Mapping of a Circle in the Complex Plane

[amr21]

I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings:
a) w=e^(ipi)z
b) w = 2z
c) w = 2e^(ipi)z
d) w = z + 2 + 2i

I have managed to complete the question for a square and a rectangle as the points are easy to map as they are corners. What value for z do I use for the circle? I'm unsure how to begin.
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[I like Serena]

I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings:
a) w=e^(ipi)z
b) w = 2z
c) w = 2e^(ipi)z
d) w = z + 2 + 2i

I have managed to complete the question for a square and a rectangle as the points are easy to map as they are corners. What value for z do I use for the circle? I'm unsure how to begin.

Hi again amr21,

The set of points on the circle with centre (-4,0) and radius 1 is $\{z \in \mathbb C : |z+4|=1\}$.
For (a) we have $w=e^{i\pi}z = -z\ \Rightarrow\ z=-w$, giving us the set $\{w\in \mathbb C : |-w+4|=1\}$.
Which object would that be? (Wondering)

 

[amr21]

Hi again amr21,

The set of points on the circle with centre (-4,0) and radius 1 is $\{z \in \mathbb C : |z+4|=1\}$.
For (a) we have $w=e^{i\pi}z = -z\ \Rightarrow\ z=-w$, giving us the set $\{w\in \mathbb C : |-w+4|=1\}$.
Which object would that be? (Wondering)

Is it the same circle reflected over the y-axis, so the mid-point is (4,0)?

 

[I like Serena]

Is it the same circle reflected over the y-axis, so the mid-point is (4,0)?

Yep. (Nod)

 

[amr21]

Yep. (Nod)

Is the set of points for part (b) $\{w \in \mathbb C : |(w/2)+4|=1\}$ ? Could you explain how to plot the w/2 part?

part (c) I believe the set is $\{w \in \mathbb C : |(-w/2)+4|=1\}$. Once again could you explain the -w/2 part

part (d) do you flip the circle over the x-axis?

part (e) I'm unsure how to create a set of values from this mapping

 

[I like Serena]

Is the set of points for part (b) $\{w \in \mathbb C : |(w/2)+4|=1\}$ ? Could you explain how to plot the w/2 part?

Starting with this one.
Let's multiply both sides of the equation by 2:
$$|(w/2)+4|=1 \quad\Rightarrow\quad 2|w/2+4|=2\quad\Rightarrow\quad |w+8|=2$$
So we get:
$$\{w \in \mathbb C : |w+8|=2\}$$
Can we tell which object this is?

part (c) I believe the set is $\{w \in \mathbb C : |(-w/2)+4|=1\}$. Once again could you explain the -w/2 part

part (d) do you flip the circle over the x-axis?

part (e) I'm unsure how to create a set of values from this mapping

 

[amr21]

Starting with this one.
Let's multiply both sides of the equation by 2:
$$|(w/2)+4|=1 \quad\Rightarrow\quad 2|w/2+4|=2\quad\Rightarrow\quad |w+8|=2$$
So we get:
$$\{w \in \mathbb C : |w+8|=2\}$$
Can we tell which object this is?

Is it a circle with centre (-8,0) and a radius of 2?

 

[I like Serena]

Is it a circle with centre (-8,0) and a radius of 2?

Yep. I think you're getting the drift. :)