Which Is One Of The Transformations Applied To The Graph Of $f(x)=x^2$ To Produce The Graph Of $p(x)=-50+14x-x^2$?A. A Shift Left 7 Units B. A Shift Up 7 Units C. A Shift Right 1 Unit D. A Shift Down 1 Unit

Feb 26, 2025 by ADMIN 216 views

**Transformations of Quadratic Functions: A Comparative Analysis**

Introduction

In mathematics, transformations of functions are essential concepts that help us understand how functions can be manipulated to produce new functions. One of the most common types of functions that undergo transformations is the quadratic function, which is represented by the equation $f(x) = x^2$ . In this article, we will explore the transformations applied to the graph of $f(x) = x^2$ to produce the graph of $p(x) = -50 + 14x - x^2$ . We will analyze the given options and determine which one is the correct transformation.

Understanding the Original Function

The original function is $f(x) = x^2$ . This is a quadratic function that opens upwards, and its graph is a parabola that is symmetric about the y-axis. The vertex of the parabola is at the origin (0, 0).

Understanding the Transformed Function

The transformed function is $p(x) = -50 + 14x - x^2$ . To understand the transformations applied to the original function, we need to compare the two functions. The transformed function has a negative coefficient for the $x^2$ term, which means that the parabola opens downwards. The vertex of the parabola has been shifted to the right and downwards.

Comparing the Two Functions

To determine the transformations applied to the original function, we need to compare the two functions. The transformed function has a negative constant term (-50), which means that the graph of the transformed function has been shifted downwards by 50 units. The transformed function also has a linear term (14x), which means that the graph of the transformed function has been shifted to the right by 1 unit.

Analyzing the Options

Now that we have analyzed the transformations applied to the original function, let's analyze the given options:

A. A shift left 7 units B. A shift up 7 units C. A shift right 1 unit D. A shift down 1 unit

Based on our analysis, we can conclude that the correct transformation is:

C. A shift right 1 unit

The graph of the transformed function has been shifted to the right by 1 unit, and the vertex of the parabola has been shifted to the right and downwards.

Conclusion

In conclusion, the transformations applied to the graph of $f(x) = x^2$ to produce the graph of $p(x) = -50 + 14x - x^2$ are a shift to the right by 1 unit and a shift downwards by 50 units. The correct option is C. A shift right 1 unit.

References

[1] Algebra II for Dummies, by Mary Jane Sterling
[2] Calculus for Dummies, by Mark Ryan
[3] Mathematics for Dummies, by Mary Jane Sterling

Additional Resources

Khan Academy: Quadratic Functions
Mathway: Quadratic Functions
Wolfram Alpha: Quadratic Functions

Frequently Asked Questions

Q: What is the difference between a shift left and a shift right? A: A shift left means that the graph of the function is shifted to the left, while a shift right means that the graph of the function is shifted to the right.

Q: What is the difference between a shift up and a shift down? A: A shift up means that the graph of the function is shifted upwards, while a shift down means that the graph of the function is shifted downwards.

Introduction

In our previous article, we explored the transformations applied to the graph of $f(x) = x^2$ to produce the graph of $p(x) = -50 + 14x - x^2$ . We analyzed the given options and determined that the correct transformation is a shift to the right by 1 unit and a shift downwards by 50 units. In this article, we will provide a Q&A guide to help you understand the transformations of quadratic functions.

Q&A Guide

Q: What is a transformation of a function? A: A transformation of a function is a change in the graph of the function that results in a new function. Transformations can include shifts, stretches, compressions, and reflections.

Q: What are the different types of transformations? A: The different types of transformations are:

Shifts: horizontal and vertical shifts
Stretches: horizontal and vertical stretches
Compressions: horizontal and vertical compressions
Reflections: reflections about the x-axis and y-axis

Q: How do I determine the transformations applied to a function? A: To determine the transformations applied to a function, you need to compare the original function with the transformed function. You can do this by analyzing the coefficients of the terms in the function.

Q: What is a horizontal shift? A: A horizontal shift is a transformation that moves the graph of the function to the left or right. If the coefficient of the x-term is positive, the graph is shifted to the right. If the coefficient of the x-term is negative, the graph is shifted to the left.

Q: What is a vertical shift? A: A vertical shift is a transformation that moves the graph of the function up or down. If the constant term is positive, the graph is shifted upwards. If the constant term is negative, the graph is shifted downwards.

Q: How do I determine the direction of a horizontal shift? A: To determine the direction of a horizontal shift, you need to analyze the coefficient of the x-term. If the coefficient is positive, the graph is shifted to the right. If the coefficient is negative, the graph is shifted to the left.

Q: How do I determine the direction of a vertical shift? A: To determine the direction of a vertical shift, you need to analyze the constant term. If the constant term is positive, the graph is shifted upwards. If the constant term is negative, the graph is shifted downwards.

Q: What is a stretch? A: A stretch is a transformation that changes the width or height of the graph of the function. A horizontal stretch changes the width of the graph, while a vertical stretch changes the height of the graph.

Q: What is a compression? A: A compression is a transformation that changes the width or height of the graph of the function. A horizontal compression changes the width of the graph, while a vertical compression changes the height of the graph.

Q: How do I determine the type of stretch or compression? A: To determine the type of stretch or compression, you need to analyze the coefficient of the x-term. If the coefficient is greater than 1, the graph is stretched. If the coefficient is less than 1, the graph is compressed.

Q: What is a reflection? A: A reflection is a transformation that flips the graph of the function about the x-axis or y-axis.

Q: How do I determine the type of reflection? A: To determine the type of reflection, you need to analyze the sign of the coefficient of the x-term. If the coefficient is positive, the graph is reflected about the x-axis. If the coefficient is negative, the graph is reflected about the y-axis.

Conclusion

In conclusion, transformations of quadratic functions are essential concepts that help us understand how functions can be manipulated to produce new functions. By analyzing the coefficients of the terms in the function, we can determine the transformations applied to the function. We hope that this Q&A guide has helped you understand the transformations of quadratic functions.

References

[1] Algebra II for Dummies, by Mary Jane Sterling
[2] Calculus for Dummies, by Mark Ryan
[3] Mathematics for Dummies, by Mary Jane Sterling

Additional Resources

Khan Academy: Quadratic Functions
Mathway: Quadratic Functions
Wolfram Alpha: Quadratic Functions

Frequently Asked Questions

Q: How do I determine the transformations applied to a function? A: To determine the transformations applied to a function, you need to compare the original function with the transformed function. You can do this by analyzing the coefficients of the terms in the function.