Successive transformations of a parabola in order

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Consider the parabola $y = x^2 - 4x + 1$. Apply, in the given order, the following transformations (each to the curve obtained at the previous step):

• (a) translation by $\vec v = (-1; 3)$;
• (b) dilation by factor $2$ along the $y$-axis;
• (c) rotation by $90^\circ$ counterclockwise about the origin.

Solution

Base: $y=(x-2)^2-3$.

(a) With $x\to x+1,\ y\to y-3$: $y=(x-1)^2$.

(b) Dilation along $y$ ($y\to y/2$): $y=2(x-1)^2$.

(c) Rotation $90^\circ$ counterclockwise $(x,y)\to(-y,x)$, i.e. $x\to y,\ y\to -x$: $x=-2(y-1)^2$. A parabola with horizontal axis, vertex $(0;1)$, opening left.

$\boxed{\text{(a) } y=(x-1)^2;\ \text{(b) } y=2(x-1)^2;\ \text{(c) } x=-2(y-1)^2}$