To determine which functions have the same transformations as \( f(x) = 2 \csc(x - 1) - 3 \), we need to analyze the characteristics of the given function.
- Transformation Overview:
- The term \( \csc(x - 1) \) indicates a horizontal shift to the right by 1 unit.
- The coefficient 2 indicates a vertical stretch by a factor of 2.
- The term \( -3 \) indicates a vertical shift downward by 3 units.
Now let's analyze each of the provided functions:
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\( f(x) = 2(x - 1)^2 - 3 \)
- This is a quadratic function, and its transformations involve a horizontal shift (right by 1) and a vertical stretch by 2, followed by a vertical shift down by 3.
- Same transformations? No, because it is not a cosecant function.
-
\( f(x) = \frac{2(x - 1)}{x - 3} \)
- This function is a rational function. It has a transformation to the right by 1 and perhaps some scaling, but it does not exhibit the same oscillatory behavior as a cosecant function.
- Same transformations? No.
-
\( f(x) = 2\sin(x - 3) \)
- This function is a sine function, which is different from the cosecant function (the cosecant is the reciprocal of sine). The sine function is shifted right by 3 and vertically stretched by 2.
- Same transformations? No.
-
\( f(x) = 2e^{x - 1} - 3 \)
- This function is an exponential function, which also has a different form than the cosecant function. It is shifted right by 1 and down by 3 and scaled vertically.
- Same transformations? No.
-
\( (x - 1)^2 + (y - 3)^2 = 2 \)
- This is the equation of a circle centered at \( (1, 3) \) with a radius of \( \sqrt{2} \). It has shifts similar to those seen in the original function, but it does not exhibit any vertical stretches or oscillatory behavior.
- Same transformations? No.
In conclusion, none of the provided options exhibit the same transformations as \( f(x) = 2 \csc(x - 1) - 3 \).
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