Question

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 \(y = \sin(x)\)