Question
What is the vertical shift in f(x)=log
3
(
81
x
6
) as compared to its parent function?
A. 4 units up
B. 6 units down
C. 4 units down
D. 6 units up
3
(
81
x
6
) as compared to its parent function?
A. 4 units up
B. 6 units down
C. 4 units down
D. 6 units up
Answers
Answered by
GPT-4o mini
To determine the vertical shift of the function \( f(x) = \log_{3}(81x^6) \) compared to its parent function \( g(x) = \log_{3}(x) \), we will first need to analyze how the function \( f(x) \) is transformed from the parent function.
Starting with the expression \( 81x^6 \):
1. **Rewrite \( 81 \)**: We know that \( 81 = 3^4 \). Therefore, we can rewrite \( f(x) \) as:
\[
f(x) = \log_{3}(3^4 x^6) = \log_{3}(3^4) + \log_{3}(x^6)
\]
Using the properties of logarithms:
\[
f(x) = 4 + 6\log_{3}(x)
\]
2. **Break it down**: This separates out into:
\[
f(x) = 4 + 6\log_{3}(x) = 6\log_{3}(x) + 4
\]
Here, the term \( 6\log_{3}(x) \) indicates a vertical stretch by a factor of 6, while the \( +4 \) indicates a vertical shift.
3. **Determine the vertical shift**: The presence of \( +4 \) indicates that the function \( f(x) \) is shifted 4 units upward compared to its parent function \( g(x) = \log_{3}(x) \).
Therefore, the correct answer is:
**A. 4 units up**
Starting with the expression \( 81x^6 \):
1. **Rewrite \( 81 \)**: We know that \( 81 = 3^4 \). Therefore, we can rewrite \( f(x) \) as:
\[
f(x) = \log_{3}(3^4 x^6) = \log_{3}(3^4) + \log_{3}(x^6)
\]
Using the properties of logarithms:
\[
f(x) = 4 + 6\log_{3}(x)
\]
2. **Break it down**: This separates out into:
\[
f(x) = 4 + 6\log_{3}(x) = 6\log_{3}(x) + 4
\]
Here, the term \( 6\log_{3}(x) \) indicates a vertical stretch by a factor of 6, while the \( +4 \) indicates a vertical shift.
3. **Determine the vertical shift**: The presence of \( +4 \) indicates that the function \( f(x) \) is shifted 4 units upward compared to its parent function \( g(x) = \log_{3}(x) \).
Therefore, the correct answer is:
**A. 4 units up**