Sketch at least one cycle of the graph of the function yequalssine left parenthesis StartFraction pi Over 3 EndFraction x plus StartFraction 5 pi Over 3 EndFraction right parenthesis
. Determine the​ period, phase​ shift, and range of the function. Identify five points on the graph.

To analyze the given function:

\[
y = \sin \left( \frac{\pi}{3}x + \frac{5\pi}{3} \right)
\]

### Step 1: **Identify key parameters of the sine function**

The general form of a sine function is:

\[
y = A \sin(Bx + C) + D
\]

Where:
- \(A\) is the amplitude
- \(B\) affects the period
- \(C\) affects the phase shift
- \(D\) is the vertical shift (if any)

### Step 2: **Determine the period**
The period \(T\) of a sine function is given by:

\[
T = \frac{2\pi}{|B|}
\]

For the function \(y = \sin \left( \frac{\pi}{3}x + \frac{5\pi}{3} \right)\), the value of \(B\) is \( \frac{\pi}{3} \), so the period is:

\[
T = \frac{2\pi}{\left| \frac{\pi}{3} \right|} = \frac{2\pi}{\frac{\pi}{3}} = 6
\]

Thus, the **period** of the function is **6**.

### Step 3: **Determine the phase shift**
The phase shift is given by:

\[
\text{Phase shift} = -\frac{C}{B}
\]

In the given function, \(C = \frac{5\pi}{3}\) and \(B = \frac{\pi}{3}\). Therefore, the phase shift is:

\[
\text{Phase shift} = -\frac{\frac{5\pi}{3}}{\frac{\pi}{3}} = -5
\]

Thus, the **phase shift** is **5 units to the left**.

### Step 4: **Determine the range**
The range of the sine function is determined by its amplitude. Since no vertical shift is indicated (the value of \(D\) is 0), the range remains the same as a standard sine function, i.e.,:

\[
\text{Range}

Sketch at least one cycle of the graph of the function y = sin((π/3)x + 5π/3). Determine the period, phase shift, and range of the function. Identify five points on the graph.

Learn how to sketch one cycle of the sine function y = sin((π/3)x + 5π/3), determine its period, phase shift, and range, and identify five key points on the graph. This trigonometry problem involves key concepts such as graphing trigonometric functions and calculating period and phase shifts, suitable for students in Grades 10-12.

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