To determine the correct graph for the function $y = \frac{1}{2} \cos\left(\frac{4}{3}x\right)$, let's analyze its properties, such as amplitude, period, and phase shift.

1. Amplitude

The amplitude of the function is given by the coefficient in front of the cosine: $\text{Amplitude} = \left|\frac{1}{2}\right| = 0.5$

This means the maximum and minimum values of the function are $0.5$ and $-0.5$, respectively.

2. Period

The period of a cosine function is calculated using the formula: $\text{Period} = \frac{2\pi}{B}$ where $B = \frac{4}{3}$. Plugging this value in: $\text{Period} = \frac{2\pi}{\frac{4}{3}} = \frac{3\pi}{2}$

So, the function repeats every $\frac{3\pi}{2}$, or approximately 4.71 units.

3. Phase Shift and Vertical Shift

The given function $y = \frac{1}{2} \cos\left(\frac{4}{3}x\right)$ has:

• No phase shift since there is no horizontal translation term.
• No vertical shift, meaning it oscillates symmetrically around the x-axis.

Evaluating the Options

1. Amplitude: We know the amplitude should be 0.5.
2. Period: The period should be approximately $4.71$. We must find the graph that fits these criteria.

Comparison of Options

• Option A: Has a period of approximately 6.3, which is longer than $4.71$. Therefore, this is incorrect.
• Option B: Shows a period of approximately 2.4, which is shorter than $4.71$. This is also incorrect.
• Option C: Shows a period of approximately 1.6, which again is much shorter than $4.71$. Incorrect.
• Option D: Matches the correct range and period. A period of $\frac{3\pi}{2}$ (or approximately 4.71) matches perfectly with the properties of $y = \frac{1}{2} \cos\left(\frac{4}{3}x\right)$. The amplitude also aligns with 0.5.

Thus, the correct answer is Option D.

If you have any further questions or need details, feel free to ask!

Further Questions:

1. What are the steps to determine the phase shift of a trigonometric function?
2. How do changes in amplitude affect the graph of a trigonometric function?
3. How can we determine the vertical shift of a trigonometric function from its equation?
4. Why is the period formula $\frac{2\pi}{B}$ for sine and cosine functions?
5. How do we transform the graph of $\cos(x)$ when the coefficient of $x$ changes?

Tip: Always start by finding the amplitude and period when analyzing trigonometric functions. These are the key features that determine the shape and repetition of the graph.