Determine The Period Of Each Function.21. Y = 5 Sec ⁡ ( 2 X Y = 5 \sec(2x Y = 5 Sec ( 2 X ]

Introduction

In mathematics, the period of a function is the distance along the x-axis over which the function repeats itself. It is an essential concept in understanding the behavior and properties of various mathematical functions. In this article, we will focus on determining the period of the function $y = 5 \sec(2x)$.

What is the Period of a Function?

The period of a function is the length of the interval over which the function repeats itself. It is denoted by the symbol 'T' and is measured in units of the x-axis. The period of a function can be thought of as the distance between two consecutive points on the graph of the function that have the same y-value.

Types of Periodic Functions

There are several types of periodic functions, including:

• Trigonometric functions: These are functions that involve trigonometric ratios such as sine, cosine, and tangent. Examples of trigonometric functions include $y = \sin(x)$ and $y = \cos(x)$.
• Exponential functions: These are functions that involve exponential terms such as $e^x$ and $e^{-x}$.
• Polynomial functions: These are functions that involve polynomial terms such as $x^2$ and $x^3$.

Determining the Period of the Function $y = 5 \sec(2x)$

To determine the period of the function $y = 5 \sec(2x)$, we need to understand the properties of the secant function. The secant function is the reciprocal of the cosine function, and its period is given by the formula:

$$T = \frac{2\pi}{b}$$

where 'b' is the coefficient of the x-term in the function.

In the case of the function $y = 5 \sec(2x)$, the coefficient of the x-term is 2. Therefore, the period of the function is given by:

$$T = \frac{2\pi}{2} = \pi$$

Properties of the Period of the Function $y = 5 \sec(2x)$

The period of the function $y = 5 \sec(2x)$ is $\pi$. This means that the function repeats itself every $\pi$ units along the x-axis. The graph of the function will have a repeating pattern every $\pi$ units, with the same y-values at corresponding points on the graph.

Graph of the Function $y = 5 \sec(2x)$

The graph of the function $y = 5 \sec(2x)$ is a periodic function that repeats itself every $\pi$ units along the x-axis. The graph will have a repeating pattern of peaks and troughs, with the same y-values at corresponding points on the graph.

Conclusion

In conclusion, the period of the function $y = 5 \sec(2x)$ is $\pi$. This means that the function repeats itself every $\pi$ units along the x-axis. The graph of the function will have a repeating pattern every $\pi$ units, with the same y-values at corresponding points on the graph.

References

• Mathematics Handbook. (2019). Springer.
• Calculus: Early Transcendentals. (2018). James Stewart.

Frequently Asked Questions

• What is the period of a function? The period of a function is the distance along the x-axis over which the function repeats itself.
• How do you determine the period of a function? To determine the period of a function, you need to understand the properties of the function and use the formula for the period.
• What is the period of the function $y = 5 \sec(2x)$? The period of the function $y = 5 \sec(2x)$ is $\pi$.
  Determine the Period of Each Function: Q&A =====================================

Introduction

In our previous article, we discussed the concept of the period of a function and how to determine it for the function $y = 5 \sec(2x)$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the period of a function?

A: The period of a function is the distance along the x-axis over which the function repeats itself.

Q: How do you determine the period of a function?

A: To determine the period of a function, you need to understand the properties of the function and use the formula for the period. For trigonometric functions, the period is given by the formula:

$$T = \frac{2\pi}{b}$$

where 'b' is the coefficient of the x-term in the function.

Q: What is the period of the function $y = \sin(x)$?

A: The period of the function $y = \sin(x)$ is $2\pi$. This means that the function repeats itself every $2\pi$ units along the x-axis.

Q: What is the period of the function $y = \cos(x)$?

A: The period of the function $y = \cos(x)$ is also $2\pi$. This means that the function repeats itself every $2\pi$ units along the x-axis.

Q: What is the period of the function $y = \tan(x)$?

A: The period of the function $y = \tan(x)$ is $\pi$. This means that the function repeats itself every $\pi$ units along the x-axis.

Q: How do you determine the period of a function with a coefficient of x?

A: To determine the period of a function with a coefficient of x, you need to use the formula:

$$T = \frac{2\pi}{b}$$

where 'b' is the coefficient of the x-term in the function.

Q: What is the period of the function $y = 5 \sec(2x)$?

A: The period of the function $y = 5 \sec(2x)$ is $\pi$. This means that the function repeats itself every $\pi$ units along the x-axis.

Q: How do you graph a function with a period?

A: To graph a function with a period, you need to plot the function over a single period and then repeat the graph over the entire x-axis.

Q: What is the significance of the period of a function?

A: The period of a function is an essential concept in understanding the behavior and properties of the function. It helps to identify the repeating pattern of the function and can be used to determine the function's amplitude and frequency.

Conclusion

In conclusion, the period of a function is an essential concept in understanding the behavior and properties of the function. By understanding the period of a function, you can determine its repeating pattern and use it to graph the function. We hope that this Q&A section has helped to clarify any doubts and provide additional information on the topic.

References

• Mathematics Handbook. (2019). Springer.
• Calculus: Early Transcendentals. (2018). James Stewart.

Frequently Asked Questions

• What is the period of a function? The period of a function is the distance along the x-axis over which the function repeats itself.
• How do you determine the period of a function? To determine the period of a function, you need to understand the properties of the function and use the formula for the period.
• What is the period of the function $y = 5 \sec(2x)$? The period of the function $y = 5 \sec(2x)$ is $\pi$.