Introduction to Transformations in Mathematics

Oct 13, 2025 by windailys.com 127 views

Understanding the Relationship Between R(x) and the Graph of f(x) = x^2

Introduction to Transformations in Mathematics

In mathematics, transformations play a crucial role in understanding the behavior of functions and their graphs. A transformation is a change in the position, size, or shape of a function's graph. In this article, we will explore the relationship between the function R(x) and the graph of f(x) = x^2, focusing on how R(x) is related to the graph of f(x) = x^2.

The Function R(x) = -0.5(x-250)^2 + 31,250

The function R(x) = -0.5(x-250)^2 + 31,250 is a quadratic function that can be analyzed to understand its relationship with the graph of f(x) = x^2. To begin, let's break down the function R(x) into its components:

The coefficient -0.5 represents a vertical compression of the graph of f(x) = x^2.
The term (x-250) represents a horizontal shift of the graph of f(x) = x^2 to the right by 250 units.
The constant term +31,250 represents a vertical shift of the graph of f(x) = x^2 upwards by 31,250 units.

Vertical Stretching and Reflection

The graph of R(x) = -0.5(x-250)^2 + 31,250 is the graph of f(x) = x^2 vertically stretched, reflected across the x-axis, and shifted upwards. To understand this, let's analyze the components of R(x) in more detail:

The coefficient -0.5 represents a vertical compression of the graph of f(x) = x^2, but since it is negative, it actually represents a vertical stretch.
The term (x-250) represents a horizontal shift of the graph of f(x) = x^2 to the right by 250 units.
The constant term +31,250 represents a vertical shift of the graph of f(x) = x^2 upwards by 31,250 units.

Comparing the Graphs of R(x) and f(x) = x^2

To visualize the relationship between the graphs of R(x) and f(x) = x^2, let's compare their characteristics:

Vertex: The vertex of the graph of R(x) is located at (250, 31,250), while the vertex of the graph of f(x) = x^2 is located at (0, 0).
Axis of Symmetry: The axis of symmetry of the graph of R(x) is the vertical line x = 250, while the axis of symmetry of the graph of f(x) = x^2 is the vertical line x = 0.
Shape: The graph of R(x) is a parabola that is vertically stretched, reflected across the x-axis, and shifted upwards, while the graph of f(x) = x^2 is a parabola that opens upwards.

Conclusion

In conclusion, the graph of R(x) = -0.5(x-250)^2 + 31,250 is the graph of f(x) = x^2 vertically stretched, reflected across the x-axis, and shifted upwards. Understanding the relationship between R(x) and the graph of f(x) = x^2 is essential in mathematics, as it allows us to analyze and compare the characteristics of different functions and their graphs.

Key Takeaways

The function R(x) = -0.5(x-250)^2 + 31,250 is a quadratic function that can be analyzed to understand its relationship with the graph of f(x) = x^2.
The graph of R(x) is the graph of f(x) = x^2 vertically stretched, reflected across the x-axis, and shifted upwards.
Understanding the relationship between R(x) and the graph of f(x) = x^2 is essential in mathematics, as it allows us to analyze and compare the characteristics of different functions and their graphs.

Further Reading

For further reading on transformations in mathematics, we recommend the following resources:

Khan Academy: Transformations of Functions
Mathway: Transformations of Quadratic Functions
Wolfram MathWorld: Transformations of Functions

By understanding the relationship between R(x) and the graph of f(x) = x^2, we can gain a deeper understanding of transformations in mathematics and how they can be applied to analyze and compare the characteristics of different functions and their graphs.
Q&A: Understanding the Relationship Between R(x) and the Graph of f(x) = x^2

Introduction

In our previous article, we explored the relationship between the function R(x) = -0.5(x-250)^2 + 31,250 and the graph of f(x) = x^2. We analyzed the components of R(x) and how they affect the graph of f(x) = x^2. In this article, we will answer some frequently asked questions about the relationship between R(x) and the graph of f(x) = x^2.

Q: What is the effect of the coefficient -0.5 on the graph of f(x) = x^2?

A: The coefficient -0.5 represents a vertical compression of the graph of f(x) = x^2. However, since it is negative, it actually represents a vertical stretch. This means that the graph of R(x) will be taller and narrower than the graph of f(x) = x^2.

Q: How does the term (x-250) affect the graph of f(x) = x^2?

A: The term (x-250) represents a horizontal shift of the graph of f(x) = x^2 to the right by 250 units. This means that the graph of R(x) will be shifted 250 units to the right compared to the graph of f(x) = x^2.

Q: What is the effect of the constant term +31,250 on the graph of f(x) = x^2?

A: The constant term +31,250 represents a vertical shift of the graph of f(x) = x^2 upwards by 31,250 units. This means that the graph of R(x) will be shifted 31,250 units upwards compared to the graph of f(x) = x^2.

Q: How does the reflection across the x-axis affect the graph of f(x) = x^2?

A: The reflection across the x-axis means that the graph of R(x) will be a mirror image of the graph of f(x) = x^2. This means that the graph of R(x) will be upside down compared to the graph of f(x) = x^2.

Q: Can you provide an example of how to graph R(x) = -0.5(x-250)^2 + 31,250?

A: Yes, here is an example of how to graph R(x) = -0.5(x-250)^2 + 31,250:

Start by graphing the function f(x) = x^2.
Shift the graph of f(x) = x^2 to the right by 250 units to get the graph of (x-250)^2.
Multiply the graph of (x-250)^2 by -0.5 to get the graph of -0.5(x-250)^2.
Shift the graph of -0.5(x-250)^2 upwards by 31,250 units to get the graph of R(x) = -0.5(x-250)^2 + 31,250.

Q: How can I use the relationship between R(x) and the graph of f(x) = x^2 to solve problems?

A: You can use the relationship between R(x) and the graph of f(x) = x^2 to solve problems by analyzing the components of R(x) and how they affect the graph of f(x) = x^2. For example, you can use the relationship between R(x) and the graph of f(x) = x^2 to find the vertex of the graph of R(x), or to determine the axis of symmetry of the graph of R(x).

Conclusion

In conclusion, the relationship between R(x) and the graph of f(x) = x^2 is a fundamental concept in mathematics. By understanding the components of R(x) and how they affect the graph of f(x) = x^2, we can gain a deeper understanding of transformations in mathematics and how they can be applied to solve problems. We hope that this Q&A article has provided you with a better understanding of the relationship between R(x) and the graph of f(x) = x^2.