Problem 48 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=\frac{1}{2} \cos (2 x+\pi)$$

Step-by-Step Solution

Answer

Amplitude: \( A=\frac{1}{2} \), Period: \( P = \pi \), Phase Shift: \( C = -\pi \)

Step 1: Determine the amplitude

The amplitude is given by the absolute value of the coefficient of the cosine function. In this case, the amplitude is \(A = |\frac{1}{2}| = \frac{1}{2}\).

Step 2: Determine the period

The period is given by \( \frac{2\pi}{B} \). B is the coefficient of x within the function. The period in this case is \( \frac{2\pi}{2} = \pi \).

Step 3: Determine the phase shift

The phase shift is given by C, the value being added or subtracted inside the cosine function. Here, we have an addition of \( \pi \) to \( 2x \), so the phase shift to the left is \( C = -\pi \).

Step 4: Plot the function

Now, we need to sketch one period of the function using the amplitude, period, and phase shift. Remember to adjust your x and y axes according to the values of amplitude and period. Because the function is a cosine function, it starts from the maximum value at the phase shift, descends to the minimum and goes back to the maximum, all within the period calculated earlier.