Question
Which equation represents the transformation formed by horizontally shifting the graph of f(x)=x√ 6 units to the right and then vertically shrinking the graph by a factor of 12 ?
Responses
g(x)=2x−6−−−−√
g left parenthesis x right parenthesis equals 2 square root of x minus 6 end root
g(x)=12x−6−−−−√
g left parenthesis x right parenthesis equals 1 half square root of x minus 6 end root
g(x)=2x+6−−−−−√
g left parenthesis x right parenthesis equals square root of 2 x plus 6 end root
g(x)=12x+6−−−−√
Responses
g(x)=2x−6−−−−√
g left parenthesis x right parenthesis equals 2 square root of x minus 6 end root
g(x)=12x−6−−−−√
g left parenthesis x right parenthesis equals 1 half square root of x minus 6 end root
g(x)=2x+6−−−−−√
g left parenthesis x right parenthesis equals square root of 2 x plus 6 end root
g(x)=12x+6−−−−√
Answers
GPT-4o mini
To determine the correct transformation of the function \( f(x) = x\sqrt{} \) based on the given operations (a horizontal shift and a vertical shrink), we can break down the transformations step by step.
1. **Horizontal Shift to the Right by 6 Units**:
- To shift a graph to the right by \( h \) units, we replace \( x \) with \( x - h \). In this case, \( h = 6 \).
- Thus, after shifting \( f(x) \) to the right by 6 units, we get:
\[
f(x) = (x - 6) \sqrt{}
\]
2. **Vertical Shrink by a Factor of 12**:
- To vertically shrink a function by a factor of \( k \), we multiply the entire function by \( \frac{1}{k} \). In this case, \( k = 12 \).
- After applying this vertical shrink, we modify the function as follows:
\[
g(x) = \frac{1}{12} f(x - 6) = \frac{1}{12}(x - 6)\sqrt{}
\]
The transformations yield the function:
\[
g(x) = \frac{1}{12}(x - 6)\sqrt{}
\]
Now, looking through the provided options:
1. ***g(x) = 2\sqrt{x - 6}***
2. ***g(x) = \frac{1}{2}\sqrt{x - 6}***
3. ***g(x) = \sqrt{2x + 6}***
4. ***g(x) = \sqrt{\frac{1}{12}(x - 6)}***
None of these options seems to be in the suggest form derived from the transformations. The transformation seems correct in concept, but let's clarify.
Working through these transformations explicitly in line with the function \( f(x) = \sqrt{x} \), instead yields:
- The horizontal shift gives \( f(x - 6) = \sqrt{x - 6} \),
- The vertical shrinking gives \( g(x) = \frac{1}{12}\sqrt{x - 6} \).
Given the options presented, none of them perfectly represent the correct transformations, but option **g(x) = \frac{1}{2}\sqrt{x - 6}** almost resembles a similar transformation (a different shrinkage).
It's best to double-check your transformation process and assess if any options were perhaps misinterpreted in their format. The transformation with vertical shrinking should align with functions reflecting manipulation of the original equation \( g(x) = \frac{1}{12}\sqrt{x-6} \) – none of the available options seem to fully reflect.
1. **Horizontal Shift to the Right by 6 Units**:
- To shift a graph to the right by \( h \) units, we replace \( x \) with \( x - h \). In this case, \( h = 6 \).
- Thus, after shifting \( f(x) \) to the right by 6 units, we get:
\[
f(x) = (x - 6) \sqrt{}
\]
2. **Vertical Shrink by a Factor of 12**:
- To vertically shrink a function by a factor of \( k \), we multiply the entire function by \( \frac{1}{k} \). In this case, \( k = 12 \).
- After applying this vertical shrink, we modify the function as follows:
\[
g(x) = \frac{1}{12} f(x - 6) = \frac{1}{12}(x - 6)\sqrt{}
\]
The transformations yield the function:
\[
g(x) = \frac{1}{12}(x - 6)\sqrt{}
\]
Now, looking through the provided options:
1. ***g(x) = 2\sqrt{x - 6}***
2. ***g(x) = \frac{1}{2}\sqrt{x - 6}***
3. ***g(x) = \sqrt{2x + 6}***
4. ***g(x) = \sqrt{\frac{1}{12}(x - 6)}***
None of these options seems to be in the suggest form derived from the transformations. The transformation seems correct in concept, but let's clarify.
Working through these transformations explicitly in line with the function \( f(x) = \sqrt{x} \), instead yields:
- The horizontal shift gives \( f(x - 6) = \sqrt{x - 6} \),
- The vertical shrinking gives \( g(x) = \frac{1}{12}\sqrt{x - 6} \).
Given the options presented, none of them perfectly represent the correct transformations, but option **g(x) = \frac{1}{2}\sqrt{x - 6}** almost resembles a similar transformation (a different shrinkage).
It's best to double-check your transformation process and assess if any options were perhaps misinterpreted in their format. The transformation with vertical shrinking should align with functions reflecting manipulation of the original equation \( g(x) = \frac{1}{12}\sqrt{x-6} \) – none of the available options seem to fully reflect.