For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 - ¼ cos ⅔ x

Verified step by step guidance

1. Identify the standard form of the cosine function: \( y = a \cos(bx - c) + d \).
2. Compare the given function \( y = 3 - \frac{1}{4} \cos \frac{2}{3}x \) with the standard form to identify the parameters: \( a = -\frac{1}{4} \), \( b = \frac{2}{3} \), \( c = 0 \), and \( d = 3 \).
3. Determine the amplitude, which is the absolute value of \( a \): \( |a| = \left| -\frac{1}{4} \right| = \frac{1}{4} \).
4. Calculate the period of the function using the formula \( \frac{2\pi}{b} \): \( \frac{2\pi}{\frac{2}{3}} = 3\pi \).
5. Identify the vertical translation, which is \( d = 3 \), and the phase shift, which is \( \frac{c}{b} = \frac{0}{\frac{2}{3}} = 0 \).

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of cosine functions, it is determined by the coefficient in front of the cosine term. For the function y = 3 - ¼ cos (⅔ x), the amplitude is |−¼|, which equals ¼, indicating how far the wave oscillates above and below its midline.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For cosine functions, the period can be calculated using the formula 2π divided by the coefficient of x inside the cosine function. In this case, the period of y = 3 - ¼ cos (⅔ x) is 2π / (⅔) = 3π, indicating how frequently the wave repeats itself.

Vertical Translation and Phase Shift

Vertical translation refers to the upward or downward shift of the graph of a function, determined by the constant added or subtracted from the function. In y = 3 - ¼ cos (⅔ x), the vertical translation is 3 units down. Phase shift, on the other hand, indicates a horizontal shift of the graph, which is determined by any horizontal transformations applied to x. In this function, there is no phase shift since there is no addition or subtraction inside the cosine argument.