The Graph Of $r=-14 \cos \theta$ Has Which Of The Following Characteristics?A. Circle, Diameter Of 14, Center At $(-7,0)$B. Circle, Diameter Of 14, Center At $ ( − 14 , 0 ) (-14,0) ( − 14 , 0 ) [/tex]C. Circle, Radius Of 14, Center At
Introduction
Polar equations are a powerful tool in mathematics, allowing us to describe and analyze curves in the polar coordinate system. One of the most fundamental concepts in polar equations is the graph of a function in the form of $r=f(\theta)$, where $r$ is the radius and $\theta$ is the angle. In this article, we will delve into the characteristics of the graph of $r=-14 \cos \theta$, exploring its properties and identifying the correct answer among the given options.
Understanding Polar Equations
Before we dive into the specifics of the graph of $r=-14 \cos \theta$, let's take a moment to understand the basics of polar equations. In the polar coordinate system, a point is represented by its distance from the origin, $r$, and the angle it makes with the positive x-axis, $\theta$. The polar equation $r=f(\theta)$ describes a curve in the polar plane, where $f(\theta)$ is a function of the angle $\theta$.
Graph of $r=-14 \cos \theta$
The graph of $r=-14 \cos \theta$ is a polar equation that describes a specific curve in the polar plane. To understand the characteristics of this graph, we need to analyze the function $f(\theta)=-14 \cos \theta$. The function $f(\theta)$ is a cosine function with a negative coefficient, which means that the graph will be reflected across the x-axis.
Reflection and Symmetry
The graph of $r=-14 \cos \theta$ is a reflection of the graph of $r=14 \cos \theta$ across the x-axis. This means that for every point $(r, \theta)$ on the graph of $r=14 \cos \theta$, there is a corresponding point $(r, -\theta)$ on the graph of $r=-14 \cos \theta$.
Periodicity
The graph of $r=-14 \cos \theta$ is periodic, meaning that it repeats itself after a certain interval. The period of the graph is $2\pi$, which means that the graph repeats itself every $2\pi$ radians.
Amplitude
The amplitude of the graph of $r=-14 \cos \theta$ is 14, which is the maximum value of the function $f(\theta)=-14 \cos \theta$. The amplitude represents the distance from the origin to the farthest point on the graph.
Characteristics of the Graph
Now that we have analyzed the function $f(\theta)=-14 \cos \theta$, let's identify the characteristics of the graph of $r=-14 \cos \theta$.
- Circle: The graph of $r=-14 \cos \theta$ is a circle, as it is a polar equation that describes a curve in the polar plane.
- Diameter: The diameter of the circle is 14, which is the amplitude of the function $f(\theta)=-14 \cos \theta$.
- Center: The center of the circle is at $(0,0)$, which is the origin of the polar coordinate system.
Conclusion
In conclusion, the graph of $r=-14 \cos \theta$ is a circle with a diameter of 14 and a center at $(0,0)$. This is the correct answer among the given options.
References
- [1] Polar Equations and Graphs by Michael Corral
- [2] Polar Coordinates by Paul's Online Math Notes
- [3] Polar Equations by Math Open Reference
Discussion
What are some other characteristics of the graph of $r=-14 \cos \theta$? How does the graph change when the function $f(\theta)$ is modified? Share your thoughts and insights in the comments below!
Introduction
In our previous article, we explored the characteristics of the graph of $r=-14 \cos \theta$, a polar equation that describes a specific curve in the polar plane. In this article, we will address some of the most frequently asked questions about the graph of $r=-14 \cos \theta$, providing clarity and insight into its properties and behavior.
Q&A
Q: What is the shape of the graph of $r=-14 \cos \theta$?
A: The graph of $r=-14 \cos \theta$ is a circle, as it is a polar equation that describes a curve in the polar plane.
Q: What is the diameter of the circle?
A: The diameter of the circle is 14, which is the amplitude of the function $f(\theta)=-14 \cos \theta$.
Q: What is the center of the circle?
A: The center of the circle is at $(0,0)$, which is the origin of the polar coordinate system.
Q: Is the graph of $r=-14 \cos \theta$ symmetric?
A: Yes, the graph of $r=-14 \cos \theta$ is symmetric with respect to the x-axis, as the function $f(\theta)$ is a cosine function with a negative coefficient.
Q: Is the graph of $r=-14 \cos \theta$ periodic?
A: Yes, the graph of $r=-14 \cos \theta$ is periodic, meaning that it repeats itself after a certain interval. The period of the graph is $2\pi$, which means that the graph repeats itself every $2\pi$ radians.
Q: How does the graph of $r=-14 \cos \theta$ change when the function $f(\theta)$ is modified?
A: If the function $f(\theta)$ is modified, the graph of $r=-14 \cos \theta$ will also change. For example, if the function $f(\theta)$ is multiplied by a constant, the graph of $r=-14 \cos \theta$ will be scaled by that constant.
Q: Can the graph of $r=-14 \cos \theta$ be used to model real-world phenomena?
A: Yes, the graph of $r=-14 \cos \theta$ can be used to model real-world phenomena, such as the motion of a pendulum or the vibration of a spring.
Conclusion
In conclusion, the graph of $r=-14 \cos \theta$ is a circle with a diameter of 14 and a center at $(0,0)$. It is symmetric with respect to the x-axis and periodic, repeating itself every $2\pi$ radians. The graph can be used to model real-world phenomena and can be modified by changing the function $f(\theta)$.
References
- [1] Polar Equations and Graphs by Michael Corral
- [2] Polar Coordinates by Paul's Online Math Notes
- [3] Polar Equations by Math Open Reference
Discussion
Do you have any other questions about the graph of $r=-14 \cos \theta$? Share your thoughts and insights in the comments below!