Question

figure bcde was transformed using the composition ( r_{x - axis} cdot t_{2,2} ). if point b on the pre - image lies at ( (3,4) ), what are the coordinates of ( b ) on the final image?
( (5,6) )
( (5, - 6) )
( (1,2) )
( (1, - 2) )

Explanation:

Step1: Translation

First, perform the translation \(T_{2,2}\). The rule for translation \(T_{a,b}\) is \((x,y)\to(x + a,y + b)\). For \(T_{2,2}\) and the point \((3,4)\), we have \(x=3,y = 4,a = 2,b=2\).

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The point after translation is \((5,6)\).

Step2: Reflection over the \(x\) - axis

The rule for reflection over the \(x\) - axis \(r_{x - axis}\) is \((x,y)\to(x,-y)\). For the point \((5,6)\) after translation, using the reflection rule with \(x = 5,y = 6\), we get \(x''=x=5\) and \(y''=-y=-6\).

Answer:

\((5,-6)\)