Comparison with Parent Function

Oct 13, 2025 by miodeya.com 4 views

Introduction

In mathematics, a vertical shift in a function refers to a change in the position of the graph of the function along the y-axis. This type of shift can be either upward or downward, depending on the direction of the shift. In this article, we will focus on understanding the vertical shift in a logarithmic function, specifically the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$ , as compared to its parent function.

Parent Function

The parent function of a logarithmic function is the basic function without any transformations. In this case, the parent function is $f(x)=\log _3x$ . This function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=-\infty$ .

Given Function

The given function is $f(x)=\log _3\left(\frac{x^6}{81}\right)$ . To understand the vertical shift in this function, we need to compare it with its parent function.

Vertical Shift

To determine the vertical shift, we need to rewrite the given function in a form that is similar to the parent function. We can do this by using the properties of logarithms.

f(x)=\log _3\left(\frac{x^6}{81}\right)

f(x)=\log _3\left(\frac{x^6}{3^4}\right)

f(x)=\log _3\left(\frac{x^6}{3^4}\right)

Using the property of logarithms that $\log _a\left(\frac{b}{c}\right)=\log _ab-\log _ac$ , we can rewrite the function as:

f(x)=\log _3x^6-\log _33^4

f(x)=6\log _3x-4\log _33

Since $\log _33=1$ , we can simplify the function to:

f(x)=6\log _3x-4

Comparison with Parent Function

Now that we have rewritten the given function in a form that is similar to the parent function, we can compare the two functions to determine the vertical shift.

The parent function is $f(x)=\log _3x$ , and the given function is $f(x)=6\log _3x-4$ . To determine the vertical shift, we need to find the difference between the two functions.

f(x)-f(x)=6\log _3x-4-\log _3x

f(x)-f(x)=5\log _3x-4

Since the difference between the two functions is a constant term of $-4$ , we can conclude that the given function is shifted down by 4 units compared to its parent function.

Conclusion

In conclusion, the vertical shift in the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$ as compared to its parent function is 4 units down. This means that the graph of the given function is shifted down by 4 units compared to the graph of the parent function.

Answer

The correct answer is D. 4 units down.

Final Thoughts

Understanding vertical shift in logarithmic functions is an important concept in mathematics. By comparing the given function with its parent function, we can determine the vertical shift and understand how the graph of the function is affected. This knowledge can be applied to a wide range of mathematical problems and is an essential tool for mathematicians and scientists.

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "Vertical Shift" by Khan Academy
[3] "Logarithmic Functions" by Wolfram MathWorld

Additional Resources

[1] "Logarithmic Functions" by MIT OpenCourseWare
[2] "Vertical Shift" by Purplemath
[3] "Logarithmic Functions" by Mathway
Understanding Vertical Shift in Logarithmic Functions: Q&A ===========================================================

Introduction

In our previous article, we discussed the concept of vertical shift in logarithmic functions and how to determine the vertical shift in a given function. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on this topic.

Q&A

Q: What is a vertical shift in a logarithmic function?

A: A vertical shift in a logarithmic function refers to a change in the position of the graph of the function along the y-axis. This type of shift can be either upward or downward, depending on the direction of the shift.

Q: How do I determine the vertical shift in a logarithmic function?

A: To determine the vertical shift, you need to compare the given function with its parent function. You can do this by rewriting the given function in a form that is similar to the parent function using the properties of logarithms.

Q: What is the parent function of a logarithmic function?

A: The parent function of a logarithmic function is the basic function without any transformations. For example, the parent function of the function $f(x)=\log _3\left(\frac{x^6}{81}\right)$ is $f(x)=\log _3x$ .

Q: How do I rewrite a logarithmic function in a form that is similar to its parent function?

A: You can rewrite a logarithmic function in a form that is similar to its parent function using the properties of logarithms. For example, you can use the property that $\log _a\left(\frac{b}{c}\right)=\log _ab-\log _ac$ to rewrite the function.

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

A: A vertical shift refers to a change in the position of the graph of the function along the y-axis, while a horizontal shift refers to a change in the position of the graph of the function along the x-axis.

Q: Can a logarithmic function have both a vertical and a horizontal shift?

A: Yes, a logarithmic function can have both a vertical and a horizontal shift. For example, the function $f(x)=2\log _3(x-1)+1$ has both a vertical shift and a horizontal shift.

Q: How do I determine the vertical shift in a logarithmic function with a horizontal shift?

A: To determine the vertical shift in a logarithmic function with a horizontal shift, you need to compare the given function with its parent function, taking into account the horizontal shift.

Q: What is the significance of understanding vertical shift in logarithmic functions?

A: Understanding vertical shift in logarithmic functions is important because it helps you to analyze and interpret the behavior of the function. It also helps you to identify the key features of the function, such as its asymptotes and intercepts.

Conclusion

In conclusion, understanding vertical shift in logarithmic functions is an important concept in mathematics. By answering the questions in this Q&A section, you should have a better understanding of how to determine the vertical shift in a logarithmic function and how to analyze and interpret the behavior of the function.

Additional Resources

[1] "Logarithmic Functions" by MIT OpenCourseWare
[2] "Vertical Shift" by Khan Academy
[3] "Logarithmic Functions" by Wolfram MathWorld

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "Vertical Shift" by Purplemath
[3] "Logarithmic Functions" by Mathway