The transformations are shown in steps by colour-coded parabolas. The dotted black parabola is the original function.
a) blue: f(x)=6x² b) green: f(x)=6(x+3)² c) purple: f(x)=-6(2x+3)² d) red: f(x)=-6(2x+3)²-3 e) dashed blue: f(x)=-6(2(x-4)+3)²=-6(2x-5)² f) dashed red: f(x)=6(2(x+1)-5)²+5=6(2x-3)²+5.
One way to track the transformations is to follow the vertex which starts at the origin.
The original function had one zero at x=0. The transformations have a zero (non-complex) only when the vertex lies on the x axis.