Graphing Transformations Of Logarithmic Functions Log_6(x) Guide

• A. Translate each point of the graph of $f(x)$ 3 units up.
• B. Translate each point of the graph of $f(x)$ 3 units down.
• C. Reflect the graph of $f(x)$ across the x-axis.

Understanding transformations of functions is a fundamental concept in mathematics, especially when dealing with logarithmic functions. This article will serve as a comprehensive guide on how to graph transformations of the logarithmic function $f(x) = \log_6 x$. We will explore vertical translations (up and down) and reflections across the x-axis. By the end of this guide, you will be equipped with the knowledge and skills to accurately graph these transformations.

Understanding the Base Logarithmic Function: $f(x) = \log_6 x$

To effectively graph transformations of a logarithmic function, it is crucial to first understand the behavior and characteristics of the base logarithmic function itself. The function $f(x) = \log_6 x$ represents the logarithm of x to the base 6. Let's delve into its key properties:

• Domain: The domain of $f(x) = \log_6 x$ is all positive real numbers, i.e., $x > 0$. Logarithmic functions are only defined for positive arguments because you cannot raise a positive base (in this case, 6) to any power and obtain a non-positive result.
• Range: The range of $f(x) = \log_6 x$ is all real numbers. As x approaches 0 from the positive side, the logarithm approaches negative infinity, and as x increases, the logarithm also increases, covering all real values.
• Vertical Asymptote: The graph of $f(x) = \log_6 x$ has a vertical asymptote at $x = 0$. This means the function approaches the vertical line x = 0 but never actually touches it. This is because the logarithm is undefined for x = 0.
• Key Points: To graph the function, it's helpful to identify a few key points. Some common points include:
  • When $x = 1$, $f(1) = \log_6 1 = 0$ (because any number raised to the power of 0 is 1).
  • When $x = 6$, $f(6) = \log_6 6 = 1$ (because 6 raised to the power of 1 is 6).
  • When $x = \frac{1}{6}$, $f(\frac{1}{6}) = \log_6 \frac{1}{6} = -1$ (because 6 raised to the power of -1 is $\frac{1}{6}$).

Plotting these points and understanding the asymptotic behavior allows us to sketch the basic graph of $f(x) = \log_6 x$. The graph will start very close to the y-axis (the asymptote) and gradually increase as x increases, passing through the points (1, 0) and (6, 1).

Understanding these fundamental properties is crucial as we move on to exploring transformations of this logarithmic function. Each transformation will alter the position or orientation of this base graph, but the underlying principles remain the same. The vertical asymptote, the domain restriction, and the key points all play a role in how the transformed graph will look.

A. Vertical Translation: Translating the Graph 3 Units Up

Vertical translations involve shifting the entire graph of a function either upwards or downwards along the y-axis. When we translate the graph of $f(x) = \log_6 x$ three units upward, we are essentially adding 3 to the output value of the function for every input x. This results in a new function, which we can denote as $g(x)$. The transformation can be represented mathematically as:

$g(x) = f(x) + 3$

Substituting $f(x) = \log_6 x$, we get:

$g(x) = \log_6 x + 3$

This equation, $g(x) = \log_6 x + 3$, represents the transformed logarithmic function after the vertical translation. So, how does this translation affect the key features of the graph?

• Key Points: To graph $g(x)$, let's consider the key points of the original function and how they are affected by the transformation:
  • The point (1, 0) on $f(x)$ becomes (1, 0 + 3) = (1, 3) on $g(x)$.
  • The point (6, 1) on $f(x)$ becomes (6, 1 + 3) = (6, 4) on $g(x)$.
  • The point ($\frac{1}{6}$, -1) on $f(x)$ becomes ($\frac{1}{6}$, -1 + 3) = ($\frac{1}{6}$, 2) on $g(x)$.

Notice that the x-coordinates of the points remain unchanged, while the y-coordinates are increased by 3, reflecting the upward shift.

• Vertical Asymptote: The vertical asymptote of the function remains at x = 0. Vertical translations do not affect the vertical asymptotes of logarithmic functions because they only shift the graph up or down, not left or right.
• Domain and Range: The domain of $g(x)$ remains the same as $f(x)$, which is $x > 0$. The range, however, is still all real numbers. The upward shift does not restrict the values that the function can take, it only changes their position on the y-axis.

To sketch the graph of $g(x) = \log_6 x + 3$, we plot the transformed key points and draw a smooth curve that follows the shape of the logarithmic function, approaching the vertical asymptote at x = 0. The graph will be identical to the graph of $f(x) = \log_6 x$, but shifted three units upwards.

B. Vertical Translation: Translating the Graph 3 Units Down

Now, let's consider the case where we translate the graph of $f(x) = \log_6 x$ three units downward. This transformation is analogous to the upward translation, but instead of adding 3 to the output values, we subtract 3. The new function, which we can denote as $h(x)$, is given by:

$h(x) = f(x) - 3$

Substituting $f(x) = \log_6 x$, we get:

$h(x) = \log_6 x - 3$

This equation represents the transformed logarithmic function after the downward vertical translation. Let's analyze the effects of this transformation on the key features of the graph.

• Key Points: Similar to the upward translation, we examine how the key points of the original function are transformed:
  • The point (1, 0) on $f(x)$ becomes (1, 0 - 3) = (1, -3) on $h(x)$.
  • The point (6, 1) on $f(x)$ becomes (6, 1 - 3) = (6, -2) on $h(x)$.
  • The point ($\frac{1}{6}$, -1) on $f(x)$ becomes ($\frac{1}{6}$, -1 - 3) = ($\frac{1}{6}$, -4) on $h(x)$.

As before, the x-coordinates remain unchanged, but the y-coordinates are now decreased by 3, reflecting the downward shift.

• Vertical Asymptote: The vertical asymptote remains at x = 0. Vertical translations only shift the graph vertically, so the vertical asymptote, which is a vertical line, remains unaffected.
• Domain and Range: The domain of $h(x)$ remains the same as $f(x)$, which is $x > 0$. The range is still all real numbers, as the downward shift does not limit the possible output values.

To graph $h(x) = \log_6 x - 3$, we plot the transformed key points and sketch a smooth curve that follows the shape of the logarithmic function, approaching the vertical asymptote at x = 0. The graph will be identical to the graph of $f(x) = \log_6 x$, but shifted three units downwards.

C. Reflection Across the x-axis

Reflecting a function across the x-axis involves taking the mirror image of the graph with respect to the x-axis. This transformation changes the sign of the y-coordinate for each point on the graph. Mathematically, reflecting $f(x)$ across the x-axis results in a new function, which we'll denote as $k(x)$, given by:

$k(x) = -f(x)$

Substituting $f(x) = \log_6 x$, we get:

$k(x) = -\log_6 x$

This equation, $k(x) = -\log_6 x$, represents the transformed logarithmic function after the reflection across the x-axis. Let's examine how this transformation affects the key features of the graph.

• Key Points: To understand the effect of the reflection, we consider how the key points of the original function are transformed:
  • The point (1, 0) on $f(x)$ becomes (1, -0) = (1, 0) on $k(x)$. Notice that this point remains unchanged because it lies on the x-axis.
  • The point (6, 1) on $f(x)$ becomes (6, -1) on $k(x)$.
  • The point ($\frac{1}{6}$, -1) on $f(x)$ becomes ($\frac{1}{6}$, -(-1)) = ($\frac{1}{6}$, 1) on $k(x)$.

The x-coordinates remain the same, while the y-coordinates change their sign. Positive y-coordinates become negative, and negative y-coordinates become positive.

• Vertical Asymptote: The vertical asymptote remains at x = 0. Reflections across the x-axis do not affect vertical asymptotes because they only flip the graph vertically.
• Domain and Range: The domain of $k(x)$ remains the same as $f(x)$, which is $x > 0$. However, the range is still all real numbers. The reflection does not limit the possible output values; it only inverts them.

To sketch the graph of $k(x) = -\log_6 x$, we plot the transformed key points and draw a smooth curve that is the mirror image of the graph of $f(x) = \log_6 x$ with respect to the x-axis. The reflected graph will still approach the vertical asymptote at x = 0, but it will decrease as x increases, which is a characteristic difference from the original function.

Conclusion

In this guide, we have explored how to graph transformations of the logarithmic function $f(x) = \log_6 x$, specifically vertical translations and reflections across the x-axis. We saw that vertical translations shift the graph up or down, changing the y-coordinates of the points while leaving the vertical asymptote and domain unchanged. Reflections across the x-axis invert the graph vertically, changing the sign of the y-coordinates but also preserving the vertical asymptote and domain.

By understanding these transformations, you can effectively graph a variety of logarithmic functions. Remember to always start by understanding the base function and then apply the transformations step by step. Identifying key points and the vertical asymptote will help you accurately sketch the transformed graph. Mastering these techniques provides a solid foundation for further exploration of logarithmic functions and their applications in various fields of mathematics and beyond.