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 3 2 𝑖 .

  • Aa translation by 2 3
  • Ba translation by 3 2
  • Ca translation by 3 2
  • Da translation by 2 3
  • Ea translation by 3 2

Q2:

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

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

Q3:

Find an equation for the image of | 𝑧 + 3 2 𝑖 | = 6 under the transformation of the complex plane 𝑇 𝑧 3 2 𝑧 .

  • A | | | 𝑤 + 9 2 3 𝑖 | | | = 9
  • B | | | 𝑤 + 9 2 3 𝑖 | | | = 6
  • C | | | 𝑤 + 2 4 3 𝑖 | | | = 6
  • D | 𝑤 + 3 2 𝑖 | = 9
  • E | | | 𝑤 + 2 4 3 𝑖 | | | = 4

Q4:

Find an equation for the image of | | 𝑧 + 3 + 3 𝑖 | | = 2 3 under the transformation of 𝑧 -plane to the 𝑤 -plane given by 𝑤 = 2 3 + 6 𝑖 𝑧 4 .

  • A | 𝑤 2 8 | = 1 2
  • B | | 𝑤 4 3 3 𝑖 | | = 2 4
  • C | 𝑤 2 0 | = 1 2
  • D | | 𝑤 1 6 + 1 2 3 𝑖 | | = 2 4
  • E | 𝑤 2 0 | = 2 4

Q5:

Given that I m ( 𝑧 ) = 4 , find an equation for R e ( 𝑤 ) under the transformation 𝑇 𝑧 ( 2 𝑖 ) 𝑧 that maps the 𝑧 -plane onto the 𝑤 -plane.

  • A R e ( 𝑤 ) = 2
  • B R e ( 𝑤 ) = 2
  • C R e ( 𝑤 ) = 8
  • D R e ( 𝑤 ) = 8
  • E R e ( 𝑤 ) = 1 6

Q6:

Consider the complex number 𝑧 = 2 + 2 3 𝑖 .

Write 𝑧 in exponential form.

  • A 4 𝑒
  • B 2 𝑒
  • C 2 𝑒
  • D 𝑒
  • E 4 𝑒

Find the value of ( 𝑧 ) ( 2 + 𝑖 ) .

  • A 8 ( 2 + 𝑖 )
  • B 6 4 ( 2 + 𝑖 )
  • C 8 ( 2 + 𝑖 )
  • D 8 ( 1 + 2 𝑖 )
  • E 6 4 ( 2 + 𝑖 )

Find an equation for the image of | 𝑧 3 𝑖 | = 5 under the transformation of the complex plane 𝑇 𝑧 ( 𝑧 ) 𝑧 .

  • A | 𝑤 + 3 6 𝑖 | = 6 0
  • B | 𝑤 + 1 9 2 𝑖 | = 3 2 0
  • C | 𝑤 1 9 2 𝑖 | = 3 2 0
  • D | 𝑤 3 6 𝑖 | = 6 0
  • E | 𝑤 + 3 6 𝑖 | = 6 0

Q7:

Find an equation for the image of the line | 𝑧 | = | 𝑧 + 3 𝑖 | under the transformation of the complex plane 𝑇 𝑧 ( 𝑖 1 ) 𝑧 .

  • A | 𝑤 | = | 𝑤 3 3 𝑖 |
  • B | 𝑤 | = | 𝑤 1 + 4 𝑖 |
  • C | 𝑤 | = | | | 𝑤 + 3 2 3 2 𝑖 | | |
  • D | 𝑤 | = | 𝑤 + 3 + 3 𝑖 |
  • E | 𝑤 | = | | | 𝑤 + 3 2 + 3 2 𝑖 | | |

Q8:

Find an equation for the image of the half line a r g ( 𝑧 + 3 𝑖 ) = 3 𝜋 4 under the transformation 𝑇 𝑧 3 + 𝑖 ( 𝑧 + 2 + 4 𝑖 ) 6 𝑖 .

  • A a r g ( 𝑤 2 + 2 6 𝑖 ) = 5 𝜋 4
  • B a r g ( 𝑤 2 + 2 6 𝑖 ) = 𝜋 4
  • C a r g 𝑤 1 8 + 1 3 2 𝑖 = 5 𝜋 4
  • D a r g ( 𝑤 + 8 1 0 𝑖 ) = 5 𝜋 4
  • E a r g 𝑤 1 8 + 1 3 2 𝑖 = 𝜋 4

Q9:

Find an equation for the image of | 𝑧 3 | = 2 under the transformation of the complex plane 𝑇 𝑧 ( 𝑧 ) 𝑧 , where 𝑧 = 3 2 + 3 2 𝑖 .

  • A | | | 𝑤 2 4 + 2 4 𝑖 | | | = 1 2
  • B | | | 𝑤 2 4 + 2 4 𝑖 | | | = 1 3
  • C | | 𝑤 9 2 9 2 𝑖 | | = 1 3
  • D | | 𝑤 9 2 9 2 𝑖 | | = 1 2
  • E | | 𝑤 3 3 2 3 2 𝑖 | | = 2

Q10:

In the 𝑧 -plane, a curve is given by the Cartesian equation 𝑦 = 𝑥 . The transformation 𝑇 𝑧 𝑧 2 + 4 𝑖 represents a transformation from the 𝑧 -plane to the 𝑤 -plane. Find the Cartesian equation for the image of the curve in the 𝑤 -plane.

  • A 𝑣 = ( 𝑢 2 ) + 4
  • B 𝑣 = ( 𝑢 + 4 ) + 2
  • C 𝑣 = ( 𝑢 4 ) 4
  • D 𝑣 = ( 𝑢 + 2 ) + 4
  • E 𝑣 = ( 𝑢 + 2 ) 4

Q11:

Find an equation for the image of R e ( 𝑧 ) = 3 under the transformation of the complex plane 𝑇 𝑧 𝑧 + 9 2 𝑖 .

  • A R e ( 𝑤 ) = 6
  • B R e ( 𝑤 ) = 1 2
  • C R e ( 𝑤 ) = 1 2
  • D R e ( 𝑤 ) = 6
  • E R e ( 𝑤 ) = 4

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