Question

graph the function $f(x)=\cos(x)-1$.

Explanation:

Step1: Analyze the parent function

The parent function is \( y = \cos(x) \). Its amplitude \( A = 1 \), period \( T=2\pi \), and it has key points at \( (0,1),(\frac{\pi}{2},0),(\pi, - 1),(\frac{3\pi}{2},0),(2\pi,1) \).

Step2: Apply the vertical - shift transformation

The function \( f(x)=\cos(x)-1 \) is a vertical shift of \( y = \cos(x) \) down by 1 unit.
For a general transformation \( y = f(x)+k \), when \( k=-1 \), each \( y \) - coordinate of the points on \( y = \cos(x) \) is decreased by 1.
The key points of \( y=\cos(x)-1 \) are:
When \( x = 0 \), \( y=\cos(0)-1=1 - 1=0 \);
When \( x=\frac{\pi}{2} \), \( y=\cos(\frac{\pi}{2})-1=0 - 1=-1 \);
When \( x=\pi \), \( y=\cos(\pi)-1=-1 - 1=-2 \);
When \( x=\frac{3\pi}{2} \), \( y=\cos(\frac{3\pi}{2})-1=0 - 1=-1 \);
When \( x = 2\pi \), \( y=\cos(2\pi)-1=1 - 1=0 \).

Step3: 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 coordinate plane. Then, connect these points with a smooth curve that has the same shape (cosine - wave shape) as \( y = \cos(x) \), but shifted down by 1 unit.

Answer:

Plot the key points \( (0,0),(\frac{\pi}{2},-1),(\pi,-2),(\frac{3\pi}{2},-1),(2\pi,0) \) and draw a smooth cosine - like curve through them.