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. Rewrite the function in the standard form for a cosine function: \(y = A \cos(B(x - C)) + D\). Identify each component by comparing it to the given function \(y = 3 - \frac{1}{4} \cos \frac{2}{3} x\).
2. Determine the amplitude \(A\), which is the absolute value of the coefficient in front of the cosine function. Here, the coefficient is \(-\frac{1}{4}\), so the amplitude is \(| -\frac{1}{4} |\).
3. Find the vertical translation \(D\), which is the constant term added or subtracted outside the cosine function. In this case, it is the number added to the cosine term, which is \(3\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the cosine function. Here, \(B = \frac{2}{3}\).
5. Identify the phase shift \(C\) by rewriting the argument of the cosine in the form \(B(x - C)\). Since the function is \(\cos \frac{2}{3} x\) with no subtraction inside the argument, the phase shift is \(0\).

Key Concepts

Amplitude of a Trigonometric Function

Amplitude measures the maximum distance a trigonometric function's graph deviates from its midline. For functions like y = a cos(bx + c) + d, the amplitude is the absolute value of 'a'. It determines the height of the peaks and depths of the troughs.

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), the period is calculated as 2π divided by the absolute value of 'b'. It indicates how frequently the function repeats its pattern.

Vertical Translation and Phase Shift

Vertical translation shifts the graph up or down by 'd' units, changing the midline of the function. Phase shift moves the graph horizontally and is found by solving bx + c = 0 for x, indicating how the function is shifted left or right.