Question

what are the coordinates of point (2,3) after a translation to the left of 2 units and down 5 units, and then a dilation by a factor of 0.5 about (0,0)?
(-6,-3)
(0,-1)
(-2,-1)
(0,2)

Explanation:

Step1: Calculate the coordinates after translation

When translating a point \((x,y)\) to the left by \(a\) units and down by \(b\) units, the new coordinates \((x',y')\) are given by \(x'=x - a\) and \(y'=y - b\).
For the point \((2,3)\), \(a = 2\) and \(b = 5\).
So \(x'=2-2=0\) and \(y'=3 - 5=-2\).
The coordinates after translation are \((0,-2)\).

Step2: Calculate the coordinates after dilation

When dilating a point \((x,y)\) by a factor \(k\) about the origin \((0,0)\), the new coordinates \((x'',y'')\) are given by \(x''=k\times x\) and \(y''=k\times y\).
Here \(k = 0.5\), \(x = 0\), \(y=-2\).
So \(x''=0.5\times0 = 0\) and \(y''=0.5\times(-2)=-1\).

Answer:

\((0,-1)\)