Understanding the Period of Sine Function

Oct 13, 2025 by recoveryshake.com 116 views

Introduction

When dealing with trigonometric functions, understanding the concept of period is crucial in analyzing and graphing these functions. The period of a function is the horizontal distance after which the function repeats itself. In this article, we will focus on finding the period of the graph of the given function $y=\frac{1}{2} \sin (2x) - 3$ .

Understanding the Period of Sine Function

The standard form of the sine function is $y = \sin (bx)$ , where $b$ is a constant. The period of the sine function is given by the formula $\frac{2\pi}{|b|}$ . This means that the graph of the sine function repeats itself after a horizontal distance of $\frac{2\pi}{|b|}$ .

Modifying the Period of Sine Function

In the given function $y=\frac{1}{2} \sin (2x) - 3$ , the coefficient of $x$ inside the sine function is $2$ . This means that the period of the function is affected by the value of $b$ , which is $2$ in this case.

Finding the Period of the Given Function

To find the period of the given function, we can use the formula for the period of the sine function: $\frac{2\pi}{|b|}$ . In this case, $b = 2$ , so the period of the function is $\frac{2\pi}{|2|} = \pi$ .

Conclusion

In conclusion, the period of the graph of $y=\frac{1}{2} \sin (2x) - 3$ is $\pi$ . This means that the graph of the function repeats itself after a horizontal distance of $\pi$ .

Example Use Case

To illustrate the concept of period, let's consider an example. Suppose we want to graph the function $y=\frac{1}{2} \sin (2x) - 3$ over a horizontal distance of $2\pi$ . Since the period of the function is $\pi$ , we can expect the graph to repeat itself after a horizontal distance of $\pi$ . This means that the graph will have two complete periods over a horizontal distance of $2\pi$ .

Tips and Tricks

When dealing with trigonometric functions, it's essential to understand the concept of period. Here are some tips and tricks to help you find the period of a function:

Use the formula for the period of the sine function: $\frac{2\pi}{|b|}$ .
Identify the coefficient of $x$ inside the sine function.
Plug in the value of $b$ into the formula for the period.
Simplify the expression to find the period of the function.

Common Mistakes

When finding the period of a function, it's common to make mistakes. Here are some common mistakes to avoid:

Forgetting to use the absolute value of $b$ in the formula for the period.
Not identifying the coefficient of $x$ inside the sine function.
Not simplifying the expression to find the period of the function.

Final Thoughts

In conclusion, the period of the graph of $y=\frac{1}{2} \sin (2x) - 3$ is $\pi$ . Understanding the concept of period is crucial in analyzing and graphing trigonometric functions. By following the tips and tricks outlined in this article, you can find the period of any function and graph it accurately.

Frequently Asked Questions

What is the period of the graph of $y=\frac{1}{2} \sin (2x) - 3$ $y = \frac{1}{2} sin (2 x) - 3$ ?
- The period of the graph of $y=\frac{1}{2} \sin (2x) - 3$ is $\pi$ .
How do I find the period of a function?
- Use the formula for the period of the sine function: $\frac{2\pi}{|b|}$ .
What is the coefficient of $x$ $x$ inside the sine function?
- The coefficient of $x$ inside the sine function is $2$ .
How do I simplify the expression to find the period of the function?
- Simplify the expression by dividing $2\pi$ by the absolute value of $b$ .

Introduction

In our previous article, we discussed the concept of period in trigonometric functions and how to find the period of a given function. In this article, we will continue to explore the topic of period and answer some frequently asked questions.

Q&A

Q1: What is the period of the graph of $y=\sin (3x)$ ?

A1: The period of the graph of $y=\sin (3x)$ is $\frac{2\pi}{|3|} = \frac{2\pi}{3}$ .

Q2: How do I find the period of a function in the form $y=a\sin (bx)$ ?

A2: To find the period of a function in the form $y=a\sin (bx)$ , use the formula for the period of the sine function: $\frac{2\pi}{|b|}$ .

Q3: What is the coefficient of $x$ inside the sine function?

A3: The coefficient of $x$ inside the sine function is $b$ .

Q4: How do I simplify the expression to find the period of the function?

A4: Simplify the expression by dividing $2\pi$ by the absolute value of $b$ .

Q5: What is the period of the graph of $y=\cos (2x)$ ?

A5: The period of the graph of $y=\cos (2x)$ is $\frac{2\pi}{|2|} = \pi$ .

Q6: How do I find the period of a function in the form $y=a\cos (bx)$ ?

A6: To find the period of a function in the form $y=a\cos (bx)$ , use the formula for the period of the cosine function: $\frac{2\pi}{|b|}$ .

Q7: What is the difference between the period of the sine function and the period of the cosine function?

A7: The period of the sine function is $\frac{2\pi}{|b|}$ , while the period of the cosine function is also $\frac{2\pi}{|b|}$ . The only difference is that the cosine function has a phase shift of $\frac{\pi}{2}$ .

Q8: How do I graph a function with a period of $\pi$ ?

A8: To graph a function with a period of $\pi$ , start by graphing the function over a horizontal distance of $\pi$ . Then, repeat the graph over the same horizontal distance to create a complete period.

Q9: What is the period of the graph of $y=\tan (x)$ ?

A9: The period of the graph of $y=\tan (x)$ is $\pi$ .

Q10: How do I find the period of a function in the form $y=a\tan (bx)$ ?

A10: To find the period of a function in the form $y=a\tan (bx)$ , use the formula for the period of the tangent function: $\frac{\pi}{|b|}$ .

Conclusion

In conclusion, understanding the concept of period is crucial in analyzing and graphing trigonometric functions. By following the tips and tricks outlined in this article, you can find the period of any function and graph it accurately.

Frequently Asked Questions

What is the period of the graph of $y=\sin (3x)$ $y = sin (3 x)$ ?
- The period of the graph of $y=\sin (3x)$ is $\frac{2\pi}{|3|} = \frac{2\pi}{3}$ .
How do I find the period of a function in the form $y=a\sin (bx)$ $y = a sin (b x)$ ?
- Use the formula for the period of the sine function: $\frac{2\pi}{|b|}$ .
What is the coefficient of $x$ $x$ inside the sine function?
- The coefficient of $x$ inside the sine function is $b$ .
How do I simplify the expression to find the period of the function?
- Simplify the expression by dividing $2\pi$ by the absolute value of $b$ .