Determine the type of transformation

Oct 13, 2025 by windailys.com 112 views

Introduction

In mathematics, functions are used to describe the relationship between variables. When we transform a function, we are essentially changing its shape or position on the coordinate plane. Understanding these transformations is crucial in various mathematical and real-world applications. In this article, we will explore the transformations undergone from the function $f(x)=x$ to $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ .

Horizontal Compression

The first transformation we need to consider is the horizontal compression. A horizontal compression occurs when the function is stretched or compressed horizontally. In this case, the function $f(x)=x$ is transformed into $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ . To understand this transformation, let's analyze the input of the function. The original function has an input of $x$ , while the transformed function has an input of $3x$ . This means that the function is being compressed horizontally by a factor of 3.

Vertical Shift

In addition to the horizontal compression, the function is also shifted vertically. A vertical shift occurs when the function is moved up or down on the coordinate plane. In this case, the function is shifted down by $\frac{1}{3}$ units. This is evident from the term $-\frac{1}{3}$ in the transformed function.

Reflection

However, there is no reflection involved in this transformation. A reflection occurs when the function is flipped over a line or axis. In this case, the function is not flipped over any line or axis, so there is no reflection.

Conclusion

In conclusion, the function $f(x)=x$ is transformed into $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ through a horizontal compression by a factor of 3 and a vertical shift down by $\frac{1}{3}$ units. There is no reflection involved in this transformation.

Answer

Based on the analysis above, the correct answer is:

A. Compressed horizontally and shifted down

Explanation

The correct answer is A. Compressed horizontally and shifted down because the function is compressed horizontally by a factor of 3 and shifted down by $\frac{1}{3}$ units.

Example Use Cases

Understanding transformations of functions is crucial in various mathematical and real-world applications. Here are a few example use cases:

Graphing: When graphing functions, it's essential to understand the transformations involved to accurately plot the function on the coordinate plane.
Optimization: In optimization problems, understanding transformations of functions can help us find the maximum or minimum value of a function.
Signal Processing: In signal processing, transformations of functions are used to analyze and manipulate signals.

Conclusion

Introduction

In our previous article, we explored the transformations undergone from the function $f(x)=x$ to $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ . In this article, we will answer some frequently asked questions about transformations of functions.

Q: What is a horizontal compression?

A: A horizontal compression occurs when the function is stretched or compressed horizontally. In the case of the function $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ , the function is compressed horizontally by a factor of 3.

Q: What is a vertical shift?

A: A vertical shift occurs when the function is moved up or down on the coordinate plane. In the case of the function $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ , the function is shifted down by $\frac{1}{3}$ units.

Q: What is a reflection?

A: A reflection occurs when the function is flipped over a line or axis. In the case of the function $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ , there is no reflection involved.

Q: How do I determine the type of transformation?

A: To determine the type of transformation, you need to analyze the input and output of the function. In the case of the function $f(3x)-\frac{1}{3}=3x-\frac{1}{3}$ , the input is $3x$ and the output is $3x-\frac{1}{3}$ . By analyzing the input and output, we can determine that the function is compressed horizontally by a factor of 3 and shifted down by $\frac{1}{3}$ units.

Q: What are some common types of transformations?

A: Some common types of transformations include:

Horizontal compression: When the function is stretched or compressed horizontally.
Vertical shift: When the function is moved up or down on the coordinate plane.
Reflection: When the function is flipped over a line or axis.
Horizontal stretch: When the function is stretched horizontally.
Vertical compression: When the function is compressed vertically.

Q: How do I apply transformations to a function?

A: To apply transformations to a function, you need to follow these steps:

Determine the type of transformation: Determine the type of transformation that needs to be applied to the function.
Analyze the input and output: Analyze the input and output of the function to determine the specific transformation that needs to be applied.
Apply the transformation: Apply the transformation to the function by modifying the input and output accordingly.

Q: What are some real-world applications of transformations of functions?

A: Some real-world applications of transformations of functions include:

Graphing: When graphing functions, it's essential to understand the transformations involved to accurately plot the function on the coordinate plane.
Optimization: In optimization problems, understanding transformations of functions can help us find the maximum or minimum value of a function.
Signal Processing: In signal processing, transformations of functions are used to analyze and manipulate signals.

Conclusion

In conclusion, understanding transformations of functions is crucial in various mathematical and real-world applications. By analyzing the transformations involved, we can accurately plot functions on the coordinate plane, find the maximum or minimum value of a function, and analyze and manipulate signals.