Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 4 cos(2x − π)

Verified step by step guidance

1. Identify the general form of the cosine function: \(y = A \cos(Bx - C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) relates to the phase shift.
2. Find the amplitude by taking the absolute value of \(A\). For the function \(y = 4 \cos(2x - \pi)\), the amplitude is \(|4|\).
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 2\), so substitute to find the period.
4. Determine the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\). Since the function is \(y = 4 \cos(2x - \pi)\), rewrite the inside as \(2x - \pi = 2(x - \frac{\pi}{2})\), so \(C = \pi\) and \(B = 2\).
5. To graph one period, start at the phase shift on the x-axis, then plot points over one full period length calculated in step 3, using the amplitude to mark maximum and minimum values of the cosine wave.

Key Concepts

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of a trigonometric function, representing the height from the midline to the peak. For functions like y = a cos(bx + c), the amplitude is |a|. In the given function y = 4 cos(2x − π), the amplitude is 4, indicating the graph oscillates between -4 and 4.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function along the x-axis. For y = cos(bx + c), the period is calculated as (2π) / |b|. In y = 4 cos(2x − π), b = 2, so the period is π, meaning the function repeats every π units.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving bx + c = 0 for x. It is given by -c/b. For y = 4 cos(2x − π), the phase shift is (π/2) units to the right, indicating the graph is shifted right by π/2 compared to the standard cosine function.