Question

which transformation or set of transformations will map a point (x, y) onto the point (2x + 1, 2y)?
a a dilation of scale factor 2 through the origin followed by a translation of right 1 unit
b a dilation of scale factor 2 through the origin followed by a translation of up 1 unit
c a dilation of scale factor 2 through the point (1, 0)
d a dilation of scale factor 2 through the point (0, 1)

Explanation:

Step1: Analyze dilation effect

A dilation of scale - factor 2 through the origin on a point $(x,y)$ gives $(2x,2y)$. The rule for dilation of scale - factor $k$ through the origin on a point $(x,y)$ is $(kx,ky)$. Here $k = 2$, so $(x,y)\to(2x,2y)$.

Step2: Analyze translation effect

To get from $(2x,2y)$ to $(2x + 1,2y)$, we use the translation rule. For a translation in the $x$ - direction by $h$ units and $y$ - direction by $v$ units, the rule is $(x,y)\to(x + h,y + v)$. Here, $h=1$ and $v = 0$, which means a translation of 1 unit to the right.

Answer:

A. a dilation of scale factor 2 through the origin followed by a translation of right 1 unit