.

The graph should be drawn on the provided grid with the above - mentioned key - points and a smooth cosine - like curve passing through them. There is no single numerical answer, but the steps above describe how to graph the function $y=\cos(x + 3\pi)+1$ (or $y=-\cos x + 1$).