Question
which functions have the same transformations as f(x)=2csc(x-1)-3 select all that apply
f(x)=2(x-1)^2-3
f(x)=(2(x-1))/(x-3)
f(x)=2sin(x-3)
f(x)=2e^(x-1)-3
(x-1)^2+(y-3)^2=2
f(x)=2(x-1)^2-3
f(x)=(2(x-1))/(x-3)
f(x)=2sin(x-3)
f(x)=2e^(x-1)-3
(x-1)^2+(y-3)^2=2
Answers
Answer
Based on your answer, at least 1 is the same. We are not trying to find the same shape/function, just the transformations used
Answered by
GPT-4o mini
To determine which functions have the same transformations as \( f(x) = 2\csc(x-1) - 3 \), we need to analyze the transformations involved in this specific function.
1. **Cosecant function transformation**: The basic function here is \( \csc(x) \), which is the reciprocal of the sine function. The transformation involves:
- **Horizontal shifting**: The \( (x-1) \) shifts the graph to the right by 1 unit.
- **Vertical stretching**: The factor of 2 stretches the graph vertically.
- **Vertical shifting**: The \( -3 \) shifts the graph down by 3 units.
Now let's analyze each of the given functions to see if they have the same transformations.
1. **\( f(x) = 2(x-1)^2 - 3 \)**:
- This is a parabola, and it has a vertical stretch (2) and a vertical shift (-3) and a horizontal shift (+1), but it is not the same transformation as the cosecant function.
- **Not the same.**
2. **\( f(x) = \frac{2(x-1)}{x-3} \)**:
- This is a rational function. The vertical stretch is present, but the transformations do not correlate to cosecant or any sine transformation.
- **Not the same.**
3. **\( f(x) = 2 \sin(x-3) \)**:
- This function has a vertical stretch of 2 and a horizontal shift to the right by 3. However, it’s a sine function and not a cosecant function, plus it has a different horizontal shift than the original function.
- **Not the same.**
4. **\( f(x) = 2e^{(x-1)} - 3 \)**:
- This function is an exponential function, with a vertical stretch and a vertical shift but does not relate in any way to sine or cosecant.
- **Not the same.**
5. **\( (x-1)^2 + (y-3)^2 = 2 \)**:
- This equation represents a circle centered at (1,3) with a radius of \( \sqrt{2} \). Although it involves the transformations of \( x-1 \) and \( y-3 \), it does not represent the same transformations as the original cosecant function.
- **Not the same.**
In conclusion, none of the provided functions match the transformations of \( f(x) = 2\csc(x-1) - 3 \). Thus, the answer is:
**None of the functions provided have the same transformations as \( f(x) = 2\csc(x-1) - 3 \).**
1. **Cosecant function transformation**: The basic function here is \( \csc(x) \), which is the reciprocal of the sine function. The transformation involves:
- **Horizontal shifting**: The \( (x-1) \) shifts the graph to the right by 1 unit.
- **Vertical stretching**: The factor of 2 stretches the graph vertically.
- **Vertical shifting**: The \( -3 \) shifts the graph down by 3 units.
Now let's analyze each of the given functions to see if they have the same transformations.
1. **\( f(x) = 2(x-1)^2 - 3 \)**:
- This is a parabola, and it has a vertical stretch (2) and a vertical shift (-3) and a horizontal shift (+1), but it is not the same transformation as the cosecant function.
- **Not the same.**
2. **\( f(x) = \frac{2(x-1)}{x-3} \)**:
- This is a rational function. The vertical stretch is present, but the transformations do not correlate to cosecant or any sine transformation.
- **Not the same.**
3. **\( f(x) = 2 \sin(x-3) \)**:
- This function has a vertical stretch of 2 and a horizontal shift to the right by 3. However, it’s a sine function and not a cosecant function, plus it has a different horizontal shift than the original function.
- **Not the same.**
4. **\( f(x) = 2e^{(x-1)} - 3 \)**:
- This function is an exponential function, with a vertical stretch and a vertical shift but does not relate in any way to sine or cosecant.
- **Not the same.**
5. **\( (x-1)^2 + (y-3)^2 = 2 \)**:
- This equation represents a circle centered at (1,3) with a radius of \( \sqrt{2} \). Although it involves the transformations of \( x-1 \) and \( y-3 \), it does not represent the same transformations as the original cosecant function.
- **Not the same.**
In conclusion, none of the provided functions match the transformations of \( f(x) = 2\csc(x-1) - 3 \). Thus, the answer is:
**None of the functions provided have the same transformations as \( f(x) = 2\csc(x-1) - 3 \).**
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.