Question

what is the image of point ( b(3,2) ) after these three transformations?
i. a translation 1 unit to the left and 2 units down;
ii. a reflection in the ( y )-axis; and
iii. a ( 90^{circ} ) counterclockwise rotation about the origin

Explanation:

Step1: Translation

For a translation \(1\) unit to the left (\(x\) - coordinate decreases by \(1\)) and \(2\) units down (\(y\) - coordinate decreases by \(2\)), if the original point is \(B(3,2)\), the new coordinates \((x_1,y_1)\) are given by:
\(x_1=3 - 1=2\), \(y_1=2-2 = 0\). So the point after translation is \((2,0)\).

Step2: Reflection in the \(y\) - axis

The rule for reflection in the \(y\) - axis is \((x,y)\to(-x,y)\). For the point \((2,0)\), after reflection in the \(y\) - axis, the new coordinates \((x_2,y_2)\) are \((- 2,0)\).

Step3: \(90^{\circ}\) counter - clockwise rotation about the origin

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). For the point \((-2,0)\), substituting \(x=-2\) and \(y = 0\) into the rotation formula:
\(x_3=0\), \(y_3=-2\).

Answer:

\((0,-2)\)