Suppose F ( X ) = X 2 F(x)=x^2 F ( X ) = X 2 And G ( X ) = ( 1 3 X ) 2 G(x)=\left(\frac{1}{3} X\right)^2 G ( X ) = ( 3 1 ​ X ) 2 . Which Statement Best Compares The Graph Of G ( X G(x G ( X ] With The Graph Of F ( X F(x F ( X ]?A. The Graph Of G ( X G(x G ( X ] Is The Graph Of F ( X F(x F ( X ] Vertically

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Introduction

In mathematics, functions are used to describe the relationship between variables. When comparing the graphs of two functions, we need to consider their characteristics, such as the shape, size, and position. In this article, we will compare the graphs of two functions, $f(x)=x^2$ and $g(x)=\left(\frac{1}{3} x\right)^2$, and determine which statement best compares the graph of $g(x)$ with the graph of $f(x)$.

Understanding the Functions

Function $f(x)$

The function $f(x)=x^2$ is a quadratic function that represents a parabola. The graph of this function is a U-shaped curve that opens upwards. The vertex of the parabola is at the origin (0, 0), and the axis of symmetry is the y-axis.

Function $g(x)$

The function $g(x)=\left(\frac{1}{3} x\right)^2$ is also a quadratic function, but it is a transformation of the function $f(x)$. The graph of this function is a scaled version of the graph of $f(x)$. The coefficient $\frac{1}{3}$ in front of $x$ represents a horizontal compression of the graph of $f(x)$ by a factor of 3.

Comparing the Graphs

To compare the graphs of $f(x)$ and $g(x)$, we need to consider their characteristics. The graph of $g(x)$ is a scaled version of the graph of $f(x)$, which means that it has the same shape as the graph of $f(x)$ but is compressed horizontally by a factor of 3.

Vertical Stretch or Compression

The graph of $g(x)$ is not a vertical stretch or compression of the graph of $f(x)$. The coefficient $\frac{1}{3}$ in front of $x$ represents a horizontal compression, not a vertical compression.

Horizontal Stretch or Compression

The graph of $g(x)$ is a horizontal compression of the graph of $f(x)$ by a factor of 3. This means that the graph of $g(x)$ is narrower than the graph of $f(x)$.

Reflection

The graph of $g(x)$ is not a reflection of the graph of $f(x)$. The graph of $g(x)$ is a scaled version of the graph of $f(x)$, not a reflection.

Conclusion

In conclusion, the graph of $g(x)$ is a horizontal compression of the graph of $f(x)$ by a factor of 3. This means that the graph of $g(x)$ is narrower than the graph of $f(x)$. Therefore, the statement that best compares the graph of $g(x)$ with the graph of $f(x)$ is:

The graph of $g(x)$ is the graph of $f(x)$ horizontally compressed by a factor of 3.

Final Answer

The final answer is that the graph of $g(x)$ is the graph of $f(x)$ horizontally compressed by a factor of 3.

Discussion

This problem requires the student to understand the concept of function transformations and how to compare the graphs of two functions. The student needs to analyze the characteristics of the two functions and determine which statement best compares the graph of $g(x)$ with the graph of $f(x)$.

Key Concepts

• Function transformations
• Horizontal compression
• Graph comparison

Mathematical Operations

• Multiplication
• Exponentiation

Mathematical Concepts

• Quadratic functions
• Parabolas
• Axis of symmetry

Real-World Applications

• Engineering
• Physics
• Computer Science

References

• [1] "Functions and Graphs" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman
  Q&A: Comparing the Graphs of $f(x)$ and $g(x)$ =====================================================

Introduction

In our previous article, we compared the graphs of two functions, $f(x)=x^2$ and $g(x)=\left(\frac{1}{3} x\right)^2$, and determined that the graph of $g(x)$ is a horizontal compression of the graph of $f(x)$ by a factor of 3. In this article, we will answer some frequently asked questions about comparing the graphs of $f(x)$ and $g(x)$.

Q: What is the difference between the graphs of $f(x)$ and $g(x)$?

A: The graph of $g(x)$ is a horizontal compression of the graph of $f(x)$ by a factor of 3. This means that the graph of $g(x)$ is narrower than the graph of $f(x)$.

Q: How do I determine if the graph of $g(x)$ is a vertical stretch or compression of the graph of $f(x)$?

A: To determine if the graph of $g(x)$ is a vertical stretch or compression of the graph of $f(x)$, you need to look at the coefficient in front of $x$. If the coefficient is greater than 1, the graph of $g(x)$ is a vertical stretch of the graph of $f(x)$. If the coefficient is less than 1, the graph of $g(x)$ is a vertical compression of the graph of $f(x)$.

Q: How do I determine if the graph of $g(x)$ is a reflection of the graph of $f(x)$?

A: To determine if the graph of $g(x)$ is a reflection of the graph of $f(x)$, you need to look at the sign of the coefficient in front of $x$. If the sign is negative, the graph of $g(x)$ is a reflection of the graph of $f(x)$.

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

A: The axis of symmetry of the graph of $g(x)$ is the y-axis.

Q: How do I graph the functions $f(x)$ and $g(x)$?

A: To graph the functions $f(x)$ and $g(x)$, you can use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to draw the graphs by hand.

Q: What are some real-world applications of comparing the graphs of $f(x)$ and $g(x)$?

A: Some real-world applications of comparing the graphs of $f(x)$ and $g(x)$ include:

• Engineering: Comparing the graphs of $f(x)$ and $g(x)$ can help engineers design and optimize systems.
• Physics: Comparing the graphs of $f(x)$ and $g(x)$ can help physicists understand the behavior of physical systems.
• Computer Science: Comparing the graphs of $f(x)$ and $g(x)$ can help computer scientists develop algorithms and data structures.

Conclusion

In conclusion, comparing the graphs of $f(x)$ and $g(x)$ is an important concept in mathematics and has many real-world applications. By understanding how to compare the graphs of these two functions, you can develop your problem-solving skills and apply them to a variety of fields.

Final Answer

The final answer is that the graph of $g(x)$ is a horizontal compression of the graph of $f(x)$ by a factor of 3.

Discussion

This problem requires the student to understand the concept of function transformations and how to compare the graphs of two functions. The student needs to analyze the characteristics of the two functions and determine which statement best compares the graph of $g(x)$ with the graph of $f(x)$.

Key Concepts

• Function transformations
• Horizontal compression
• Graph comparison

Mathematical Operations

• Multiplication
• Exponentiation

Mathematical Concepts

• Quadratic functions
• Parabolas
• Axis of symmetry

Real-World Applications

• Engineering
• Physics
• Computer Science

References

• [1] "Functions and Graphs" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman