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 3βˆ’2𝑖.

  • Aa translation by ο”βˆ’3βˆ’2
  • Ba translation by 2βˆ’3
  • Ca translation by ο”βˆ’32
  • Da translation by ο”βˆ’23
  • Ea translation by 3βˆ’2

Q2:

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

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

Q3:

Find an equation for the image of |𝑧+3βˆ’2𝑖|=6 under the transformation of the complex plane π‘‡βˆΆπ‘§β†¦32𝑧.

  • A|𝑀+3βˆ’2𝑖|=9
  • B|||𝑀+2βˆ’43𝑖|||=4
  • C|||𝑀+92βˆ’3𝑖|||=6
  • D|||𝑀+92βˆ’3𝑖|||=9
  • E|||𝑀+2βˆ’43𝑖|||=6

Q4:

Find an equation for the image of ||𝑧+√3+3𝑖||=2√3 under the transformation of 𝑧-plane to the 𝑀-plane given by 𝑀=ο€»βˆ’2√3+6π‘–ο‡π‘§βˆ’4.

  • A|π‘€βˆ’28|=12
  • B||π‘€βˆ’4βˆ’βˆš3βˆ’3𝑖||=24
  • C|π‘€βˆ’20|=12
  • D|π‘€βˆ’20|=24
  • E||π‘€βˆ’16+12√3𝑖||=24

Q5:

Given that Im(𝑧)=4, find an equation for Re(𝑀) under the transformation π‘‡βˆΆπ‘§β†¦(2𝑖)𝑧 that maps the 𝑧-plane onto the 𝑀-plane.

  • ARe(𝑀)=8
  • BRe(𝑀)=16
  • CRe(𝑀)=βˆ’2
  • DRe(𝑀)=βˆ’8
  • ERe(𝑀)=2

Q6:

Consider the complex number 𝑧=2+2√3π‘–οŠ§.

Write π‘§οŠ§ in exponential form.

  • A4𝑒οŽ₯
  • B2𝑒οŽ₯
  • C2π‘’ο‘½οŽ’οƒ
  • Dπ‘’ο‘½οŽ’οƒ
  • E4π‘’ο‘½οŽ’οƒ

Find the value of (𝑧)(2+𝑖).

  • Aβˆ’8(1+2𝑖)
  • Bβˆ’64(2+𝑖)
  • C64(2+𝑖)
  • Dβˆ’8(2+𝑖)
  • E8(2+𝑖)

Find an equation for the image of |π‘§βˆ’3𝑖|=5 under the transformation of the complex plane π‘‡βˆΆπ‘§β†¦(𝑧)π‘§οŠ§οŠ©.

  • A|𝑀+36𝑖|=60
  • B|π‘€βˆ’192𝑖|=320
  • C|𝑀+36𝑖|=60
  • D|π‘€βˆ’36𝑖|=60
  • E|𝑀+192𝑖|=320

Q7:

Find an equation for the image of the line |𝑧|=|𝑧+3𝑖| under the transformation of the complex plane π‘‡βˆΆπ‘§β†¦(π‘–βˆ’1)𝑧.

  • A|𝑀|=|𝑀+3+3𝑖|
  • B|𝑀|=|π‘€βˆ’1+4𝑖|
  • C|𝑀|=|||𝑀+32βˆ’32𝑖|||
  • D|𝑀|=|||𝑀+32+32𝑖|||
  • E|𝑀|=|π‘€βˆ’3βˆ’3𝑖|

Q8:

Find an equation for the image of the half line arg(𝑧+3𝑖)=3πœ‹4 under the transformation π‘‡βˆΆπ‘§β†¦ο€»βˆš3+𝑖(𝑧+2+4𝑖)βˆ’6π‘–οŠ©.

  • Aarg(π‘€βˆ’2+26𝑖)=πœ‹4
  • Bargο€Όπ‘€βˆ’18+132π‘–οˆ=πœ‹4
  • Carg(𝑀+8βˆ’10𝑖)=5πœ‹4
  • Dargο€Όπ‘€βˆ’18+132π‘–οˆ=5πœ‹4
  • Earg(π‘€βˆ’2+26𝑖)=5πœ‹4

Q9:

Find an equation for the image of |π‘§βˆ’3|=2 under the transformation of the complex plane π‘‡βˆΆπ‘§β†¦(𝑧)π‘§οŠ§, where 𝑧=3√2+3√2π‘–οŠ§.

  • A||π‘€βˆ’3βˆ’3√2βˆ’3√2𝑖||=2
  • B||π‘€βˆ’9√2βˆ’9√2𝑖||=13
  • C||π‘€βˆ’9√2βˆ’9√2𝑖||=12
  • D|||π‘€βˆ’βˆš24+√24𝑖|||=13
  • E|||π‘€βˆ’βˆš24+√24𝑖|||=12

Q10:

In the 𝑧-plane, a curve is given by the Cartesian equation 𝑦=π‘₯. The transformation π‘‡βˆΆπ‘§β†¦π‘§βˆ’2+4𝑖 represents a transformation from the 𝑧-plane to the 𝑀-plane. Find the Cartesian equation for the image of the curve in the 𝑀-plane.

  • A𝑣=(𝑒+2)βˆ’4
  • B𝑣=(π‘’βˆ’4)βˆ’4
  • C𝑣=(𝑒+2)+4
  • D𝑣=(𝑒+4)+2
  • E𝑣=(π‘’βˆ’2)+4

Q11:

Find an equation for the image of Re(𝑧)=βˆ’3 under the transformation of the complex plane π‘‡βˆΆπ‘§β†¦π‘§+9βˆ’2𝑖.

  • ARe(𝑀)=4
  • BRe(𝑀)=βˆ’12
  • CRe(𝑀)=12
  • DRe(𝑀)=6
  • ERe(𝑀)=βˆ’6

Q12:

A line is given by the equation |π‘§βˆ’1βˆ’3𝑖|=|π‘§βˆ’6βˆ’π‘–|. Find an equation for the image of the line under the transformation π‘‡βˆΆπ‘§β†¦βˆ’2π‘§βˆ’π‘–.

  • A|π‘€βˆ’2βˆ’5𝑖|=|π‘€βˆ’12βˆ’π‘–|
  • B|||π‘€βˆ’12βˆ’2𝑖|||=|||π‘€βˆ’72βˆ’12𝑖|||
  • C|||𝑀+12+2𝑖|||=|||𝑀+72+12𝑖|||
  • D|𝑀+2+5𝑖|=|𝑀+12+𝑖|
  • E|𝑀+2+7𝑖|=|𝑀+12+3𝑖|

Q13:

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their product with 5ο€»πœ‹2+π‘–πœ‹2cossin.

  • Aa dilation with center the origin and scale factor 5 combined with a rotation by an angle of πœ‹2 clockwise about the origin
  • Ba dilation with center the origin and scale factor 52 combined with a rotation by an angle of πœ‹2 counterclockwise about the origin
  • Ca dilation with center the origin and scale factor 5 combined with a rotation by an angle of πœ‹2 counterclockwise about the origin
  • Da dilation with center the origin and scale factor 15 combined with a rotation by an angle of πœ‹2 clockwise about the origin
  • Ea dilation with center the origin and scale factor 15 combined with a rotation by an angle of πœ‹2 counterclockwise about the origin

Q14:

The geometric transformation that occurs when numbers in the complex plane are mapped to their product with 𝑧=53βˆ’2𝑖 is a dilation with center the origin followed by a rotation about the origin by an angle less than 180∘. Is this rotation clockwise or counterclockwise?

  • Acounterclockwise
  • Bclockwise

Q15:

Find an equation for the image of arg(𝑧)=πœƒ under the transformation π‘‡βˆΆπ‘§β†¦π‘’π‘§οƒοŽŠ that maps the 𝑧-plane to 𝑀-plane.

  • Aarg(𝑀)=πœ‘βˆ’πœƒ
  • Barg(𝑀)=πœƒ+πœ‘
  • Carg(𝑀)=πœƒ+πœ‘2
  • Darg(𝑀)=βˆšπœƒ+πœ‘οŠ¨οŠ¨
  • Earg(𝑀)=πœƒβˆ’πœ‘

Q16:

Find an equation for the image of |𝑧|=2 under the transformation of the complex plane π‘‡βˆΆπ‘§β†¦π‘§+1+𝑖.

  • A|𝑀|=2+√2
  • B|𝑀+1+𝑖|=2
  • C|π‘€βˆ’1βˆ’π‘–|=2
  • D|π‘€βˆ’1+𝑖|=2
  • E|𝑀|=√2

Q17:

Find an equation for the image of |π‘§βˆ’2|=3 under the transformation of 𝑧-plane to the 𝑀-plane given by 𝑀=𝑖(2𝑧+2).

  • A|π‘€βˆ’2𝑖|=10
  • B|π‘€βˆ’4𝑖|=6
  • C|π‘€βˆ’6𝑖|=6
  • D|𝑀+2𝑖|=10
  • E|𝑀+6𝑖|=6

Q18:

Find an equation for the image of the half-line arg(π‘§βˆ’4+5𝑖)=πœ‹2 under the transformation π‘‡βˆΆπ‘§β†¦π‘–π‘§βˆ’3βˆ’π‘–.

  • Aarg(π‘€βˆ’2βˆ’3𝑖)=πœ‹
  • Barg(2π‘€βˆ’7βˆ’4𝑖)=πœ‹
  • Carg(𝑀+2+3𝑖)=πœ‹
  • Darg(𝑀+2+3𝑖)=πœ‹2
  • Earg(π‘€βˆ’1βˆ’6𝑖)=πœ‹2

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