Question

use a vertical shift to graph the function.
( y = 2 sin \frac { 1 } { 4 } x + 1 )
use the graphing tool to graph the function.
(for any answer boxes shown with the grapher, type an exact
answer. type the word pi to insert the symbol ( pi ) as needed.)

Explanation:

Step1: Analyze the parent function

The parent function of \(y = 2\sin\frac{1}{4}x+1\) is \(y=\sin x\). The general form of a sinusoidal function is \(y = A\sin(Bx - C)+D\). Here, \(A = 2\), \(B=\frac{1}{4}\), \(C = 0\), \(D = 1\).

Step2: Determine the amplitude

The amplitude is \(|A|=2\). It represents the maximum distance of the function from its mid - line.

Step3: Determine the period

The period of a sinusoidal function \(y=\sin(Bx)\) is \(T=\frac{2\pi}{|B|}\). Substituting \(B = \frac{1}{4}\), we get \(T=\frac{2\pi}{\frac{1}{4}}=8\pi\).

Step4: Determine the vertical shift

The vertical shift is \(D = 1\). The mid - line of the function \(y = 2\sin\frac{1}{4}x+1\) is \(y = 1\).

To graph \(y = 2\sin\frac{1}{4}x+1\):

• Start by drawing the mid - line \(y = 1\).
• Mark key points for the parent function \(y=\sin x\) over one period \([0,8\pi]\) (since the period \(T = 8\pi\)). For \(y=\sin x\), key points are \((0,0)\), \((\frac{\pi}{2},1)\), \((\pi,0)\), \((\frac{3\pi}{2},- 1)\), \((2\pi,0)\). For \(y = 2\sin\frac{1}{4}x+1\), when \(x = 0\), \(y=2\sin(0)+1=1\); when \(x = 2\pi\), \(y=2\sin(\frac{2\pi}{4})+1=2\sin(\frac{\pi}{2})+1=2\times1 + 1=3\); when \(x = 4\pi\), \(y=2\sin(\frac{4\pi}{4})+1=2\sin(\pi)+1=1\); when \(x = 6\pi\), \(y=2\sin(\frac{6\pi}{4})+1=2\sin(\frac{3\pi}{2})+1=2\times(-1)+1=-1\); when \(x = 8\pi\), \(y=2\sin(2\pi)+1=1\).

Plot these key points \((0,1)\), \((2\pi,3)\), \((4\pi,1)\), \((6\pi,-1)\), \((8\pi,1)\) and connect them with a smooth sinusoidal curve.

Answer:

Graph the function \(y = 2\sin\frac{1}{4}x+1\) with amplitude \(2\), period \(8\pi\), and vertical shift \(1\) (mid - line \(y = 1\)) by plotting key points \((0,1)\), \((2\pi,3)\), \((4\pi,1)\), \((6\pi,-1)\), \((8\pi,1)\) and connecting them with a smooth sinusoidal curve.