Asked by bloozoo
Which of the following functions represents the function f(x)=log2x if it is shifted 5 units up, 7 units to the right, and is shrunk vertically by a factor of 13 ?
Option #1: g(x)=3log2(x+5)−7
Option #2: g(x)=13log2(x+7)+5
Option #3: g(x)=13log2(x−7)+5
(1 point)
Option #1: g(x)=3log2(x+5)−7
Option #2: g(x)=13log2(x+7)+5
Option #3: g(x)=13log2(x−7)+5
(1 point)
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine the correct transformation of the function \( f(x) = \log_2 x \) based on the given conditions, we must apply the transformations step-by-step.
1. **Shift 7 units to the right**: This shifts the graph horizontally. The transformation for shifting \( f(x) \) to the right by 7 units is:
\[
f(x) \to f(x - 7) = \log_2(x - 7).
\]
2. **Shrink vertically by a factor of 13**: This transformation multiplies the function by \( \frac{1}{13} \):
\[
f(x - 7) \to \frac{1}{13} f(x - 7) = \frac{1}{13} \log_2(x - 7).
\]
3. **Shift 5 units up**: Adding 5 to the function gives:
\[
\frac{1}{13} f(x - 7) + 5 = \frac{1}{13} \log_2(x - 7) + 5.
\]
Thus, the transformed function after all the transformations is:
\[
g(x) = \frac{1}{13} \log_2(x - 7) + 5.
\]
None of the given options match exactly, as we derived a vertical shrinking by a factor of \(\frac{1}{13}\). We can examine the options, keeping in mind we to multiply the output of the log function by \(\frac{1}{13}\) instead of \(13\).
Let's analyze each option:
- **Option #1**: \( g(x) = 3\log_2(x+5) - 7 \) does not represent the correct transformations because it has the wrong transformations and shifts.
- **Option #2**: \( g(x) = 13\log_2(x+7) + 5 \) shifts in the wrong direction, and uses the wrong factor.
- **Option #3**: \( g(x) = 13\log_2(x−7) + 5 \) applies an incorrect vertical stretch (should be a shrink, not stretch).
Therefore, none of the options provided correctly represent the transformations applied to \( f(x) = \log_2 x \). The key error is that we are supposed to shrink by a factor of 13 (leading to \( \frac{1}{13} \)) while all given options incorrectly would expand the log function vertically instead of shrinking it.
Hence, based on the transformations described, the correct function would be:
\[
g(x) = \frac{1}{13} \log_2(x - 7) + 5.
\]
Since this function is not among the provided options, none is correct.
1. **Shift 7 units to the right**: This shifts the graph horizontally. The transformation for shifting \( f(x) \) to the right by 7 units is:
\[
f(x) \to f(x - 7) = \log_2(x - 7).
\]
2. **Shrink vertically by a factor of 13**: This transformation multiplies the function by \( \frac{1}{13} \):
\[
f(x - 7) \to \frac{1}{13} f(x - 7) = \frac{1}{13} \log_2(x - 7).
\]
3. **Shift 5 units up**: Adding 5 to the function gives:
\[
\frac{1}{13} f(x - 7) + 5 = \frac{1}{13} \log_2(x - 7) + 5.
\]
Thus, the transformed function after all the transformations is:
\[
g(x) = \frac{1}{13} \log_2(x - 7) + 5.
\]
None of the given options match exactly, as we derived a vertical shrinking by a factor of \(\frac{1}{13}\). We can examine the options, keeping in mind we to multiply the output of the log function by \(\frac{1}{13}\) instead of \(13\).
Let's analyze each option:
- **Option #1**: \( g(x) = 3\log_2(x+5) - 7 \) does not represent the correct transformations because it has the wrong transformations and shifts.
- **Option #2**: \( g(x) = 13\log_2(x+7) + 5 \) shifts in the wrong direction, and uses the wrong factor.
- **Option #3**: \( g(x) = 13\log_2(x−7) + 5 \) applies an incorrect vertical stretch (should be a shrink, not stretch).
Therefore, none of the options provided correctly represent the transformations applied to \( f(x) = \log_2 x \). The key error is that we are supposed to shrink by a factor of 13 (leading to \( \frac{1}{13} \)) while all given options incorrectly would expand the log function vertically instead of shrinking it.
Hence, based on the transformations described, the correct function would be:
\[
g(x) = \frac{1}{13} \log_2(x - 7) + 5.
\]
Since this function is not among the provided options, none is correct.
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