Questions: Determine the amplitude, period, any vertical translation, and any phase shift of the given graph. y=4 cos (x/5-pi/5)

Transcript text: Determine the amplitude, period, any vertical translation, and any phase shift of the given graph. \[ y=4 \cos \left(\frac{x}{5}-\frac{\pi}{5}\right) \]

Solution

Solution Steps

Step 1: Amplitude

The amplitude of the function is the absolute value of \(A\), which is \(|4|\). Therefore, the amplitude is \( 4 \).

Step 2: Period

The period of the function is calculated using the formula \( rac{2\pi}{|B|}\), where \(B = 0.2\). Thus, the period is \( 31.42 \).

Step 3: Phase Shift

The phase shift is determined by the formula \( rac{C}{B}\), where \(C = -0.628\) and \(B = 0.2\). Therefore, the phase shift is \( -3.14 \).

Step 4: Vertical Translation

The vertical translation of the function is given by \(D\), which is \( 0 \).

Final Answer:

The function \(y = 4 \cdot \mathrm{cos}( 0.2(x + 0.628)) + 0\) has an amplitude of \( 4 \), a period of \( 31.42 \), a phase shift of \( -3.14 \), and a vertical translation of \( 0 \).

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