# Worksheet: Möbius Transformations

In this worksheet, we will practice interpreting möbius transformation in the complex plane.

**Q1: **

Consider the MΓΆbius transformations and , where or 0.

Write an expression for the composition .

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**Q2: **

A transformation that maps the -plane to the -plane is defined by , where .

Find an equation for the image of under the transformation.

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Find an equation for the image of .

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Find a Cartesian equation for the image of .

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Find a Cartesian equation for the image of .

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**Q3: **

A transformation that maps the -plane to the -plane is defined by .

Find the Cartesian equation for the image of .

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Find the Cartesian equation for the image of .

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**Q4: **

A transformation, , that maps the -plane to the -plane is given by , where .

Find a Cartesian equation for the image of the imaginary axis under the transformation .

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Hence, find the image of the region under the transformation .

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**Q5: **

Which of the following MΓΆbius transformations maps to the real axis?

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**Q6: **

Suppose is a linear transformation with . What is the value of ?

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**Q7: **

Suppose is a linear transformation. What is the value of ?

**Q8: **

Which of the following MΓΆbius transformations, , maps to 0, 0 to , and has ?

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**Q9: **

Find the equation for the image of under the transformation , , which maps the -plane to the -plane.

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