# 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 .

- A
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- E

**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 ?

- A
- BThis depends on the definition 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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**Q10: **

A transformation, , that maps the -plane to the -plane is given by , where . Find the image of the region under the transformation .

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**Q11: **

Given that , simplify the expression , where .

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**Q12: **

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

Find the Cartesian equation of the image of under the transformation.

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Find the Cartesian equation of the image of under the transformation.

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**Q13: **

Consider the two MΓΆbius transformations and . Write an expression for the composition in terms of , , , , , , , and and determine whether the resulting transformation is a MΓΆbius transformation.

- A is not a MΓΆbius transformation.
- B is a MΓΆbius transformation.
- C is not a MΓΆbius transformation.
- D is not a MΓΆbius transformation.
- E is a MΓΆbius transformation.

**Q14: **

Write an expression for the composition , where and are the MΓΆbius transformations and , where or .

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**Q15: **

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

Find the Cartesian equation of the image of under the transformation.

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- E

**Q16: **

A transformation, , that maps the -plane to the -plane is given by , where . Find the image of the region under the transformation .

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