Analyze The Effect On The Graph Of $f(x) = \log_2 X$ When $f(x$\] Is Replaced By $-2f(x$\].A. The Graph Reflects Across The $x$-axis And Vertically Compresses By A Factor Of $\frac{1}{2}$.B. The Graph Reflects

Introduction

In mathematics, particularly in the field of functions and graphing, understanding how transformations affect the graph of a function is crucial. One common transformation is replacing a function with a scaled version of itself. In this article, we will analyze the effect of replacing $f(x) = \log_2 x$ with $-2f(x)$ on the graph of the function.

Understanding the Original Function

The original function is $f(x) = \log_2 x$. This is a logarithmic function with base 2, which means that for every input $x$, the function returns the power to which 2 must be raised to produce $x$. The graph of this function is a logarithmic curve that increases as $x$ increases.

Replacing $f(x)$ with $-2f(x)$

When we replace $f(x)$ with $-2f(x)$, we are essentially scaling the function by a factor of $-2$. This means that for every input $x$, the new function returns $-2$ times the value of the original function.

Analyzing the Effect on the Graph

To analyze the effect of this transformation on the graph, let's consider the following:

• Reflection across the x-axis: When we multiply the function by $-2$, we are essentially reflecting the graph across the x-axis. This is because the negative sign in front of the function changes the sign of the y-values, effectively flipping the graph upside down.
• Vertical compression: In addition to reflecting the graph across the x-axis, the factor of $-2$ also compresses the graph vertically by a factor of $\frac{1}{2}$. This means that for every input $x$, the new function returns a value that is half the value of the original function.

Conclusion

In conclusion, replacing $f(x) = \log_2 x$ with $-2f(x)$ results in a graph that reflects across the x-axis and vertically compresses by a factor of $\frac{1}{2}$. This transformation changes the shape and position of the graph, but it does not change the overall shape of the logarithmic curve.

Example

To illustrate this, let's consider an example. Suppose we want to find the value of $-2f(x)$ when $x = 8$. We can start by finding the value of $f(x)$ when $x = 8$.

$f(8) = \log_2 8 = 3$

Now, we can multiply this value by $-2$ to find the value of $-2f(x)$ when $x = 8$.

$-2f(8) = -2(3) = -6$

This means that when $x = 8$, the value of $-2f(x)$ is $-6$.

Graphical Representation

The following graph shows the original function $f(x) = \log_2 x$ and the transformed function $-2f(x)$.

Introduction

In our previous article, we analyzed the effect of replacing $f(x) = \log_2 x$ with $-2f(x)$ on the graph of the function. We found that this transformation results in a graph that reflects across the x-axis and vertically compresses by a factor of $\frac{1}{2}$. In this article, we will answer some frequently asked questions about this transformation.

Q: What is the effect of replacing $f(x)$ with $-2f(x)$ on the graph of $f(x) = \log_2 x$?

A: The graph of $-2f(x)$ reflects across the x-axis and vertically compresses by a factor of $\frac{1}{2}$.

Q: Why does the graph reflect across the x-axis?

A: The graph reflects across the x-axis because the negative sign in front of the function changes the sign of the y-values, effectively flipping the graph upside down.

Q: Why does the graph vertically compress by a factor of $\frac{1}{2}$?

A: The graph vertically compresses by a factor of $\frac{1}{2}$ because the factor of $-2$ in front of the function reduces the y-values by half.

Q: How does this transformation affect the shape of the graph?

A: This transformation changes the shape of the graph by reflecting it across the x-axis and compressing it vertically. However, it does not change the overall shape of the logarithmic curve.

Q: Can you provide an example of how this transformation affects the graph?

A: Let's consider an example. Suppose we want to find the value of $-2f(x)$ when $x = 8$. We can start by finding the value of $f(x)$ when $x = 8$.

$f(8) = \log_2 8 = 3$

Now, we can multiply this value by $-2$ to find the value of $-2f(x)$ when $x = 8$.

$-2f(8) = -2(3) = -6$

This means that when $x = 8$, the value of $-2f(x)$ is $-6$.

Q: How can I visualize this transformation?

A: You can visualize this transformation by plotting the graph of $f(x) = \log_2 x$ and the graph of $-2f(x)$ on the same coordinate plane. The graph of $-2f(x)$ will be a reflection of the graph of $f(x)$ across the x-axis, and it will be vertically compressed by a factor of $\frac{1}{2}$.

Q: What are some real-world applications of this transformation?

A: This transformation has many real-world applications, such as:

• Signal processing: This transformation is used in signal processing to filter out noise and compress signals.
• Image processing: This transformation is used in image processing to compress and enhance images.
• Data analysis: This transformation is used in data analysis to compress and visualize large datasets.

Conclusion

In conclusion, replacing $f(x) = \log_2 x$ with $-2f(x)$ results in a graph that reflects across the x-axis and vertically compresses by a factor of $\frac{1}{2}$. This transformation changes the shape and position of the graph, but it does not change the overall shape of the logarithmic curve. We hope this article has helped you understand this transformation and its applications.