What Is The Period Of The Function Y = 2 Sin ⁡ X Y=2 \sin X Y = 2 Sin X ?A. − 1 , 1 {-1,1} − 1 , 1 B. All Real Numbers C. 2 Π 2 \pi 2 Π D. Π \pi Π

$$

Introduction

When dealing with trigonometric functions, understanding their properties is crucial for solving problems and making predictions. One of the essential properties of a trigonometric function is its period, which represents the distance along the x-axis over which the function repeats itself. In this article, we will explore the concept of the period of a function and apply it to the specific case of the function $y=2 \sin x$.

What is the Period of a Function?

The period of a function is the length of the interval over which the function repeats itself. In other words, it is the distance along the x-axis over which the function's graph repeats its pattern. For example, the graph of the sine function repeats every $2 \pi$ units, which means that the period of the sine function is $2 \pi$.

The Period of the Function $y=2 \sin x$

To find the period of the function $y=2 \sin x$, we need to understand how the period of the sine function is affected by the coefficient 2. The coefficient 2 is a vertical stretch factor, which means that it stretches the graph of the sine function vertically by a factor of 2. However, this vertical stretch does not affect the period of the function.

Why Does the Vertical Stretch Not Affect the Period?

The period of a function is determined by the distance along the x-axis over which the function repeats itself, not by the vertical stretch or compression of the graph. Since the vertical stretch factor 2 does not affect the distance along the x-axis over which the function repeats itself, it does not affect the period of the function.

Conclusion

In conclusion, the period of the function $y=2 \sin x$ is the same as the period of the sine function, which is $2 \pi$. This is because the vertical stretch factor 2 does not affect the period of the function. Therefore, the correct answer is:

The Correct Answer

The period of the function $y=2 \sin x$ is $2 \pi$.

Comparison of Options

Let's compare the options given in the problem:

• Option A: $[-1,1]$ - This is not the correct answer because the period of the function is not a finite interval.
• Option B: All real numbers - This is not the correct answer because the period of the function is a specific value, not all real numbers.
• Option C: $2 \pi$ - This is the correct answer because the period of the function $y=2 \sin x$ is indeed $2 \pi$.
• Option D: $\pi$ - This is not the correct answer because the period of the function is $2 \pi$, not $\pi$.

Final Answer

The final answer is $\boxed{2 \pi}$.

Additional Information

• The period of a function is a fundamental concept in mathematics and is used to describe the repeating pattern of a function.
• The period of a function is determined by the distance along the x-axis over which the function repeats itself.
• The vertical stretch or compression of a graph does not affect the period of the function.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Mathematics for the Nonmathematician" by Morris Kline, 1967.

Related Topics

• Period of a function
• Trigonometric functions
• Vertical stretch and compression

Tags

• Period of a function
• Trigonometric functions
• Vertical stretch and compression
• Mathematics
• Calculus
• Trigonometry
  

$$

Introduction

In our previous article, we discussed the concept of the period of a function and applied it to the specific case of the function $y=2 \sin x$. In this article, we will answer some frequently asked questions related to the period of the function $y=2 \sin x$.

Q1: What is the period of the function $y=2 \sin x$?

A1: The period of the function $y=2 \sin x$ is $2 \pi$. This is because the vertical stretch factor 2 does not affect the period of the function.

Q2: Why does the vertical stretch not affect the period?

A2: The period of a function is determined by the distance along the x-axis over which the function repeats itself, not by the vertical stretch or compression of the graph. Since the vertical stretch factor 2 does not affect the distance along the x-axis over which the function repeats itself, it does not affect the period of the function.

Q3: What is the difference between the period and the range of a function?

A3: The period of a function is the length of the interval over which the function repeats itself, while the range of a function is the set of all possible output values of the function. For example, the period of the sine function is $2 \pi$, while the range of the sine function is $[-1,1]$.

Q4: How do you find the period of a function?

A4: To find the period of a function, you need to understand the properties of the function and how it repeats itself. For trigonometric functions, the period is usually given by a specific formula, such as $2 \pi$ for the sine function.

Q5: Can the period of a function be changed by a vertical stretch or compression?

A5: No, the period of a function cannot be changed by a vertical stretch or compression. The period of a function is determined by the distance along the x-axis over which the function repeats itself, not by the vertical stretch or compression of the graph.

Q6: What is the period of the function $y=3 \sin x$?

A6: The period of the function $y=3 \sin x$ is also $2 \pi$. This is because the vertical stretch factor 3 does not affect the period of the function.

Q7: How does the period of a function relate to its graph?

A7: The period of a function is the length of the interval over which the function repeats itself, which means that the graph of the function will repeat its pattern over this interval.

Q8: Can the period of a function be negative?

A8: No, the period of a function cannot be negative. The period of a function is always a positive value, representing the length of the interval over which the function repeats itself.

Q9: How do you determine the period of a function with a horizontal shift?

A9: To determine the period of a function with a horizontal shift, you need to understand how the shift affects the function's graph. For example, if the function $y=\sin x$ is shifted horizontally by $2 \pi$, its period will remain the same, $2 \pi$.

Q10: Can the period of a function be zero?

A10: No, the period of a function cannot be zero. The period of a function is always a positive value, representing the length of the interval over which the function repeats itself.

Conclusion

In conclusion, the period of the function $y=2 \sin x$ is $2 \pi$, and it is not affected by the vertical stretch factor 2. We have also answered some frequently asked questions related to the period of the function $y=2 \sin x$.

Additional Information

• The period of a function is a fundamental concept in mathematics and is used to describe the repeating pattern of a function.
• The period of a function is determined by the distance along the x-axis over which the function repeats itself.
• The vertical stretch or compression of a graph does not affect the period of the function.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Mathematics for the Nonmathematician" by Morris Kline, 1967.

Related Topics

• Period of a function
• Trigonometric functions
• Vertical stretch and compression
• Mathematics
• Calculus
• Trigonometry

Tags

• Period of a function
• Trigonometric functions
• Vertical stretch and compression
• Mathematics
• Calculus
• Trigonometry