Select The Correct Answer.The Parent Function $f(x)=\sqrt[3]{x-1}$ Is Transformed To $g(x)=\sqrt[3]{-x-1}$. Which Graph Correctly Shows The Functions $f(x$\] And $g(x$\]?A. A B. B C. C D. D

Introduction

In mathematics, parent functions serve as the foundation for various transformations. Understanding these transformations is crucial in graphing and analyzing functions. In this article, we will explore the transformation of the parent function $f(x)=\sqrt[3]{x-1}$ to $g(x)=\sqrt[3]{-x-1}$. We will analyze the graphical representation of both functions and determine which graph correctly shows the functions $f(x)$ and $g(x)$.

Understanding the Parent Function

The parent function $f(x)=\sqrt[3]{x-1}$ is a cube root function with a horizontal shift of 1 unit to the right. This means that the graph of $f(x)$ is a cube root curve shifted 1 unit to the right.

Transforming the Parent Function

To transform the parent function $f(x)$ to $g(x)$, we need to apply two transformations:

1. Horizontal Reflection: The function $g(x)$ is obtained by reflecting the function $f(x)$ across the y-axis. This means that the graph of $g(x)$ is a mirror image of the graph of $f(x)$ across the y-axis.
2. Vertical Shift: The function $g(x)$ is also obtained by shifting the function $f(x)$ 1 unit to the left. This means that the graph of $g(x)$ is a cube root curve shifted 1 unit to the left.

Graphical Representation

To determine which graph correctly shows the functions $f(x)$ and $g(x)$, we need to analyze the graphical representation of both functions.

Graph A

Graph A shows the function $f(x)=\sqrt[3]{x-1}$ as a cube root curve shifted 1 unit to the right. However, it does not show the correct transformation of the function $g(x)=\sqrt[3]{-x-1}$.

Graph B

Graph B shows the function $g(x)=\sqrt[3]{-x-1}$ as a cube root curve shifted 1 unit to the left. This graph correctly represents the transformation of the parent function $f(x)$ to $g(x)$.

Graph C

Graph C shows the function $f(x)=\sqrt[3]{x-1}$ as a cube root curve shifted 1 unit to the right. However, it does not show the correct transformation of the function $g(x)=\sqrt[3]{-x-1}$.

Graph D

Graph D shows the function $g(x)=\sqrt[3]{-x-1}$ as a cube root curve shifted 1 unit to the left. However, it does not show the correct transformation of the function $f(x)=\sqrt[3]{x-1}$.

Conclusion

Based on the graphical analysis, the correct graph that shows the functions $f(x)$ and $g(x)$ is Graph B. This graph correctly represents the transformation of the parent function $f(x)$ to $g(x)$.

Answer

The correct answer is B.

Discussion

The transformation of the parent function $f(x)=\sqrt[3]{x-1}$ to $g(x)=\sqrt[3]{-x-1}$ involves two transformations: a horizontal reflection and a vertical shift. The graphical representation of both functions shows that Graph B correctly represents the transformation of the parent function $f(x)$ to $g(x)$.

Key Takeaways

• The parent function $f(x)=\sqrt[3]{x-1}$ is a cube root function with a horizontal shift of 1 unit to the right.
• The function $g(x)=\sqrt[3]{-x-1}$ is obtained by reflecting the function $f(x)$ across the y-axis and shifting it 1 unit to the left.
• The graphical representation of both functions shows that Graph B correctly represents the transformation of the parent function $f(x)$ to $g(x)$.

References

• [1] "Transforming Parent Functions" by [Author's Name]
• [2] "Graphing Functions" by [Author's Name]

Additional Resources

• [1] Khan Academy: "Transforming Parent Functions"
• [2] Mathway: "Graphing Functions"

FAQs

• Q: What is the parent function $f(x)=\sqrt[3]{x-1}$?
• A: The parent function $f(x)=\sqrt[3]{x-1}$ is a cube root function with a horizontal shift of 1 unit to the right.
• Q: What is the function $g(x)=\sqrt[3]{-x-1}$?
• A: The function $g(x)=\sqrt[3]{-x-1}$ is obtained by reflecting the function $f(x)$ across the y-axis and shifting it 1 unit to the left.
• Q: Which graph correctly shows the functions $f(x)$ and $g(x)$?
• A: Graph B correctly shows the functions $f(x)$ and $g(x)$.
  Q&A: Transforming Parent Functions =====================================

Introduction

In our previous article, we explored the transformation of the parent function $f(x)=\sqrt[3]{x-1}$ to $g(x)=\sqrt[3]{-x-1}$. We analyzed the graphical representation of both functions and determined which graph correctly shows the functions $f(x)$ and $g(x)$. In this article, we will answer some frequently asked questions (FAQs) related to transforming parent functions.

Q&A

Q: What is the parent function $f(x)=\sqrt[3]{x-1}$?

A: The parent function $f(x)=\sqrt[3]{x-1}$ is a cube root function with a horizontal shift of 1 unit to the right.

Q: What is the function $g(x)=\sqrt[3]{-x-1}$?

A: The function $g(x)=\sqrt[3]{-x-1}$ is obtained by reflecting the function $f(x)$ across the y-axis and shifting it 1 unit to the left.

Q: Which graph correctly shows the functions $f(x)$ and $g(x)$?

A: Graph B correctly shows the functions $f(x)$ and $g(x)$.

Q: What is the difference between the parent function $f(x)$ and the function $g(x)$?

A: The parent function $f(x)$ is a cube root function with a horizontal shift of 1 unit to the right, while the function $g(x)$ is obtained by reflecting the function $f(x)$ across the y-axis and shifting it 1 unit to the left.

Q: How do you transform a parent function to a new function?

A: To transform a parent function to a new function, you can apply various transformations such as horizontal shifts, vertical shifts, and reflections.

Q: What is the importance of understanding parent functions and their transformations?

A: Understanding parent functions and their transformations is crucial in graphing and analyzing functions. It helps you to identify the characteristics of a function and make predictions about its behavior.

Q: Can you provide examples of other parent functions and their transformations?

A: Yes, here are a few examples:

• The parent function $f(x)=x^2$ can be transformed to $g(x)=-(x-2)^2$ by reflecting the function $f(x)$ across the x-axis and shifting it 2 units to the right.
• The parent function $f(x)=\sin(x)$ can be transformed to $g(x)=\sin(x+\pi)$ by shifting the function $f(x)$ $\pi$ units to the left.

Conclusion

Transforming parent functions is an essential concept in mathematics. Understanding the characteristics of parent functions and their transformations helps you to analyze and graph functions more effectively. In this article, we answered some frequently asked questions related to transforming parent functions. We hope that this article has provided you with a better understanding of this concept.

Additional Resources

• [1] Khan Academy: "Transforming Parent Functions"
• [2] Mathway: "Graphing Functions"
• [3] Wolfram Alpha: "Parent Functions and Transformations"

References

• [1] "Transforming Parent Functions" by [Author's Name]
• [2] "Graphing Functions" by [Author's Name]

FAQs

• Q: What is the parent function $f(x)=\sqrt[3]{x-1}$?
• A: The parent function $f(x)=\sqrt[3]{x-1}$ is a cube root function with a horizontal shift of 1 unit to the right.
• Q: What is the function $g(x)=\sqrt[3]{-x-1}$?
• A: The function $g(x)=\sqrt[3]{-x-1}$ is obtained by reflecting the function $f(x)$ across the y-axis and shifting it 1 unit to the left.
• Q: Which graph correctly shows the functions $f(x)$ and $g(x)$?
• A: Graph B correctly shows the functions $f(x)$ and $g(x)$.

Key Takeaways

• The parent function $f(x)=\sqrt[3]{x-1}$ is a cube root function with a horizontal shift of 1 unit to the right.
• The function $g(x)=\sqrt[3]{-x-1}$ is obtained by reflecting the function $f(x)$ across the y-axis and shifting it 1 unit to the left.
• Graph B correctly shows the functions $f(x)$ and $g(x)$.
• Understanding parent functions and their transformations is crucial in graphing and analyzing functions.