Question

What is the vertical shift in f(x)=log3(x^6/81)
as compared to its parent function?(1 point)
Responses

4 units down
4 units down

4 units up
4 units up

6 units up
6 units up

6 units down

Answer 1

To determine the vertical shift of \( f(x) = \log_3\left(\frac{x^6}{81}\right) \) compared to its parent function \( g(x) = \log_3(x) \), we can first simplify the function \( f(x) \).

Using the properties of logarithms, we can rewrite \( f(x) \):

\[
f(x) = \log_3\left(\frac{x^6}{81}\right) = \log_3(x^6) - \log_3(81)
\]

We know from logarithm properties:

\[
\log_3(x^6) = 6 \log_3(x)
\]

Next, we calculate \( \log_3(81) \). Since \( 81 = 3^4 \), we have:

\[
\log_3(81) = \log_3(3^4) = 4
\]

Now we can substitute this back into our function:

\[
f(x) = 6 \log_3(x) - 4
\]

From the equation \( f(x) = 6 \log_3(x) - 4 \), it’s clear that:

1. The transformation \( 6 \log_3(x) \) represents a vertical stretch of the parent function \( g(x) = \log_3(x) \).
2. The \(-4\) in the equation indicates a vertical shift downward by 4 units.

Thus, the vertical shift in \( f(x) \) compared to its parent function \( g(x) \) is **4 units down**.

The correct answer is **4 units down**.