Graph The Function $f(x) = \cos \left(\frac{x}{2}\right$\].

Introduction

In mathematics, graphing functions is an essential skill that helps us visualize and understand the behavior of mathematical equations. One of the most fundamental functions in mathematics is the cosine function, which is a periodic function that oscillates between -1 and 1. In this article, we will focus on graphing the function $f(x) = \cos \left(\frac{x}{2}\right)$, which is a variation of the standard cosine function.

Understanding the Cosine Function

The cosine function is a trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function is periodic, meaning that it repeats itself at regular intervals. The standard cosine function is defined as $f(x) = \cos x$, where $x$ is the angle in radians.

Graphing the Function $f(x) = \cos \left(\frac{x}{2}\right)$

To graph the function $f(x) = \cos \left(\frac{x}{2}\right)$, we need to understand the properties of the cosine function and how it is affected by the factor $\frac{1}{2}$. The factor $\frac{1}{2}$ is a horizontal stretch factor, which means that the graph of the function will be stretched horizontally by a factor of 2.

Properties of the Graph

The graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ has several properties that are worth noting:

• Period: The period of the graph is $2\pi$, which means that the graph repeats itself every $2\pi$ units.
• Amplitude: The amplitude of the graph is 1, which means that the graph oscillates between -1 and 1.
• Phase Shift: The graph has a phase shift of $\pi$, which means that the graph is shifted to the right by $\pi$ units.
• Horizontal Stretch: The graph has a horizontal stretch factor of 2, which means that the graph is stretched horizontally by a factor of 2.

Graphing the Function

To graph the function $f(x) = \cos \left(\frac{x}{2}\right)$, we can use the following steps:

1. Plot the Midline: The midline of the graph is the horizontal line $y = 0$. Plot this line on the graph.
2. Plot the Maximum and Minimum Points: The maximum and minimum points of the graph occur at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$, respectively. Plot these points on the graph.
3. Plot the Periodic Points: The periodic points of the graph occur at $x = \frac{\pi}{2} + 2\pi k$ and $x = \frac{3\pi}{2} + 2\pi k$, where $k$ is an integer. Plot these points on the graph.
4. Connect the Points: Connect the points that you have plotted to form the graph of the function.

Example Graph

Here is an example graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$:

Conclusion

Graphing the function $f(x) = \cos \left(\frac{x}{2}\right)$ is a straightforward process that involves understanding the properties of the cosine function and how it is affected by the factor $\frac{1}{2}$. By following the steps outlined in this article, you can graph the function and visualize its behavior.

Applications of the Graph

The graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ has several applications in mathematics and science. Some of these applications include:

• Modeling Periodic Phenomena: The graph of the function can be used to model periodic phenomena such as the motion of a pendulum or the vibration of a spring.
• Solving Trigonometric Equations: The graph of the function can be used to solve trigonometric equations such as $\cos \left(\frac{x}{2}\right) = 0$.
• Analyzing Data: The graph of the function can be used to analyze data that exhibits periodic behavior.

Final Thoughts

Q&A: Graphing the Cosine Function

Q: What is the period of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$?

A: The period of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ is $2\pi$, which means that the graph repeats itself every $2\pi$ units.

Q: What is the amplitude of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$?

A: The amplitude of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ is 1, which means that the graph oscillates between -1 and 1.

Q: What is the phase shift of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$?

A: The graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ has a phase shift of $\pi$, which means that the graph is shifted to the right by $\pi$ units.

Q: What is the horizontal stretch factor of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$?

A: The graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ has a horizontal stretch factor of 2, which means that the graph is stretched horizontally by a factor of 2.

Q: How do I graph the function $f(x) = \cos \left(\frac{x}{2}\right)$?

A: To graph the function $f(x) = \cos \left(\frac{x}{2}\right)$, you can use the following steps:

1. Plot the Midline: The midline of the graph is the horizontal line $y = 0$. Plot this line on the graph.
2. Plot the Maximum and Minimum Points: The maximum and minimum points of the graph occur at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$, respectively. Plot these points on the graph.
3. Plot the Periodic Points: The periodic points of the graph occur at $x = \frac{\pi}{2} + 2\pi k$ and $x = \frac{3\pi}{2} + 2\pi k$, where $k$ is an integer. Plot these points on the graph.
4. Connect the Points: Connect the points that you have plotted to form the graph of the function.

Q: What are some applications of the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$?

A: The graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ has several applications in mathematics and science, including:

• Modeling Periodic Phenomena: The graph of the function can be used to model periodic phenomena such as the motion of a pendulum or the vibration of a spring.
• Solving Trigonometric Equations: The graph of the function can be used to solve trigonometric equations such as $\cos \left(\frac{x}{2}\right) = 0$.
• Analyzing Data: The graph of the function can be used to analyze data that exhibits periodic behavior.

Q: How can I use the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to model real-world phenomena?

A: You can use the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to model real-world phenomena such as:

• The motion of a pendulum: The graph of the function can be used to model the motion of a pendulum, which exhibits periodic behavior.
• The vibration of a spring: The graph of the function can be used to model the vibration of a spring, which exhibits periodic behavior.
• The behavior of a population: The graph of the function can be used to model the behavior of a population, which may exhibit periodic behavior.

Q: How can I use the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to solve trigonometric equations?

A: You can use the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to solve trigonometric equations such as $\cos \left(\frac{x}{2}\right) = 0$. To do this, you can:

• Plot the graph: Plot the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$.
• Find the x-intercepts: Find the x-intercepts of the graph, which occur at $x = \frac{\pi}{2} + 2\pi k$ and $x = \frac{3\pi}{2} + 2\pi k$, where $k$ is an integer.
• Solve the equation: Solve the equation $\cos \left(\frac{x}{2}\right) = 0$ by finding the x-intercepts of the graph.

Q: How can I use the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to analyze data?

A: You can use the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to analyze data that exhibits periodic behavior. To do this, you can:

• Plot the data: Plot the data on a graph.
• Find the period: Find the period of the data, which is the time it takes for the data to repeat itself.
• Compare the data to the graph: Compare the data to the graph of the function $f(x) = \cos \left(\frac{x}{2}\right)$ to see if it exhibits periodic behavior.

Conclusion

Graphing the function $f(x) = \cos \left(\frac{x}{2}\right)$ is a fundamental skill that is essential for understanding the behavior of mathematical equations. By following the steps outlined in this article, you can graph the function and visualize its behavior. The graph of the function has several applications in mathematics and science, and it can be used to model periodic phenomena, solve trigonometric equations, and analyze data.