# 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 .

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Q4:

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

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Q5:

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

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Q6:

Consider the complex number .

Write in exponential form.

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Find the value of .

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

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Q7:

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

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Q8:

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

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Q9:

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

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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.

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Q11:

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

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Q12:

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

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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.

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Q16:

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

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Q17:

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

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• E

Q18:

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

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• E