Which Is The Correct Equation For The Graph Of { F(x) $}$, A Transformation Of The Graph Of { G(x) = \log_2 X $}$?A. { F(x) = -\log_2(x + 2) $}$B. { F(x) = \log_2(x - 2) + 2 $} C . \[ C. \[ C . \[ F(x) = -\log_2 X + 2

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Introduction

In mathematics, graph transformations are essential concepts that help us understand how functions behave under various operations. When dealing with logarithmic functions, it's crucial to grasp the properties of these functions and how they can be transformed. In this article, we will explore the correct equation for the graph of { f(x) $}$, a transformation of the graph of { g(x) = \log_2 x $}$.

Logarithmic Functions

A logarithmic function is a function that is the inverse of an exponential function. The general form of a logarithmic function is { y = \log_b x $}$, where { b $}$ is the base of the logarithm. In this case, we are dealing with a base-2 logarithmic function, { g(x) = \log_2 x $}$.

Properties of Logarithmic Functions

Before we dive into the transformations, let's recall some essential properties of logarithmic functions:

  • Domain: The domain of a logarithmic function is all positive real numbers.
  • Range: The range of a logarithmic function is all real numbers.
  • One-to-One: Logarithmic functions are one-to-one functions, meaning that each output value corresponds to exactly one input value.

Transformations of Logarithmic Functions

Now that we have a good understanding of logarithmic functions, let's explore the transformations. We will consider three possible transformations:

Vertical Shift

A vertical shift is a transformation that moves the graph of a function up or down. In the case of a logarithmic function, a vertical shift can be represented by the equation { f(x) = g(x) + c $}$, where { c $}$ is a constant.

Horizontal Shift

A horizontal shift is a transformation that moves the graph of a function left or right. In the case of a logarithmic function, a horizontal shift can be represented by the equation { f(x) = g(x - h) $}$, where { h $}$ is a constant.

Reflection

A reflection is a transformation that flips the graph of a function over a line. In the case of a logarithmic function, a reflection can be represented by the equation { f(x) = -g(x) $}$.

Analyzing the Options

Now that we have a good understanding of the transformations, let's analyze the options:

Option A: { f(x) = -\log_2(x + 2) $}$

This option represents a reflection of the graph of { g(x) = \log_2 x $}$ over the x-axis, followed by a horizontal shift of 2 units to the left.

Option B: { f(x) = \log_2(x - 2) + 2 $}$

This option represents a horizontal shift of 2 units to the right, followed by a vertical shift of 2 units up.

Option C: { f(x) = -\log_2 x + 2 $}$

This option represents a reflection of the graph of { g(x) = \log_2 x $}$ over the x-axis, followed by a vertical shift of 2 units up.

Conclusion

In conclusion, the correct equation for the graph of { f(x) $}$, a transformation of the graph of { g(x) = \log_2 x $}$, is { f(x) = -\log_2 x + 2 $}$. This equation represents a reflection of the graph of { g(x) = \log_2 x $}$ over the x-axis, followed by a vertical shift of 2 units up.

Final Answer

Introduction

In our previous article, we explored the correct equation for the graph of { f(x) $}$, a transformation of the graph of { g(x) = \log_2 x $}$. In this article, we will answer some frequently asked questions related to graph transformations in mathematics.

Q&A

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

A: A vertical shift is a transformation that moves the graph of a function up or down, while a horizontal shift is a transformation that moves the graph of a function left or right.

Q: How do I determine the type of transformation that has occurred?

A: To determine the type of transformation that has occurred, you need to analyze the equation of the transformed function. If the equation is in the form { f(x) = g(x) + c $}$, it represents a vertical shift. If the equation is in the form { f(x) = g(x - h) $}$, it represents a horizontal shift.

Q: Can a function undergo multiple transformations?

A: Yes, a function can undergo multiple transformations. For example, a function can undergo a vertical shift, followed by a horizontal shift, and then a reflection.

Q: How do I graph a transformed function?

A: To graph a transformed function, you need to graph the original function and then apply the transformations. For example, if you want to graph the function { f(x) = -\log_2 x + 2 $}$, you would first graph the function { g(x) = \log_2 x $}$ and then apply the transformations: reflection over the x-axis and vertical shift of 2 units up.

Q: What is the importance of understanding graph transformations?

A: Understanding graph transformations is essential in mathematics because it helps you to analyze and solve problems involving functions. It also helps you to visualize and understand the behavior of functions.

Q: Can I use graph transformations to solve real-world problems?

A: Yes, graph transformations can be used to solve real-world problems. For example, in economics, graph transformations can be used to model the behavior of supply and demand curves. In physics, graph transformations can be used to model the behavior of motion and energy.

Q: How do I determine the domain and range of a transformed function?

A: To determine the domain and range of a transformed function, you need to analyze the equation of the transformed function. If the equation is in the form { f(x) = g(x) + c $}$, the domain and range remain the same. If the equation is in the form { f(x) = g(x - h) $}$, the domain and range are affected by the horizontal shift.

Q: Can I use graph transformations to solve optimization problems?

A: Yes, graph transformations can be used to solve optimization problems. For example, in business, graph transformations can be used to model the behavior of profit and loss curves, and to determine the optimal price and quantity to produce.

Conclusion

In conclusion, understanding graph transformations is essential in mathematics because it helps you to analyze and solve problems involving functions. It also helps you to visualize and understand the behavior of functions. By answering these frequently asked questions, we hope to have provided you with a better understanding of graph transformations and their applications.

Final Answer

The final answer is: Graph transformations are a powerful tool in mathematics that can be used to analyze and solve problems involving functions.