The Parent Function F ( X ) = X 3 F(x)=\sqrt[3]{x} F ( X ) = 3 X ​ Is Transformed To G ( X ) = 2 F ( X − 3 G(x)=2 F(x-3 G ( X ) = 2 F ( X − 3 ]. Which Is The Graph Of G ( X G(x G ( X ]?

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Introduction

In mathematics, transformations of functions are essential concepts that help us understand how different functions are related to each other. The parent function $f(x)=\sqrt[3]{x}$ is a fundamental function in mathematics, and its transformation to $g(x)=2f(x-3)$ is a crucial concept to grasp. In this article, we will delve into the world of function transformations and explore the graph of $g(x)$.

Understanding the Parent Function

The parent function $f(x)=\sqrt[3]{x}$ is a cubic root function, which means that it takes the cube root of the input value $x$. This function is a fundamental building block in mathematics, and its graph is a key concept to understand.

Graph of $f(x)$

The graph of $f(x)=\sqrt[3]{x}$ is a cubic root function that has a few key characteristics:

• The graph passes through the origin $(0,0)$, which means that when $x=0$, $f(x)=0$.
• The graph is increasing, which means that as $x$ increases, $f(x)$ also increases.
• The graph has a horizontal asymptote at $y=0$, which means that as $x$ approaches infinity, $f(x)$ approaches 0.

Understanding the Transformation

The transformation of $f(x)$ to $g(x)=2f(x-3)$ involves two key components:

• Horizontal shift: The function $g(x)$ is shifted 3 units to the right, which means that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 3 units to the right.
• Vertical stretch: The function $g(x)$ is stretched vertically by a factor of 2, which means that the graph of $g(x)$ is twice as tall as the graph of $f(x)$.

Graph of $g(x)$

To understand the graph of $g(x)$, we need to combine the horizontal shift and vertical stretch of the graph of $f(x)$.

• The horizontal shift of 3 units to the right means that the graph of $g(x)$ is shifted 3 units to the right compared to the graph of $f(x)$.
• The vertical stretch by a factor of 2 means that the graph of $g(x)$ is twice as tall as the graph of $f(x)$.

Combining the Transformations

To combine the horizontal shift and vertical stretch, we need to consider the following:

• The graph of $g(x)$ is shifted 3 units to the right, which means that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 3 units to the right.
• The graph of $g(x)$ is stretched vertically by a factor of 2, which means that the graph of $g(x)$ is twice as tall as the graph of $f(x)$.

Conclusion

In conclusion, the graph of $g(x)=2f(x-3)$ is a transformation of the parent function $f(x)=\sqrt[3]{x}$. The graph of $g(x)$ is a combination of a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2. Understanding the graph of $g(x)$ is essential in mathematics, and it has numerous applications in various fields.

References

• [1] "Function Transformations" by Khan Academy
• [2] "Graphing Functions" by Math Open Reference
• [3] "Cubic Root Function" by Wolfram MathWorld

Further Reading

• "Function Transformations: A Comprehensive Guide"
• "Graphing Functions: A Step-by-Step Guide"
• "Cubic Root Function: Properties and Applications"

FAQs

• Q: What is the parent function $f(x)=\sqrt[3]{x}$?
• A: The parent function $f(x)=\sqrt[3]{x}$ is a cubic root function that takes the cube root of the input value $x$.
• Q: What is the transformation of $f(x)$ to $g(x)=2f(x-3)$?
• A: The transformation of $f(x)$ to $g(x)=2f(x-3)$ involves a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2.
• Q: What is the graph of $g(x)$?
• A: The graph of $g(x)$ is a combination of a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2.
  

Introduction

In our previous article, we explored the parent function $f(x)=\sqrt[3]{x}$ and its transformation to $g(x)=2f(x-3)$. In this article, we will answer some frequently asked questions about the parent function transformation.

Q&A

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

A: The parent function $f(x)=\sqrt[3]{x}$ is a cubic root function that takes the cube root of the input value $x$. This function is a fundamental building block in mathematics, and its graph is a key concept to understand.

Q: What is the transformation of $f(x)$ to $g(x)=2f(x-3)$?

A: The transformation of $f(x)$ to $g(x)=2f(x-3)$ involves a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 3 units to the right and stretched vertically by a factor of 2.

Q: What is the graph of $g(x)$?

A: The graph of $g(x)$ is a combination of a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2. This means that the graph of $g(x)$ is twice as tall as the graph of $f(x)$ and shifted 3 units to the right.

Q: How do I graph the parent function $f(x)=\sqrt[3]{x}$?

A: To graph the parent function $f(x)=\sqrt[3]{x}$, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the graph.

Q: How do I graph the transformation $g(x)=2f(x-3)$?

A: To graph the transformation $g(x)=2f(x-3)$, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the graph.

Q: What are some real-world applications of the parent function transformation?

A: The parent function transformation has numerous real-world applications, including:

• Modeling population growth
• Analyzing financial data
• Understanding chemical reactions
• Studying electrical circuits

Q: How do I determine the horizontal shift and vertical stretch of a function?

A: To determine the horizontal shift and vertical stretch of a function, you can use the following steps:

• Identify the parent function
• Identify the transformation (e.g. horizontal shift, vertical stretch)
• Apply the transformation to the parent function
• Graph the resulting function

Q: What are some common mistakes to avoid when graphing the parent function transformation?

A: Some common mistakes to avoid when graphing the parent function transformation include:

• Failing to identify the parent function
• Failing to identify the transformation
• Applying the transformation incorrectly
• Failing to graph the resulting function

Conclusion

In conclusion, the parent function transformation is a fundamental concept in mathematics that has numerous real-world applications. By understanding the parent function transformation, you can model population growth, analyze financial data, understand chemical reactions, and study electrical circuits. We hope that this article has helped you to better understand the parent function transformation and its applications.

References

• [1] "Function Transformations" by Khan Academy
• [2] "Graphing Functions" by Math Open Reference
• [3] "Cubic Root Function" by Wolfram MathWorld

Further Reading

• "Function Transformations: A Comprehensive Guide"
• "Graphing Functions: A Step-by-Step Guide"
• "Cubic Root Function: Properties and Applications"

FAQs

• Q: What is the parent function $f(x)=\sqrt[3]{x}$?
• A: The parent function $f(x)=\sqrt[3]{x}$ is a cubic root function that takes the cube root of the input value $x$.
• Q: What is the transformation of $f(x)$ to $g(x)=2f(x-3)$?
• A: The transformation of $f(x)$ to $g(x)=2f(x-3)$ involves a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2.
• Q: What is the graph of $g(x)$?
• A: The graph of $g(x)$ is a combination of a horizontal shift of 3 units to the right and a vertical stretch by a factor of 2.