Question
Which function would represent a horizontal compression of j(x)=2√x-4+1
by a factor of 1/5
and reflection over the line x=4
?(1 point)
Responses
q(x)=−2√x-9+1
q left parenthesis x right parenthesis equals negative 2 Start Root x minus 9 End Root plus 1
k(x)=2√-5(x-4)+1
k left parenthesis x right parenthesis equals 2 Start Root negative 5 left parenthesis x minus 4 right parenthesis End Root plus 1
w(x)=−10√x-4+1
w left parenthesis x right parenthesis equals negative 10 Start Root x minus 4 End Root plus 1
m(x)=2√-1/5(x-4)+1
by a factor of 1/5
and reflection over the line x=4
?(1 point)
Responses
q(x)=−2√x-9+1
q left parenthesis x right parenthesis equals negative 2 Start Root x minus 9 End Root plus 1
k(x)=2√-5(x-4)+1
k left parenthesis x right parenthesis equals 2 Start Root negative 5 left parenthesis x minus 4 right parenthesis End Root plus 1
w(x)=−10√x-4+1
w left parenthesis x right parenthesis equals negative 10 Start Root x minus 4 End Root plus 1
m(x)=2√-1/5(x-4)+1
Answers
GPT-4o mini
To find the function that represents a horizontal compression of \( j(x) = 2\sqrt{x-4} + 1 \) by a factor of \( \frac{1}{5} \) and a reflection over the line \( x = 4 \), we need to break down the transformations:
1. **Horizontal Compression**: To compress the function horizontally by a factor of \( \frac{1}{5} \), we replace \( x \) in the function with \( 5x \). This is because replacing \( x \) with \( kx \) (where \( k > 1 \)) compresses the graph horizontally by a factor of \( \frac{1}{k} \).
2. **Reflection over the line \( x = 4 \)**: To reflect the function over the line \( x = 4 \), we perform a transformation that involves shifting the function horizontally. The reflection over \( x = 4 \) can be done by first shifting the function so that the line \( x = 4 \) becomes the y-axis, reflecting, and then shifting back. This means we can change \( x \) to \( 8 - x \) in the function.
Putting both transformations together, we start with your original function:
\[ j(x) = 2\sqrt{x-4} + 1 \]
Now apply the horizontal compression:
\[ j(5x) = 2\sqrt{5x - 4} + 1 \]
Next, apply the reflection over the line \( x = 4 \):
For the reflection, we replace \( x \) with \( 8 - 5x \):
\[ j(8 - 5x) = 2\sqrt{(8 - 5x) - 4} + 1 = 2\sqrt{4 - 5x} + 1 \]
We can also express it with a negative sign to indicate a reflection (by reversing the square root component):
\[ q(x) = 2\sqrt{-5(x - 4)} + 1 \]
Now we can compare this with the provided options.
The closest format corresponding to our derived expression is:
**\( k(x) = 2\sqrt{-5(x - 4)} + 1 \) which is equivalent to \( k(x) = 2\sqrt{4 - 5x} + 1 \)**.
So the function for the specified transformations is:
**\( k(x) = 2\sqrt{-5(x - 4)} + 1 \)**.
1. **Horizontal Compression**: To compress the function horizontally by a factor of \( \frac{1}{5} \), we replace \( x \) in the function with \( 5x \). This is because replacing \( x \) with \( kx \) (where \( k > 1 \)) compresses the graph horizontally by a factor of \( \frac{1}{k} \).
2. **Reflection over the line \( x = 4 \)**: To reflect the function over the line \( x = 4 \), we perform a transformation that involves shifting the function horizontally. The reflection over \( x = 4 \) can be done by first shifting the function so that the line \( x = 4 \) becomes the y-axis, reflecting, and then shifting back. This means we can change \( x \) to \( 8 - x \) in the function.
Putting both transformations together, we start with your original function:
\[ j(x) = 2\sqrt{x-4} + 1 \]
Now apply the horizontal compression:
\[ j(5x) = 2\sqrt{5x - 4} + 1 \]
Next, apply the reflection over the line \( x = 4 \):
For the reflection, we replace \( x \) with \( 8 - 5x \):
\[ j(8 - 5x) = 2\sqrt{(8 - 5x) - 4} + 1 = 2\sqrt{4 - 5x} + 1 \]
We can also express it with a negative sign to indicate a reflection (by reversing the square root component):
\[ q(x) = 2\sqrt{-5(x - 4)} + 1 \]
Now we can compare this with the provided options.
The closest format corresponding to our derived expression is:
**\( k(x) = 2\sqrt{-5(x - 4)} + 1 \) which is equivalent to \( k(x) = 2\sqrt{4 - 5x} + 1 \)**.
So the function for the specified transformations is:
**\( k(x) = 2\sqrt{-5(x - 4)} + 1 \)**.