Comparison with the Original Function

Oct 13, 2025 by miodeya.com 23 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 appearance. In this article, we will explore the transformation of the function $$f(x)=x^2$$ to $$g(x)=\frac{1}{2} f(x)$$ and analyze its graphical representation.

The Original Function: $$f(x)=x^2$$

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.

The Transformed Function: $$g(x)=\frac{1}{2} f(x)$$

To transform the function $$f(x)=x^2$$ into $$g(x)=\frac{1}{2} f(x)$$, we need to multiply the original function by $$\frac{1}{2}$$. This means that the new function will have the same shape as the original function, but it will be stretched vertically by a factor of $$\frac{1}{2}$$.

Graphical Representation

To visualize the graph of $$g(x)=\frac{1}{2} f(x)$$, we can start by plotting the graph of $$f(x)=x^2$$. Then, we can stretch the graph vertically by a factor of $$\frac{1}{2}$$ to obtain the graph of $$g(x)=\frac{1}{2} f(x)$$.

Key Features of the Graph

The graph of $$g(x)=\frac{1}{2} f(x)$$ has several key features that are similar to the graph of $$f(x)=x^2$$. These features include:

Vertex: The vertex of the graph of $$g(x)=\frac{1}{2} f(x)$$ is at the origin (0,0), just like the vertex of the graph of $$f(x)=x^2$$.
Axis of Symmetry: The axis of symmetry of the graph of $$g(x)=\frac{1}{2} f(x)$$ is the y-axis, just like the axis of symmetry of the graph of $$f(x)=x^2$$.
Shape: The graph of $$g(x)=\frac{1}{2} f(x)$$ is a U-shaped curve that opens upwards, just like the graph of $$f(x)=x^2$$.
Scale: The graph of $$g(x)=\frac{1}{2} f(x)$$ is stretched vertically by a factor of $$\frac{1}{2}$$ compared to the graph of $$f(x)=x^2$$.

Comparison with the Original Function

To compare the graph of $$g(x)=\frac{1}{2} f(x)$$ with the graph of $$f(x)=x^2$$, we can look at the following key differences:

Vertical Stretch: The graph of $$g(x)=\frac{1}{2} f(x)$$ is stretched vertically by a factor of $$\frac{1}{2}$$ compared to the graph of $$f(x)=x^2$$.
Y-Intercept: The y-intercept of the graph of $$g(x)=\frac{1}{2} f(x)$$ is $$\frac{1}{2}$$ times the y-intercept of the graph of $$f(x)=x^2$$.
Vertex Height: The vertex height of the graph of $$g(x)=\frac{1}{2} f(x)$$ is $$\frac{1}{2}$$ times the vertex height of the graph of $$f(x)=x^2$$.

Conclusion

In conclusion, the graph of $$g(x)=\frac{1}{2} f(x)$$ is a transformed version of the graph of $$f(x)=x^2$$. The transformation involves stretching the graph vertically by a factor of $$\frac{1}{2}$$. The key features of the graph of $$g(x)=\frac{1}{2} f(x)$$ include the vertex, axis of symmetry, shape, and scale. By comparing the graph of $$g(x)=\frac{1}{2} f(x)$$ with the graph of $$f(x)=x^2$$, we can see the effects of the vertical stretch on the graph.

References

[1] Graphing Functions. (n.d.). Retrieved from https://www.mathsisfun.com/graphing-functions.html
[2] Quadratic Functions. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/quadratic-functions.html

Additional Resources

Graphing Calculator: A graphing calculator can be used to visualize the graph of $$g(x)=\frac{1}{2} f(x)$$ and compare it with the graph of $$f(x)=x^2$$.
Math Software: Math software such as Mathematica or Maple can be used to create the graph of $$g(x)=\frac{1}{2} f(x)$$ and analyze its key features.

Final Thoughts

$In this article, we have explored the transformation of the function $$f(x)=x^2$$ to $$g(x)=\frac{1}{2} f(x)$$ and analyzed its graphical representation. The graph of $$g(x)=\frac{1}{2} f(x)$$ is a stretched version of the graph of $$f(x)=x^2$$, with a vertical stretch by a factor of $$\frac{1}{2}$$. By understanding the transformation of functions, we can gain a deeper appreciation for the properties of functions and their graphical representations.<br/> **Q&A: Understanding the Transformation of Functions**$

Introduction

In our previous article, we explored the transformation of the function $$f(x)=x^2$$ to $$g(x)=\frac{1}{2} f(x)$$ and analyzed its graphical representation. In this article, we will answer some frequently asked questions about the transformation of functions and provide additional insights into the properties of functions and their graphical representations.

Q: What is the effect of multiplying a function by a constant on its graph?

A: When a function is multiplied by a constant, its graph is stretched or compressed vertically by a factor of the constant. In the case of $$g(x)=\frac{1}{2} f(x)$$, the graph of $$g(x)$$ is stretched vertically by a factor of $$\frac{1}{2}$$ compared to the graph of $$f(x)$$.

Q: How does the transformation of a function affect its axis of symmetry?

A: The axis of symmetry of a function is not affected by the transformation of the function. In the case of $$g(x)=\frac{1}{2} f(x)$$, the axis of symmetry is still the y-axis, just like the axis of symmetry of $$f(x)=x^2$$.

Q: What is the effect of transforming a function on its vertex?

A: The vertex of a function is not affected by the transformation of the function. In the case of $$g(x)=\frac{1}{2} f(x)$$, the vertex is still at the origin (0,0), just like the vertex of $$f(x)=x^2$$.

Q: Can the transformation of a function change the shape of its graph?

A: No, the transformation of a function cannot change the shape of its graph. In the case of $$g(x)=\frac{1}{2} f(x)$$, the graph is still a U-shaped curve that opens upwards, just like the graph of $$f(x)=x^2$$.

Q: How can I visualize the graph of a transformed function?

A: You can visualize the graph of a transformed function by using a graphing calculator or math software. These tools can help you create the graph of the transformed function and compare it with the graph of the original function.

Q: What are some common transformations of functions?

A: Some common transformations of functions include:

Vertical Stretch: Multiplying a function by a constant to stretch its graph vertically.
Vertical Compression: Multiplying a function by a constant to compress its graph vertically.
Horizontal Stretch: Multiplying a function by a constant to stretch its graph horizontally.
Horizontal Compression: Multiplying a function by a constant to compress its graph horizontally.

Q: How can I apply transformations to functions in real-world problems?

A: Transformations of functions can be applied to real-world problems in a variety of ways. For example, you can use transformations to model population growth, economic trends, or physical phenomena.

Conclusion

In conclusion, the transformation of functions is an important concept in mathematics that can be used to model real-world problems. By understanding the effects of transformations on the graph of a function, you can gain a deeper appreciation for the properties of functions and their graphical representations. We hope that this Q&A article has provided you with a better understanding of the transformation of functions and its applications.

References

[1] Graphing Functions. (n.d.). Retrieved from https://www.mathsisfun.com/graphing-functions.html
[2] Quadratic Functions. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/quadratic-functions.html

Additional Resources

Graphing Calculator: A graphing calculator can be used to visualize the graph of a transformed function and compare it with the graph of the original function.
Math Software: Math software such as Mathematica or Maple can be used to create the graph of a transformed function and analyze its key features.

Final Thoughts

In this article, we have answered some frequently asked questions about the transformation of functions and provided additional insights into the properties of functions and their graphical representations. We hope that this article has provided you with a better understanding of the transformation of functions and its applications.