Transformations of the complex plane

Simple transformations

• w = z + a: translation by a.
• w = kz with k real and positive: enlargement, scale factor k, centre O.
• w = iz: rotation 90° anticlockwise about O; w = e^iα z rotates through α.
• w = az + b combines these: |a| is the enlargement factor and arg a is the rotation angle, then translate by b.

The general strategy

1. Make z the subject: z = g(w).

2. Substitute into the locus of z.

3. Simplify using |p/q| = |p|/|q|, or put w = u + iv and find a Cartesian equation in u and v.

Answer in terms of u and v (not x and y): the June 2025 examiners' report notes some students keeping x and y instead of u and v.

w = z²

If z = re^iθ then w = r² e^2iθ: the modulus is squared and the argument doubled.

So |z| = 2 maps to |w| = 4, and the half-line arg z = π/6 maps to arg w = π/3. The first quadrant (0 < arg z < π/2) maps to the upper half-plane (0 < arg w < π).

w = 1/z

If z = re^iθ then w = (1/r)e^-iθ. Circles |z| = r map to circles |w| = 1/r. Lines and circles not through O behave in more interesting ways: the next lesson deals with the general case w = (az + b)/(cz + d).

Worked example

Find the image of the circle |z - 1| = 2 under w = 2z - 3 + i.

1. Make z the subject: z = (w + 3 - i)/2.
2. Substitute: |(w + 3 - i)/2 - 1| = 2, so |(w + 1 - i)/2| = 2.
3. So |w + 1 - i| = 4.

Answer: A circle in the w-plane, centre -1 + i, radius 4

Check yourself

1. Describe the transformation w = iz + 2.
   • Rotation 90° anticlockwise about O, then translation by 2
   • Enlargement scale factor 2, then rotation
   • Reflection in the real axis, then translation by 2
   • Translation by 2, then rotation 90° clockwise
   Show answer

   Rotation 90° anticlockwise about O, then translation by 2. Multiplying by i adds π/2 to every argument and keeps moduli, which is a 90° anticlockwise rotation. Adding 2 translates.

2. Find the radius of the image of |z| = 3 under w = z².
   Show answer

   9. |w| = |z|² = 9, so the image is the circle |w| = 9.

3. What is the image of |z| = 2 under w = 1/z?
   • The circle |w| = 1/2
   • The circle |w| = 2
   • The circle |w| = 4
   • The line u = 1/2
   Show answer

   The circle |w| = 1/2. z = 1/w, so |1/w| = 2, |w| = 1/2.

4. Under w = 2z - 3 + i, the circle |z - 1| = 2 maps to a circle. Find its radius.
   Show answer

   4. z = (w + 3 - i)/2 gives |w + 1 - i| = 4. Radius 4 (the enlargement factor 2 doubles the radius).

5. Under w = z², find the real part of the image of z = 1 + 2i.
   Show answer

   -3. (1 + 2i)² = 1 + 4i + 4i² = -3 + 4i.

6. Under w = z², what is the image of the first quadrant, 0 < arg z < π/2?
   • The first quadrant again
   • The lower half-plane
   • The right half-plane
   • The upper half-plane, 0 < arg w < π
   Show answer

   The upper half-plane, 0 < arg w < π. Arguments double: 0 < arg w < π.

7. Under w = 1/z, what is the image of the line Re z = 1/2?
   • The line Re w = 2
   • The circle |w - 1| = 1
   • The circle |w| = 2
   • The circle |w - 1/2| = 1/2
   Show answer

   The circle |w - 1| = 1. z = 1/w = (u - iv)/(u² + v²), so Re z = u/(u² + v²) = 1/2, giving u² + v² = 2u, (u - 1)² + v² = 1.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit F2
• Pearson Edexcel IAL Further Pure Mathematics 2 scheme of work
• Pearson Edexcel IAL Further Pure Mathematics F2 (WFM02) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.