Let us assume that $\mathrm{z}$ lies on a circle with centre ${\mathrm{z}}_0$ (fixed point) and radius $\frac{1}{2}$ units.

$\Rightarrow \; \left|\text{z}-{\text{z}}_0\right|=\frac{1}{2}$

Now, $\omega =-1+4\text{z}\Rightarrow \omega +1=4\text{z}$

$\Rightarrow \; \omega +1-4{\text{z}}_0=4\text{z}-4{\text{z}}_0$

Now, taking modulus on both sides, we get,

$\left|\omega +1-4{\text{z}}_0\right|=4\left|\text{z}-{\text{z}}_0\right|\Rightarrow \left|\omega +1-4{\text{z}}_0\right|=2$

Locus of $\omega$ represents the circle having centre $\left(-1+4{\text{z}}_0\right)$ and radius $2$ units.