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