Question
What is the vertical shift in f(x)=log2(−x128) as compared to its parent function?(1 point) Responses 2 units up 2 units up 7 units down 7 units down 7 units up 7 units up 2 units down 2 units down
Answers
Answered by
GPT-4o mini
To find the vertical shift of the function \( f(x) = \log_2(-x + 128) \) compared to its parent function \( g(x) = \log_2(x) \), we need to rewrite the function in a more recognizable form.
First, we can rewrite \( f(x) \) as follows:
\[
f(x) = \log_2(-x + 128)
\]
This can be expressed as:
\[
f(x) = \log_2(128 - x)
\]
The transformation here involves both a horizontal reflection and a horizontal shift, but we can analyze the shifts of the logarithmic function.
The parent function \( g(x) = \log_2(x) \) has its vertical asymptote at \( x = 0 \). The point of vertical asymptote for \( f(x) \) occurs when the argument of the logarithm is zero, which is when:
\[
128 - x = 0 \Rightarrow x = 128
\]
The parent function is shifted to the right to the point where its vertical asymptote is now at \( x = 128 \).
However, since the base logarithmic function does not undergo any vertical shifts (i.e., a constant added to it), the actual vertical shift regarding the function does not occur in this context.
The vertical component you're looking at in the question options focuses on shifts in the \( y \)-direction inherently related to data points or translations.
Given that the overall vertical shift of the logarithmic function remains unaffected (there's no constant added or subtracted to \( f(x) \)), the function's graph is not shifted vertically by any units.
Thus, the correct response is that there **is no vertical shift** indicated in standardized options provided. If you had to choose, based on the standard transformations typically associated with logarithmic functions, you could remind others that they are indeed no vertical shifts unless stated otherwise by a constant modification. The function experiences no vertical movement based on the available responses.
**Therefore, the response regarding vertical shift should be:** **None of the above options, or there is no vertical shift.** If constrained by choices, clarify or mention it's not listed.
First, we can rewrite \( f(x) \) as follows:
\[
f(x) = \log_2(-x + 128)
\]
This can be expressed as:
\[
f(x) = \log_2(128 - x)
\]
The transformation here involves both a horizontal reflection and a horizontal shift, but we can analyze the shifts of the logarithmic function.
The parent function \( g(x) = \log_2(x) \) has its vertical asymptote at \( x = 0 \). The point of vertical asymptote for \( f(x) \) occurs when the argument of the logarithm is zero, which is when:
\[
128 - x = 0 \Rightarrow x = 128
\]
The parent function is shifted to the right to the point where its vertical asymptote is now at \( x = 128 \).
However, since the base logarithmic function does not undergo any vertical shifts (i.e., a constant added to it), the actual vertical shift regarding the function does not occur in this context.
The vertical component you're looking at in the question options focuses on shifts in the \( y \)-direction inherently related to data points or translations.
Given that the overall vertical shift of the logarithmic function remains unaffected (there's no constant added or subtracted to \( f(x) \)), the function's graph is not shifted vertically by any units.
Thus, the correct response is that there **is no vertical shift** indicated in standardized options provided. If you had to choose, based on the standard transformations typically associated with logarithmic functions, you could remind others that they are indeed no vertical shifts unless stated otherwise by a constant modification. The function experiences no vertical movement based on the available responses.
**Therefore, the response regarding vertical shift should be:** **None of the above options, or there is no vertical shift.** If constrained by choices, clarify or mention it's not listed.