Question
Which function represents a vertical stretch by a factor of 1.85 with a reflection over the line y=−1 on function w(x)=−3(x+2)−−−−−−−√−1 ?(1 point) Responses t(x)=1.85+3(x+2)−−−−−−−√−1 t left parenthesis x right parenthesis equals 1.85 plus Start Root 3 left parenthesis x plus 2 right parenthesis End Root minus 1 h(x)=3(x+2)√1.85−1 h left parenthesis x right parenthesis equals Start Fraction Start Root 3 left parenthesis x plus 2 right parenthesis End Root over 1.85 End Fraction minus 1 k(x)=4.85(x+2)−−−−−−−−−√−1 k left parenthesis x right parenthesis equals Start Root 4.85 left parenthesis x plus 2 right parenthesis End Root minus 1 r(x)=1.853(x+2)−−−−−−−√−1
Answers
Answered by
GPT-4o mini
To transform the function \( w(x) = -3\sqrt{x+2} - 1 \) by applying a vertical stretch by a factor of 1.85 and reflecting it over the line \( y = -1 \), we can follow these steps:
1. **Reflection over \( y = -1 \)**: This means we need to take \( w(x) \) and add 2 to it (since reflecting over \( y = -1 \) involves adding 2 to the output of the function), which can be expressed as:
\[
w_{reflected}(x) = -1 - (-3\sqrt{x+2} - 1) = 3\sqrt{x+2} + 1
\]
2. **Vertical Stretch by a Factor of 1.85**: We then need to apply the vertical stretch to the reflected function:
\[
w_{stretched}(x) = 1.85 \cdot (3\sqrt{x+2} + 1)
\]
3. **Combine and Simplify**:
\[
w_{stretched}(x) = 1.85 \cdot 3\sqrt{x+2} + 1.85
\]
\[
= 5.55\sqrt{x+2} + 1.85
\]
However, none of the options provided match this exactly. Now, I'm reviewing the choices.
Let's take a look at the choices:
- \( t(x) = 1.85 + 3\sqrt{x+2} - 1 \)
- \( h(x) = 3(x+2)\sqrt{1.85} - 1 \)
- \( k(x) = 4.85\sqrt{x+2} - 1 \)
- \( r(x) = 1.853(x+2) - 1 \)
To find the correct transformation from the original:
Looking at the function transformations, just utilizing the first two transformations separately provides a new base function.
The function that interprets the stretch and reflection for the original processing is:
- **\( t(x) = 3 \times 1.85 \sqrt{x + 2} - 1 \)
Concisely recognizing the reflection and adjustment to each response, the alignment faithfully provides the structure represented by **\( t(x) \)**, simplifying:
Thus the final answer is:
**\( t(x) = 1.85 + 3\sqrt{x+2} - 1 \) - aligning with core placement.**
1. **Reflection over \( y = -1 \)**: This means we need to take \( w(x) \) and add 2 to it (since reflecting over \( y = -1 \) involves adding 2 to the output of the function), which can be expressed as:
\[
w_{reflected}(x) = -1 - (-3\sqrt{x+2} - 1) = 3\sqrt{x+2} + 1
\]
2. **Vertical Stretch by a Factor of 1.85**: We then need to apply the vertical stretch to the reflected function:
\[
w_{stretched}(x) = 1.85 \cdot (3\sqrt{x+2} + 1)
\]
3. **Combine and Simplify**:
\[
w_{stretched}(x) = 1.85 \cdot 3\sqrt{x+2} + 1.85
\]
\[
= 5.55\sqrt{x+2} + 1.85
\]
However, none of the options provided match this exactly. Now, I'm reviewing the choices.
Let's take a look at the choices:
- \( t(x) = 1.85 + 3\sqrt{x+2} - 1 \)
- \( h(x) = 3(x+2)\sqrt{1.85} - 1 \)
- \( k(x) = 4.85\sqrt{x+2} - 1 \)
- \( r(x) = 1.853(x+2) - 1 \)
To find the correct transformation from the original:
Looking at the function transformations, just utilizing the first two transformations separately provides a new base function.
The function that interprets the stretch and reflection for the original processing is:
- **\( t(x) = 3 \times 1.85 \sqrt{x + 2} - 1 \)
Concisely recognizing the reflection and adjustment to each response, the alignment faithfully provides the structure represented by **\( t(x) \)**, simplifying:
Thus the final answer is:
**\( t(x) = 1.85 + 3\sqrt{x+2} - 1 \) - aligning with core placement.**