The Parent Cosine Function

Oct 13, 2025 by windailys.com 7 views

Introduction

In mathematics, the study of transformations is a crucial aspect of understanding various functions, including the cosine function. The parent cosine function is a fundamental function that undergoes various transformations to produce different functions. In this article, we will explore the transformations of the parent cosine function and select the correct statements in a given passage.

The Parent Cosine Function

The parent cosine function is defined as:

${ f(x) = \cos x }$

This function has a period of $2\pi$ , an amplitude of 1, and a phase shift of 0.

Transformations of the Parent Cosine Function

The parent cosine function undergoes various transformations to produce different functions. These transformations include:

Horizontal Shift: The function is shifted horizontally by a certain amount.
Vertical Shift: The function is shifted vertically by a certain amount.
Horizontal Compression: The function is compressed horizontally by a certain amount.
Horizontal Stretch: The function is stretched horizontally by a certain amount.
Vertical Compression: The function is compressed vertically by a certain amount.
Vertical Stretch: The function is stretched vertically by a certain amount.
Reflection: The function is reflected about the x-axis or the y-axis.

The Given Passage

The given passage is:

"Hillary's teacher asked her to write a description of the transformations to the parent cosine function that would result in this function:

${ h(x) = -3 \cos (2x - \pi) + 4 }$

Which statements in the following list are correct?

The function is shifted horizontally by $\frac{\pi}{2}$ units to the right.
The function is shifted vertically by 4 units up.
The function is compressed horizontally by a factor of 2.
The function is stretched vertically by a factor of 3.
The function is reflected about the x-axis.
The function is reflected about the y-axis.
The function has a period of $\frac{\pi}{2}$ .
The function has an amplitude of 3.

Selecting the Correct Statements

To select the correct statements, we need to analyze the given function and identify the transformations that have been applied.

Horizontal Shift: The function is shifted horizontally by $\pi$ units to the left, not to the right. Therefore, statement 1 is incorrect.
Vertical Shift: The function is shifted vertically by 4 units up. Therefore, statement 2 is correct.
Horizontal Compression: The function is compressed horizontally by a factor of 2. Therefore, statement 3 is correct.
Vertical Stretch: The function is stretched vertically by a factor of 3. Therefore, statement 4 is correct.
Reflection: The function is not reflected about the x-axis or the y-axis. Therefore, statements 5 and 6 are incorrect.
Period: The function has a period of $\frac{\pi}{2}$ , not $\pi$ . Therefore, statement 7 is incorrect.
Amplitude: The function has an amplitude of 3, not 1. Therefore, statement 8 is correct.

Conclusion

In conclusion, the correct statements in the given passage are:

The function is shifted vertically by 4 units up.
The function is compressed horizontally by a factor of 2.
The function is stretched vertically by a factor of 3.
The function has an amplitude of 3.

Introduction

In our previous article, we explored the transformations of the parent cosine function and selected the correct statements in a given passage. In this article, we will continue to delve deeper into the world of transformations and answer some frequently asked questions.

Q: What is the parent cosine function?

A: The parent cosine function is a fundamental function defined as:

${ f(x) = \cos x }$

This function has a period of $2\pi$ , an amplitude of 1, and a phase shift of 0.

Q: What are the different types of transformations that can be applied to the parent cosine function?

A: The parent cosine function undergoes various transformations to produce different functions. These transformations include:

Horizontal Shift: The function is shifted horizontally by a certain amount.
Vertical Shift: The function is shifted vertically by a certain amount.
Horizontal Compression: The function is compressed horizontally by a certain amount.
Horizontal Stretch: The function is stretched horizontally by a certain amount.
Vertical Compression: The function is compressed vertically by a certain amount.
Vertical Stretch: The function is stretched vertically by a certain amount.
Reflection: The function is reflected about the x-axis or the y-axis.

Q: How do I determine the period of a transformed cosine function?

A: To determine the period of a transformed cosine function, you need to identify the coefficient of the x-term inside the cosine function. If the coefficient is positive, the period is the same as the parent cosine function. If the coefficient is negative, the period is half the value of the parent cosine function.

For example, if the function is:

${ h(x) = \cos (2x) }$

The period is $\frac{\pi}{2}$ .

Q: How do I determine the amplitude of a transformed cosine function?

A: To determine the amplitude of a transformed cosine function, you need to identify the coefficient of the cosine function. If the coefficient is positive, the amplitude is the same as the parent cosine function. If the coefficient is negative, the amplitude is the negative of the parent cosine function.

For example, if the function is:

${ h(x) = -3 \cos (2x) }$

The amplitude is 3.

Q: Can a transformed cosine function have a phase shift?

A: Yes, a transformed cosine function can have a phase shift. The phase shift is determined by the constant term inside the cosine function.

For example, if the function is:

${ h(x) = \cos (2x + \pi) }$

The phase shift is $\frac{\pi}{2}$ .

Q: How do I graph a transformed cosine function?

A: To graph a transformed cosine function, you need to follow these steps:

Graph the parent cosine function: Start by graphing the parent cosine function.
Apply the transformations: Apply the transformations to the parent cosine function, such as horizontal shift, vertical shift, horizontal compression, horizontal stretch, vertical compression, vertical stretch, and reflection.
Graph the transformed function: Graph the transformed function.

Conclusion

In conclusion, understanding the transformations of the parent cosine function is crucial in mathematics. By applying the transformations, we can produce different functions and graph them accordingly. We hope that this Q&A article has helped you to better understand the transformations of the parent cosine function.

Frequently Asked Questions

What is the parent cosine function?
What are the different types of transformations that can be applied to the parent cosine function?
How do I determine the period of a transformed cosine function?
How do I determine the amplitude of a transformed cosine function?
Can a transformed cosine function have a phase shift?
How do I graph a transformed cosine function?