graph the function ( f(x)=sin(x)+2 ).

Explanation:

Step1: Analyze the parent function

The parent function is (y = \sin(x)). It has an amplitude of (1), a period of (2\pi), and key points at ((0,0)), ((\frac{\pi}{2},1)), ((\pi,0)), ((\frac{3\pi}{2}, - 1)), ((2\pi,0))

Step2: Apply the vertical shift

The function (y=\sin(x)+2) is a vertical shift of (y = \sin(x)) up by (2) units. For each point ((x,y)) on (y=\sin(x)), the corresponding point on (y=\sin(x)+2) is ((x,y + 2))

• When (x = 0), (y=\sin(0)+2=0 + 2=2)
• When (x=\frac{\pi}{2}), (y=\sin(\frac{\pi}{2})+2=1 + 2=3)
• When (x=\pi), (y=\sin(\pi)+2=0+2 = 2)
• When (x=\frac{3\pi}{2}), (y=\sin(\frac{3\pi}{2})+2=-1 + 2=1)
• When (x = 2\pi), (y=\sin(2\pi)+2=0+2=2)

Step3: Plot the key points and draw the graph

Plot the points ((0,2)), ((\frac{\pi}{2},3)), ((\pi,2)), ((\frac{3\pi}{2},1)), ((2\pi,2)) and then sketch the sine - wave passing through these points. The graph of (y = \sin(x)+2) oscillates between (y = 1) and (y = 3) with a period of (2\pi)

Answer:

The graph of (y=\sin(x)+2) is a sine - wave with amplitude (1), period (2\pi), and vertical shift (2) units up from the graph of (y = \sin(x))