Question

sketch the graph of the function. $y = \cos(\frac{x}{2}) + 1$

Explanation:

Step1: Determine the amplitude and vertical shift

The general form of a cosine function is \(y = A\cos(Bx - C)+D\). For \(y=\cos(\frac{x}{2}) + 1\), \(A = 1\) (amplitude), \(D=1\) (vertical shift). The amplitude \(|A| = 1\) means the graph oscillates \(1\) unit above and below the mid - line \(y = 1\).

Step2: Find the period

The period formula for \(y = A\cos(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). Here \(B=\frac{1}{2}\), so \(T=\frac{2\pi}{\frac{1}{2}}=4\pi\).

Step3: Identify key points

For the basic cosine function \(y = \cos(x)\), key points are \((0,1)\), \((\frac{\pi}{2},0)\), \((\pi,- 1)\), \((\frac{3\pi}{2},0)\), \((2\pi,1)\). For \(y=\cos(\frac{x}{2})+1\):

• When \(x = 0\), \(y=\cos(0)+1=1 + 1=2\).
• When \(x = 2\pi\), \(y=\cos(\pi)+1=-1 + 1=0\).
• When \(x = 4\pi\), \(y=\cos(2\pi)+1=1 + 1=2\).
• When \(x=-2\pi\), \(y=\cos(-\pi)+1=-1 + 1=0\).
• When \(x = - 4\pi\), \(y=\cos(-2\pi)+1=1 + 1=2\).

Plot these key points \((-4\pi,2)\), \((-2\pi,0)\), \((0,2)\), \((2\pi,0)\), \((4\pi,2)\) and connect them with a smooth cosine - shaped curve.

Answer:

Sketch the graph by plotting the key points \((-4\pi,2)\), \((-2\pi,0)\), \((0,2)\), \((2\pi,0)\), \((4\pi,2)\) and drawing a smooth cosine - like curve with period \(4\pi\) and mid - line \(y = 1\).