Questions: Consider the equation (y=6-5 cos left(frac12 x+fracpi3right)). Identify the amplitude, period, phase shift, vertical shift, and midline.

Solution Steps

The amplitude of the function \( y = 6 - 5 \cos\left(\frac{1}{2} x + \frac{\pi}{3}\right) \) is given by the absolute value of the coefficient in front of the cosine function. Thus, we have: \[ \text{Amplitude} = | -5 | = 5 \]

The period of a cosine function is calculated using the formula \( \text{Period} = \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the cosine. In this case, \( b = \frac{1}{2} \), so: \[ \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi \]

The phase shift is determined by solving the equation \( \frac{1}{2} x + \frac{\pi}{3} = 0 \) for \( x \): \[ \frac{1}{2} x = -\frac{\pi}{3} \implies x = -\frac{2\pi}{3} \approx -2.0944 \] Thus, the phase shift is: \[ \text{Phase Shift} = -2.0944 \]

The vertical shift is the constant added to the function, which is: \[ \text{Vertical Shift} = 6 \]

The midline of the function corresponds to the vertical shift, therefore: \[ \text{Midline} = 6 \]

Final Answer