Which Of The Following Is The Graph Of $y = -3 \cdot \cos(3x$\]?

Introduction

Trigonometric functions are a fundamental part of mathematics, and understanding their graphs is crucial for solving problems in various fields, including physics, engineering, and mathematics. In this article, we will focus on graphing the trigonometric function $y = -3 \cdot \cos(3x)$ and explore its properties.

What is the Graph of $y = -3 \cdot \cos(3x)$?

The graph of $y = -3 \cdot \cos(3x)$ is a cosine function that has been stretched, compressed, and reflected. The function $y = \cos(x)$ is a standard cosine function that oscillates between $-1$ and $1$. When we multiply the function by $-3$, we are stretching the function vertically by a factor of $3$. This means that the amplitude of the function will increase by a factor of $3$, resulting in a function that oscillates between $-3$ and $3$.

Understanding the Effect of the Coefficient $-3$

The coefficient $-3$ in front of the cosine function has a significant impact on the graph. When we multiply the function by $-3$, we are essentially reflecting the function across the x-axis. This means that the function will now oscillate between $-3$ and $3$, but with a negative sign. This reflection will result in a function that is symmetric about the origin.

Understanding the Effect of the Coefficient $3$ Inside the Cosine Function

The coefficient $3$ inside the cosine function has a significant impact on the graph. When we multiply the function by $3$, we are essentially compressing the function horizontally by a factor of $3$. This means that the function will now oscillate more rapidly, resulting in a function that completes one full cycle in a shorter distance.

Graphing the Function

To graph the function $y = -3 \cdot \cos(3x)$, we can start by graphing the standard cosine function $y = \cos(x)$. We can then multiply the function by $-3$ to stretch it vertically and reflect it across the x-axis. Finally, we can multiply the function by $3$ to compress it horizontally.

Properties of the Graph

The graph of $y = -3 \cdot \cos(3x)$ has several properties that are worth noting. The function is periodic, meaning that it repeats itself at regular intervals. The period of the function is $\frac{2\pi}{3}$, which means that the function will complete one full cycle in this distance. The function is also symmetric about the origin, meaning that it will reflect itself across the x-axis.

Real-World Applications

The graph of $y = -3 \cdot \cos(3x)$ has several real-world applications. For example, the function can be used to model the motion of a pendulum. The function can also be used to model the oscillations of a spring.

Conclusion

In conclusion, the graph of $y = -3 \cdot \cos(3x)$ is a cosine function that has been stretched, compressed, and reflected. The function has several properties, including periodicity and symmetry about the origin. The function has several real-world applications, including modeling the motion of a pendulum and the oscillations of a spring.

References

• [1] "Trigonometric Functions" by Math Open Reference
• [2] "Graphing Trigonometric Functions" by Purplemath
• [3] "Cosine Function" by Wolfram MathWorld

Additional Resources

• [1] "Trigonometric Functions" by Khan Academy
• [2] "Graphing Trigonometric Functions" by IXL
• [3] "Cosine Function" by Mathway
  Frequently Asked Questions: Understanding the Graph of $y = -3 \cdot \cos(3x)$ ================================================================================

Q: What is the period of the graph of $y = -3 \cdot \cos(3x)$?

A: The period of the graph of $y = -3 \cdot \cos(3x)$ is $\frac{2\pi}{3}$. This means that the function will complete one full cycle in this distance.

Q: What is the amplitude of the graph of $y = -3 \cdot \cos(3x)$?

A: The amplitude of the graph of $y = -3 \cdot \cos(3x)$ is $3$. This means that the function will oscillate between $-3$ and $3$.

Q: Is the graph of $y = -3 \cdot \cos(3x)$ symmetric about the origin?

A: Yes, the graph of $y = -3 \cdot \cos(3x)$ is symmetric about the origin. This means that the function will reflect itself across the x-axis.

Q: How can I graph the function $y = -3 \cdot \cos(3x)$?

A: To graph the function $y = -3 \cdot \cos(3x)$, you can start by graphing the standard cosine function $y = \cos(x)$. You can then multiply the function by $-3$ to stretch it vertically and reflect it across the x-axis. Finally, you can multiply the function by $3$ to compress it horizontally.

Q: What are some real-world applications of the graph of $y = -3 \cdot \cos(3x)$?

A: Some real-world applications of the graph of $y = -3 \cdot \cos(3x)$ include modeling the motion of a pendulum and the oscillations of a spring.

Q: How can I use the graph of $y = -3 \cdot \cos(3x)$ to solve problems?

A: You can use the graph of $y = -3 \cdot \cos(3x)$ to solve problems by using it to model real-world situations. For example, you can use the graph to model the motion of a pendulum and then use it to solve problems related to the pendulum's motion.

Q: What are some common mistakes to avoid when graphing the function $y = -3 \cdot \cos(3x)$?

A: Some common mistakes to avoid when graphing the function $y = -3 \cdot \cos(3x)$ include:

• Not taking into account the coefficient $-3$ in front of the cosine function
• Not taking into account the coefficient $3$ inside the cosine function
• Not using the correct period and amplitude of the function

Q: How can I use technology to graph the function $y = -3 \cdot \cos(3x)$?

A: You can use technology such as graphing calculators or computer software to graph the function $y = -3 \cdot \cos(3x)$. This can be a useful tool for visualizing the graph and understanding its properties.

Q: What are some additional resources for learning more about the graph of $y = -3 \cdot \cos(3x)$?

A: Some additional resources for learning more about the graph of $y = -3 \cdot \cos(3x)$ include:

• Online tutorials and videos
• Textbooks and workbooks
• Online communities and forums

Conclusion

In conclusion, the graph of $y = -3 \cdot \cos(3x)$ is a cosine function that has been stretched, compressed, and reflected. The function has several properties, including periodicity and symmetry about the origin. The function has several real-world applications, including modeling the motion of a pendulum and the oscillations of a spring. By understanding the graph of $y = -3 \cdot \cos(3x)$, you can use it to solve problems and model real-world situations.