Write A Cosine Function That Has A Midline Of Y = 4 Y = 4 Y = 4 , An Amplitude Of 5, And A Period Of 4 7 \frac{4}{7} 7 4 .${ F(x) = }$

Mar 13, 2025 by ADMIN 138 views

**Cosine Function with Customized Parameters**

Introduction

In mathematics, a cosine function is a fundamental concept in trigonometry that describes the relationship between the angle and the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The standard form of a cosine function is given by:

${ f(x) = a \cos (bx + c) + d }$

where:

a is the amplitude of the function
b is the frequency of the function
c is the phase shift of the function
d is the vertical shift or midline of the function

In this article, we will discuss how to write a cosine function with a midline of $y = 4$ , an amplitude of 5, and a period of $\frac{4}{7}$ .

Understanding the Parameters

To write a cosine function with the given parameters, we need to understand the relationship between the parameters and the standard form of the cosine function.

Midline (d): The midline of the function is the vertical shift or the average value of the function. In this case, the midline is $y = 4$ .
Amplitude (a): The amplitude of the function is the maximum value that the function can attain. In this case, the amplitude is 5.
Period (T): The period of the function is the time taken by the function to complete one full cycle. In this case, the period is $\frac{4}{7}$ .

Writing the Cosine Function

To write the cosine function, we need to use the following formula:

${ f(x) = a \cos \left( \frac{2\pi}{T} (x - c) \right) + d }$

where:

a is the amplitude of the function
T is the period of the function
c is the phase shift of the function
d is the vertical shift or midline of the function

Substituting the given values, we get:

${ f(x) = 5 \cos \left( \frac{2\pi}{\frac{4}{7}} (x - c) \right) + 4 }$

Simplifying the expression, we get:

${ f(x) = 5 \cos \left( \frac{7\pi}{4} (x - c) \right) + 4 }$

Finding the Phase Shift

To find the phase shift, we need to use the following formula:

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

Substituting the given values, we get:

${ c = -\frac{\frac{4}{7}}{2\pi} }$

Simplifying the expression, we get:

${ c = -\frac{2}{7\pi} }$

Writing the Final Cosine Function

Substituting the value of c into the expression for f(x), we get:

${ f(x) = 5 \cos \left( \frac{7\pi}{4} (x + \frac{2}{7\pi}) \right) + 4 }$

Simplifying the expression, we get:

${ f(x) = 5 \cos \left( \frac{7\pi}{4} x + \frac{2}{\pi} \right) + 4 }$

Conclusion

In this article, we discussed how to write a cosine function with a midline of $y = 4$ , an amplitude of 5, and a period of $\frac{4}{7}$ . We used the standard form of the cosine function and the given parameters to write the final cosine function. The final cosine function is given by:

${ f(x) = 5 \cos \left( \frac{7\pi}{4} x + \frac{2}{\pi} \right) + 4 }$

This function can be used to model a variety of real-world phenomena, such as sound waves, light waves, and population growth.

Example Use Cases

The cosine function can be used to model a variety of real-world phenomena, such as:

Sound Waves: The cosine function can be used to model the sound waves produced by a guitar string or a drumhead.
Light Waves: The cosine function can be used to model the light waves produced by a laser or a light bulb.
Population Growth: The cosine function can be used to model the population growth of a species over time.

Code Implementation

The cosine function can be implemented in a variety of programming languages, such as Python, MATLAB, or C++. Here is an example implementation in Python:

import numpy as np
def cosine_function(x):
return 5 * np.cos((7 * np.pi / 4) * x + 2 / np.pi) + 4
x = np.linspace(-10, 10, 1000)
y = cosine_function(x)
import matplotlib.pyplot as plt
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Cosine Function')
plt.show()

This code implements the cosine function and plots the function over a range of x values. The resulting plot shows the characteristic shape of the cosine function.

Introduction

In the previous article, we discussed how to write a cosine function with a midline of $y = 4$ , an amplitude of 5, and a period of $\frac{4}{7}$ . In this article, we will answer some frequently asked questions about the cosine function.

Q: What is the amplitude of the cosine function?

A: The amplitude of the cosine function is the maximum value that the function can attain. In this case, the amplitude is 5.

Q: What is the period of the cosine function?

A: The period of the cosine function is the time taken by the function to complete one full cycle. In this case, the period is $\frac{4}{7}$ .

Q: What is the midline of the cosine function?

A: The midline of the cosine function is the vertical shift or the average value of the function. In this case, the midline is $y = 4$ .

Q: How do I find the phase shift of the cosine function?

A: To find the phase shift, you need to use the following formula:

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

where:

T is the period of the function
c is the phase shift of the function

Q: How do I write the cosine function in a programming language?

A: The cosine function can be implemented in a variety of programming languages, such as Python, MATLAB, or C++. Here is an example implementation in Python:

import numpy as np
def cosine_function(x):
return 5 * np.cos((7 * np.pi / 4) * x + 2 / np.pi) + 4
x = np.linspace(-10, 10, 1000)
y = cosine_function(x)
import matplotlib.pyplot as plt
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Cosine Function')
plt.show()

Q: What is the significance of the cosine function in real-world applications?

A: The cosine function has a wide range of applications in real-world phenomena, such as:

Sound Waves: The cosine function can be used to model the sound waves produced by a guitar string or a drumhead.
Light Waves: The cosine function can be used to model the light waves produced by a laser or a light bulb.
Population Growth: The cosine function can be used to model the population growth of a species over time.

Q: How do I plot the cosine function?

A: The cosine function can be plotted using a variety of tools, such as graphing calculators, computer software, or programming languages. Here is an example implementation in Python:

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 1000)
y = 5 * np.cos((7 * np.pi / 4) * x + 2 / np.pi) + 4
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Cosine Function')
plt.show()

Q: What is the relationship between the cosine function and the sine function?

A: The cosine function and the sine function are related by the following identity:

${ \cos(x) = \sin(\frac{\pi}{2} - x) }$

This identity shows that the cosine function and the sine function are essentially the same function, but with a phase shift of $\frac{\pi}{2}$ .

Q: How do I use the cosine function to model real-world phenomena?

A: The cosine function can be used to model a wide range of real-world phenomena, such as sound waves, light waves, and population growth. To use the cosine function to model a real-world phenomenon, you need to:

Identify the parameters of the phenomenon, such as the amplitude, period, and phase shift.
Use the cosine function to model the phenomenon, using the identified parameters.
Plot the function to visualize the phenomenon.

Conclusion

In this article, we answered some frequently asked questions about the cosine function. We discussed the amplitude, period, and midline of the cosine function, as well as how to find the phase shift and write the function in a programming language. We also discussed the significance of the cosine function in real-world applications and how to plot the function.