Identify the amplitude, period, and phase shift, and graph the function. $$ y=-2 \sin \left(\frac{1}{4} x+\frac{\pi}{8}\right) $$

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Step 1: Identify the amplitude from the function $$ y = -2 \sin \left(\frac{1}{4} x + \frac{\pi}{8}ight) $$. The amplitude is the absolute value of the coefficient in front of the sine function, which is 2.

Step 2: Determine the period using the formula for the period of a sine function, $$ 	ext{Period} = \frac{2\pi}{b} $$, where $$ b $$ is the coefficient of $$ x $$. Here, $$ b = \frac{1}{4} $$, so the period is $$ \frac{2\pi}{\frac{1}{4}} = 8\pi $$.

Step 3: Calculate the phase shift using the formula $$ 	ext{Phase Shift} = -\frac{c}{b} $$, where $$ c $$ is the constant added to $$ x $$. In this case, $$ c = \frac{\pi}{8} $$ and $$ b = \frac{1}{4} $$. Thus, the phase shift is $$ -\frac{\frac{\pi}{8}}{\frac{1}{4}} = -\frac{\pi}{8} 	imes 4 = -\pi/2 $$.

Step 4: Summarize the findings: Amplitude is 2, period is $$ 8\pi $$, and phase shift is $$ -\pi/8 $$.

Step 5: To graph the function, start by marking the amplitude on the y-axis, then plot the key points based on the period and phase shift, ensuring to reflect the negative amplitude by flipping the sine wave vertically.Amplitude: 2 Period: 8π Phase Shift: -π/8

Identify the amplitude, period, and phase shift, and graph the function. \[ y=-2 \sin \left(\frac{1}{4} x+\frac{\pi}{8}\right) \]

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