Finding the Starting Point and End Point of One Cycle

Oct 13, 2025 by windailys.com 75 views

Introduction

In this article, we will delve into the analysis of the given trigonometric function f(x) = 1 - 2 cos (x/2 + π). We will start by finding the starting point and end point of one cycle, followed by determining the period and quadrant length. Additionally, we will identify the maximum, minimum, and middle values of the function. Finally, we will sketch the graph of the function and discuss its key characteristics.

Finding the Starting Point and End Point of One Cycle

To find the starting point and end point of one cycle, we need to determine the values of x that correspond to the maximum and minimum values of the function. The function f(x) = 1 - 2 cos (x/2 + π) is a cosine function with a period of 2π. However, the given function has a phase shift of π, which affects the starting point and end point of one cycle.

The starting point of one cycle occurs when the cosine function reaches its maximum value, which is 1. This happens when the argument of the cosine function is equal to 0, i.e., x/2 + π = 0. Solving for x, we get x = -2π.

The end point of one cycle occurs when the cosine function reaches its minimum value, which is -1. This happens when the argument of the cosine function is equal to π, i.e., x/2 + π = π. Solving for x, we get x = 0.

Finding the Period and Quadrant Length

The period of the function f(x) = 1 - 2 cos (x/2 + π) is the distance between two consecutive points on the graph that have the same y-coordinate. Since the function is a cosine function with a period of 2π, the period of the given function is also 2π.

The quadrant length of the function is the distance between two consecutive points on the graph that have the same x-coordinate. Since the function is a cosine function, the quadrant length is equal to the period divided by 2, which is π.

Finding the Maximum, Minimum, and Middle Values

To find the maximum, minimum, and middle values of the function, we need to analyze the graph of the function. The maximum value of the function occurs when the cosine function reaches its maximum value, which is 1. This happens when the argument of the cosine function is equal to 0, i.e., x/2 + π = 0. Solving for x, we get x = -2π.

The minimum value of the function occurs when the cosine function reaches its minimum value, which is -1. This happens when the argument of the cosine function is equal to π, i.e., x/2 + π = π. Solving for x, we get x = 0.

The middle value of the function occurs when the cosine function reaches its middle value, which is 0. This happens when the argument of the cosine function is equal to π/2, i.e., x/2 + π = π/2. Solving for x, we get x = π.

Sketching the Graph

To sketch the graph of the function, we need to plot the points on the graph that correspond to the maximum, minimum, and middle values of the function. We also need to draw the graph of the cosine function and shift it by π units to the left.

The graph of the function f(x) = 1 - 2 cos (x/2 + π) is a cosine function with a period of 2π and a phase shift of π. The graph has a maximum value of 1, a minimum value of -3, and a middle value of -1.

Conclusion

In conclusion, we have analyzed the given trigonometric function f(x) = 1 - 2 cos (x/2 + π) and found its starting point and end point of one cycle, period, quadrant length, maximum, minimum, and middle values. We have also sketched the graph of the function and discussed its key characteristics.

Key Takeaways

The starting point of one cycle occurs when the cosine function reaches its maximum value, which is 1.
The end point of one cycle occurs when the cosine function reaches its minimum value, which is -1.
The period of the function is 2π.
The quadrant length of the function is π.
The maximum value of the function is 1.
The minimum value of the function is -3.
The middle value of the function is -1.

Future Directions

In future studies, we can explore the properties of the given function and its applications in various fields such as physics, engineering, and computer science. We can also investigate the behavior of the function under different conditions and its relationship with other mathematical functions.

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

Period: The distance between two consecutive points on the graph that have the same y-coordinate.
Quadrant length: The distance between two consecutive points on the graph that have the same x-coordinate.
Maximum value: The highest value of the function.
Minimum value: The lowest value of the function.
Middle value: The middle value of the function.

Appendix

Graph of the function: The graph of the function f(x) = 1 - 2 cos (x/2 + π) is a cosine function with a period of 2π and a phase shift of π. The graph has a maximum value of 1, a minimum value of -3, and a middle value of -1.

Q: What is the period of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The period of the function f(x) = 1 - 2 cos (x/2 + π) is 2π.

Q: What is the quadrant length of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The quadrant length of the function f(x) = 1 - 2 cos (x/2 + π) is π.

Q: What is the maximum value of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The maximum value of the function f(x) = 1 - 2 cos (x/2 + π) is 1.

Q: What is the minimum value of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The minimum value of the function f(x) = 1 - 2 cos (x/2 + π) is -3.

Q: What is the middle value of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The middle value of the function f(x) = 1 - 2 cos (x/2 + π) is -1.

Q: How do I sketch the graph of the function f(x) = 1 - 2 cos (x/2 + π)?

A: To sketch the graph of the function f(x) = 1 - 2 cos (x/2 + π), you need to plot the points on the graph that correspond to the maximum, minimum, and middle values of the function. You also need to draw the graph of the cosine function and shift it by π units to the left.

Q: What are the key characteristics of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The key characteristics of the function f(x) = 1 - 2 cos (x/2 + π) are:

The function has a period of 2π.
The function has a quadrant length of π.
The function has a maximum value of 1.
The function has a minimum value of -3.
The function has a middle value of -1.

Q: What are the applications of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The function f(x) = 1 - 2 cos (x/2 + π) has applications in various fields such as physics, engineering, and computer science. It can be used to model periodic phenomena, such as the motion of a pendulum or the vibration of a spring.

Q: How do I find the starting point and end point of one cycle of the function f(x) = 1 - 2 cos (x/2 + π)?

A: To find the starting point and end point of one cycle of the function f(x) = 1 - 2 cos (x/2 + π), you need to determine the values of x that correspond to the maximum and minimum values of the function. The starting point of one cycle occurs when the cosine function reaches its maximum value, which is 1. The end point of one cycle occurs when the cosine function reaches its minimum value, which is -1.

Q: How do I find the period and quadrant length of the function f(x) = 1 - 2 cos (x/2 + π)?

A: To find the period and quadrant length of the function f(x) = 1 - 2 cos (x/2 + π), you need to analyze the graph of the function. The period of the function is the distance between two consecutive points on the graph that have the same y-coordinate. The quadrant length of the function is the distance between two consecutive points on the graph that have the same x-coordinate.

Q: How do I find the maximum, minimum, and middle values of the function f(x) = 1 - 2 cos (x/2 + π)?

A: To find the maximum, minimum, and middle values of the function f(x) = 1 - 2 cos (x/2 + π), you need to analyze the graph of the function. The maximum value of the function occurs when the cosine function reaches its maximum value, which is 1. The minimum value of the function occurs when the cosine function reaches its minimum value, which is -1. The middle value of the function occurs when the cosine function reaches its middle value, which is 0.

Q: What are the key takeaways from the analysis of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The key takeaways from the analysis of the function f(x) = 1 - 2 cos (x/2 + π) are:

The function has a period of 2π.
The function has a quadrant length of π.
The function has a maximum value of 1.
The function has a minimum value of -3.
The function has a middle value of -1.

Q: What are the future directions for the analysis of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The future directions for the analysis of the function f(x) = 1 - 2 cos (x/2 + π) include:

Investigating the properties of the function under different conditions.
Analyzing the relationship between the function and other mathematical functions.
Exploring the applications of the function in various fields.

Q: What are the references for the analysis of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The references for the analysis of the function f(x) = 1 - 2 cos (x/2 + π) include:

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Q: What is the glossary for the analysis of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The glossary for the analysis of the function f(x) = 1 - 2 cos (x/2 + π) includes:

Period: The distance between two consecutive points on the graph that have the same y-coordinate.
Quadrant length: The distance between two consecutive points on the graph that have the same x-coordinate.
Maximum value: The highest value of the function.
Minimum value: The lowest value of the function.
Middle value: The middle value of the function.

Q: What is the appendix for the analysis of the function f(x) = 1 - 2 cos (x/2 + π)?

A: The appendix for the analysis of the function f(x) = 1 - 2 cos (x/2 + π) includes:

Graph of the function: The graph of the function f(x) = 1 - 2 cos (x/2 + π) is a cosine function with a period of 2π and a phase shift of π. The graph has a maximum value of 1, a minimum value of -3, and a middle value of -1.