https://file.aiquickdraw.com/mathbot/file/1729883294744iu2s8me8.jpg

The function provided is:

\[
\sin\left(x - \frac{\pi}{5}\right) + 5
\]

To determine the **period** of the sine function:

1. The standard sine function, \(\sin(x)\), has a period of \(2\pi\).
2. The general form of a sine function is \(\sin(bx + c) + d\), where \(b\) affects the period.

The period \(T\) of \(\sin(bx)\) is given by:

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

In this case, the argument of the sine function is:

\[
\sin\left(\frac{x - \frac{\pi}{5}}{1}\right)
\]

Since the coefficient of \(x\) is 1, the period remains \(2\pi\). The shift \(\frac{\pi}{5}\) does not affect the period—it only translates the graph horizontally.

Therefore, the period of the function \(\sin\left(x - \frac{\pi}{5}\right) + 5\) is still:

\[
T = 2\pi
\]

If you would like to know more details about graphing or other transformations involved in the function, feel free to ask!

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Here are 5 related questions:

1. How does the vertical shift affect the sine function's graph?
2. What happens if the coefficient of \(x\) inside the sine function changes?
3. How would the graph change with a negative sign in front of the sine function?
4. What is the impact of a phase shift on the graph of a sine function?
5. Can you describe how to find the amplitude of this function?

**Tip:** Always check the argument of the trigonometric function to determine if any horizontal stretching or compression occurs, as this affects the period!

What is the period of the function sin(x - π/5) + 5?

Learn how to find the period of the sine function sin(x - π/5) + 5. Understand the key concepts of sine function transformations and how the period is influenced by the coefficient of x. Suitable for students in Grades 10-12.

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