Question
Which function would represent a horizontal compression of j(x)=2x−4−−−−√+1
by a factor of 15
and reflection over the line x=4
?(1 point)
Responses
q(x)=−2x−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)=−10x−4−−−−−√+1
w left parenthesis x right parenthesis equals negative 10 Start Root x minus 4 End Root plus 1
m(x)=2−15(x−4)−−−−−−−−−√+1
by a factor of 15
and reflection over the line x=4
?(1 point)
Responses
q(x)=−2x−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)=−10x−4−−−−−√+1
w left parenthesis x right parenthesis equals negative 10 Start Root x minus 4 End Root plus 1
m(x)=2−15(x−4)−−−−−−−−−√+1
Answers
GPT-4o mini
To find the function that represents a horizontal compression by a factor of 15 and a reflection over the line \( x = 4 \) of the function \( j(x) = \sqrt{2x - 4} + 1 \), we need to perform the following steps:
1. **Horizontal Compression:** To compress the function horizontally by a factor of \( k \), we replace \( x \) with \( \frac{x}{k} \). In this case, we will replace \( x \) with \( \frac{x}{15} \).
2. **Reflection over the line \( x = 4 \):** To reflect over the line \( x = 4 \), we can take the transformation \( x \rightarrow 8 - x \), since this will reflect points across \( x = 4 \).
Now let's apply these transformations step by step on \( j(x) \):
### Step 1: Apply Horizontal Compression
Original function:
\[
j(x) = \sqrt{2x - 4} + 1
\]
If we compress horizontally by a factor of 15, we substitute \( \frac{x}{15} \) in place of \( x \):
\[
j\left(\frac{x}{15}\right) = \sqrt{2\left(\frac{x}{15}\right) - 4} + 1 = \sqrt{\frac{2x}{15} - 4} + 1
\]
### Step 2: Apply Reflection Over \( x = 4 \)
Now replace \( x \) with \( 8 - x \):
\[
j\left(8 - \frac{x}{15}\right) = \sqrt{2\left(8 - \frac{x}{15}\right) - 4} + 1
\]
Calculating the expression:
\[
= \sqrt{16 - \frac{2x}{15} - 4} + 1 = \sqrt{12 - \frac{2x}{15}} + 1
\]
Now, this form can be combined for easier representation but we want it in terms of our choices to see which matches.
Next, let's evaluate the options given:
1. \( q(x) = -\sqrt{2x - 9} + 1 \)
2. \( k(x) = 2 - \sqrt{-5(x - 4)} + 1 \)
3. \( w(x) = -\sqrt{10x - 4} + 1 \)
4. \( m(x) = 2 - \sqrt{-15(x - 4)} + 1 \)
In the reflection, we essentially want to use the \( - \) sign to indicate an inverse transformation around \( x=4 \) and note that for our original function we needed the adjustments.
When we relate these changes back, \( k(x) = 2 - \sqrt{-5(x - 4)} + 1 \) correctly signifies the parameters influencing the function downward post the transformations.
Thus, upon detailed assessment, **the correct transformation** reflecting over \( x=4 \) while keeping \( k \) condensed to \( -15 \) variable results in:
- **Final Output:** **\( m(x) = 2 - \sqrt{-15(x - 4)} + 1 \)** which **matches** our constraints of transformation.
The correct answer is \( m(x) = 2 - \sqrt{-15(x - 4)} + 1 \).
1. **Horizontal Compression:** To compress the function horizontally by a factor of \( k \), we replace \( x \) with \( \frac{x}{k} \). In this case, we will replace \( x \) with \( \frac{x}{15} \).
2. **Reflection over the line \( x = 4 \):** To reflect over the line \( x = 4 \), we can take the transformation \( x \rightarrow 8 - x \), since this will reflect points across \( x = 4 \).
Now let's apply these transformations step by step on \( j(x) \):
### Step 1: Apply Horizontal Compression
Original function:
\[
j(x) = \sqrt{2x - 4} + 1
\]
If we compress horizontally by a factor of 15, we substitute \( \frac{x}{15} \) in place of \( x \):
\[
j\left(\frac{x}{15}\right) = \sqrt{2\left(\frac{x}{15}\right) - 4} + 1 = \sqrt{\frac{2x}{15} - 4} + 1
\]
### Step 2: Apply Reflection Over \( x = 4 \)
Now replace \( x \) with \( 8 - x \):
\[
j\left(8 - \frac{x}{15}\right) = \sqrt{2\left(8 - \frac{x}{15}\right) - 4} + 1
\]
Calculating the expression:
\[
= \sqrt{16 - \frac{2x}{15} - 4} + 1 = \sqrt{12 - \frac{2x}{15}} + 1
\]
Now, this form can be combined for easier representation but we want it in terms of our choices to see which matches.
Next, let's evaluate the options given:
1. \( q(x) = -\sqrt{2x - 9} + 1 \)
2. \( k(x) = 2 - \sqrt{-5(x - 4)} + 1 \)
3. \( w(x) = -\sqrt{10x - 4} + 1 \)
4. \( m(x) = 2 - \sqrt{-15(x - 4)} + 1 \)
In the reflection, we essentially want to use the \( - \) sign to indicate an inverse transformation around \( x=4 \) and note that for our original function we needed the adjustments.
When we relate these changes back, \( k(x) = 2 - \sqrt{-5(x - 4)} + 1 \) correctly signifies the parameters influencing the function downward post the transformations.
Thus, upon detailed assessment, **the correct transformation** reflecting over \( x=4 \) while keeping \( k \) condensed to \( -15 \) variable results in:
- **Final Output:** **\( m(x) = 2 - \sqrt{-15(x - 4)} + 1 \)** which **matches** our constraints of transformation.
The correct answer is \( m(x) = 2 - \sqrt{-15(x - 4)} + 1 \).