Question

Which functions have the same transformations as f left parenthesis x right parenthesis equals 2 cosecant left parenthesis x minus 1 right parenthesis minus 3? Select all that apply.
(1 point)
Responses

f left parenthesis x right parenthesis equals 2 left parenthesis x minus 1 right parenthesis squared minus 3
Image with alt text: f left parenthesis x right parenthesis equals 2 left parenthesis x minus 1 right parenthesis squared minus 3

f left parenthesis x right parenthesis equals Start Fraction 2 left parenthesis x minus 1 right parenthesis over x minus 3 End Fraction
Image with alt text: f left parenthesis x right parenthesis equals Start Fraction 2 left parenthesis x minus 1 right parenthesis over x minus 3 End Fraction

f left parenthesis x right parenthesis equals 2 sine left parenthesis x minus 3 right parenthesis
Image with alt text: f left parenthesis x right parenthesis equals 2 sine left parenthesis x minus 3 right parenthesis

f left parenthesis x right parenthesis equals 2 e superscript x minus 1 baseline minus 3
Image with alt text: f left parenthesis x right parenthesis equals 2 e superscript x minus 1 baseline minus 3

left parenthesis x minus 1 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 2
Image with alt text: left parenthesis x minus 1 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 2

Answer 1

To determine which functions have the same transformations as \( f(x) = 2 \csc(x - 1) - 3 \), we should analyze the characteristics of each option regarding transformations.

The function \( f(x) = 2 \csc(x - 1) - 3 \) has the following transformations:
1. A vertical stretch by a factor of 2.
2. A horizontal shift to the right by 1 unit (due to \( x - 1 \)).
3. A vertical shift downward by 3 units.

Now, let’s evaluate the options based on these transformations:

1. **\( f(x) = 2 (x - 1)^2 - 3 \)**:
- This is a quadratic function, and the transformations do not match the original function’s characteristics (sine and cosecant functions can be periodic). Thus, **not a match**.

2. **\( f(x) = \frac{2(x - 1)}{x - 3} \)**:
- This is a rational function and also does not match the characteristics of the original function. Thus, **not a match**.

3. **\( f(x) = 2 \sin(x - 3) \)**:
- This has a vertical stretch of 2 and a horizontal shift, but the horizontal shift is to the right by 3 units instead of 1 and it does not include the vertical shift of -3. Thus, **not a match**.

4. **\( f(x) = 2 e^{x - 1} - 3 \)**:
- Although there is a vertical stretch of 2 and a vertical shift of -3, this is an exponential function rather than a cosecant function and does not have the same periodicity or transformations. Thus, **not a match**.

5. **\( (x - 1)^2 + (y - 3)^2 = 2 \)**:
- This represents a circle centered at (1, 3) with a radius of \( \sqrt{2} \). This is a geometric representation and does not match the characteristics of the original function of cosecant, which is periodic and transformed differently. Thus, **not a match**.

None of the functions listed match the transformations of \( f(x) = 2 \csc(x - 1) - 3 \).