# Worksheet: Transformations of the Complex Plane

In this worksheet, we will practice translating and rotating a complex number in the complex plane.

**Q1: **

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their sum with .

- Aa translation by
- Ba translation by
- Ca translation by
- Da translation by
- Ea translation by

**Q2: **

Multiplying a complex number by corresponds to a transformation composed of a rotation by about the origin and a dilation with center at the origin and positive scale factor. What kind of number is ?

- AA positive imaginary number
- BA positive real number
- CA negative real number
- DA negative imaginary number

**Q3: **

Find an equation for the image of under the transformation of the complex plane .

- A
- B
- C
- D
- E

**Q4: **

Find an equation for the image of under the transformation of -plane to the -plane given by .

- A
- B
- C
- D
- E

**Q5: **

Given that , find an equation for under the transformation that maps the -plane onto the -plane.

- A
- B
- C
- D
- E

**Q6: **

Consider the complex number .

Write in exponential form.

- A
- B
- C
- D
- E

Find the value of .

- A
- B
- C
- D
- E

Find an equation for the image of under the transformation of the complex plane .

- A
- B
- C
- D
- E

**Q7: **

Find an equation for the image of the line under the transformation of the complex plane .

- A
- B
- C
- D
- E

**Q8: **

Find an equation for the image of the half line under the transformation .

- A
- B
- C
- D
- E

**Q9: **

Find an equation for the image of under the transformation of the complex plane , where .

- A
- B
- C
- D
- E

**Q10: **

In the -plane, a curve is given by the Cartesian equation . The transformation represents a transformation from the -plane to the -plane. Find the Cartesian equation for the image of the curve in the -plane.

- A
- B
- C
- D
- E

**Q11: **

Find an equation for the image of under the transformation of the complex plane .

- A
- B
- C
- D
- E

**Q12: **

A line is given by the equation . Find an equation for the image of the line under the transformation .

- A
- B
- C
- D
- E

**Q13: **

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their product with .

- Aa dilation with center the origin and scale factor 5 combined with a rotation by an angle of clockwise about the origin
- Ba dilation with center the origin and scale factor combined with a rotation by an angle of counterclockwise about the origin
- Ca dilation with center the origin and scale factor 5 combined with a rotation by an angle of counterclockwise about the origin
- Da dilation with center the origin and scale factor combined with a rotation by an angle of clockwise about the origin
- Ea dilation with center the origin and scale factor combined with a rotation by an angle of counterclockwise about the origin

**Q14: **

The geometric transformation that occurs when numbers in the complex plane are mapped to their product with is a dilation with center the origin followed by a rotation about the origin by an angle less than . Is this rotation clockwise or counterclockwise?

- Acounterclockwise
- Bclockwise

**Q15: **

Find an equation for the image of under the transformation that maps the -plane to -plane.

- A
- B
- C
- D
- E

**Q16: **

Find an equation for the image of under the transformation of the complex plane .

- A
- B
- C
- D
- E

**Q17: **

Find an equation for the image of under the transformation of -plane to the -plane given by .

- A
- B
- C
- D
- E

**Q18: **

Find an equation for the image of the half-line under the transformation .

- A
- B
- C
- D
- E