Select The Correct Answer.Each Statement Describes A Transformation Of The Graph Of { F(x) = \sqrt{x} $}$. Which Statement Correctly Describes The Graph Of { Y = \sqrt{x-7} + 3 $}$?A. It Is The Graph Of { F $}$

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Introduction

Graph transformations are a crucial concept in mathematics, particularly in algebra and calculus. They involve changing the graph of a function in various ways, such as shifting, scaling, and reflecting. In this article, we will focus on understanding the transformation of the graph of the function { f(x) = \sqrt{x} $}$. We will analyze the given statement and determine which option correctly describes the graph of { y = \sqrt{x-7} + 3 $}$.

The Original Function

The original function is { f(x) = \sqrt{x} $}$. This function represents a square root function, which is a type of nonlinear function. The graph of this function is a curve that opens upwards, with the x-axis as its asymptote.

The Transformed Function

The transformed function is { y = \sqrt{x-7} + 3 $}$. This function represents a horizontal shift and a vertical shift of the original function. The horizontal shift is to the right by 7 units, and the vertical shift is upwards by 3 units.

Analyzing the Options

Let's analyze the options given to determine which one correctly describes the graph of the transformed function.

Option A: It is the graph of { f $}$

This option suggests that the graph of the transformed function is the same as the graph of the original function. However, we know that the transformed function has undergone a horizontal shift and a vertical shift, so this option is incorrect.

Option B: It is the graph of { f(x-7) $}$

This option suggests that the graph of the transformed function is the graph of the original function shifted 7 units to the right. This is a correct description of the horizontal shift, but it does not take into account the vertical shift.

Option C: It is the graph of { f(x-7) + 3 $}$

This option suggests that the graph of the transformed function is the graph of the original function shifted 7 units to the right and then shifted 3 units upwards. This is a correct description of both the horizontal and vertical shifts.

Option D: It is the graph of { f(x+7) - 3 $}$

This option suggests that the graph of the transformed function is the graph of the original function shifted 7 units to the left and then shifted 3 units downwards. This is an incorrect description of the horizontal and vertical shifts.

Option E: It is the graph of { f(x-7) - 3 $}$

This option suggests that the graph of the transformed function is the graph of the original function shifted 7 units to the right and then shifted 3 units downwards. This is an incorrect description of the horizontal and vertical shifts.

Conclusion

Based on our analysis, the correct option is Option C: It is the graph of { f(x-7) + 3 $}$. This option correctly describes the graph of the transformed function, which is the graph of the original function shifted 7 units to the right and then shifted 3 units upwards.

Understanding Graph Transformations

Graph transformations are an essential concept in mathematics, particularly in algebra and calculus. They involve changing the graph of a function in various ways, such as shifting, scaling, and reflecting. By understanding these transformations, we can analyze and describe the behavior of functions in different ways.

Types of Graph Transformations

There are several types of graph transformations, including:

  • Horizontal Shifts: Shifting the graph of a function to the left or right by a certain number of units.
  • Vertical Shifts: Shifting the graph of a function upwards or downwards by a certain number of units.
  • Scaling: Changing the size of the graph of a function by a certain factor.
  • Reflections: Reflecting the graph of a function across a certain line or axis.

Real-World Applications

Graph transformations have numerous real-world applications, including:

  • Physics: Graph transformations are used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Graph transformations are used to design and analyze complex systems, such as electrical circuits and mechanical systems.
  • Economics: Graph transformations are used to analyze and describe economic data, such as supply and demand curves.

Conclusion

Introduction

Graph transformations are a fundamental concept in mathematics, particularly in algebra and calculus. In our previous article, we discussed the transformation of the graph of the function { f(x) = \sqrt{x} $}$. In this article, we will provide a Q&A section to help you better understand graph transformations.

Q: What is a graph transformation?

A: A graph transformation is a change in the graph of a function, such as shifting, scaling, or reflecting.

Q: What are the different types of graph transformations?

A: There are several types of graph transformations, including:

  • Horizontal Shifts: Shifting the graph of a function to the left or right by a certain number of units.
  • Vertical Shifts: Shifting the graph of a function upwards or downwards by a certain number of units.
  • Scaling: Changing the size of the graph of a function by a certain factor.
  • Reflections: Reflecting the graph of a function across a certain line or axis.

Q: How do I determine the type of graph transformation?

A: To determine the type of graph transformation, you need to analyze the function and identify the changes made to the original function. For example, if the function is { f(x-7) + 3 $}$, it means the graph of the original function has been shifted 7 units to the right and 3 units upwards.

Q: What is the difference between a horizontal shift and a vertical shift?

A: A horizontal shift is a change in the x-coordinate of the graph, while a vertical shift is a change in the y-coordinate of the graph.

Q: How do I graph a function with a horizontal shift?

A: To graph a function with a horizontal shift, you need to shift the graph of the original function to the left or right by a certain number of units. For example, if the function is { f(x-7) $}$, you need to shift the graph of the original function 7 units to the right.

Q: How do I graph a function with a vertical shift?

A: To graph a function with a vertical shift, you need to shift the graph of the original function upwards or downwards by a certain number of units. For example, if the function is { f(x) + 3 $}$, you need to shift the graph of the original function 3 units upwards.

Q: What is the effect of scaling on a graph?

A: Scaling changes the size of the graph of a function by a certain factor. For example, if the function is { 2f(x) $}$, the graph of the original function will be scaled by a factor of 2.

Q: How do I graph a function with a reflection?

A: To graph a function with a reflection, you need to reflect the graph of the original function across a certain line or axis. For example, if the function is { -f(x) $}$, you need to reflect the graph of the original function across the x-axis.

Q: What are some real-world applications of graph transformations?

A: Graph transformations have numerous real-world applications, including:

  • Physics: Graph transformations are used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Graph transformations are used to design and analyze complex systems, such as electrical circuits and mechanical systems.
  • Economics: Graph transformations are used to analyze and describe economic data, such as supply and demand curves.

Conclusion

In conclusion, graph transformations are a fundamental concept in mathematics, particularly in algebra and calculus. By understanding these transformations, you can analyze and describe the behavior of functions in different ways. We hope this Q&A section has helped you better understand graph transformations.