Question

a shape can be transformed more than once. such a transformation is called a composition of transformations. the transformations are written together with a small o between them.

• ( r_{x = 3}circ t_{1,5} ) means translate first and then reflect.
• ( t_{(2,-4)}circ r_{yaxis} ) means first and then.
• ( r_{y - axis}circ r_{360} ) means first and then.
• ( r_{90^{circ}}circ t_{3,2} ) means first and then.

now answer the following. show all your work.

1. what is the image of ( a(-3,5) ) under ( r_{90}circ t_{1,5} )?
2. what is the image of ( p(-3,-9) ) under ( t_{(2,-4)}circ r_{y = 1} )?
3. what is the image of ( s(3,6) ) under ( r_{180^{circ}}circ t_{(1,2)} )?
4. what is the image of ( q(-2,8) ) under ( r_{y - axis}circ r_{y = 1} )?
5. graph ( \triangle abc ) with ( a(3,5), b(5,3) ) and ( c(3,-1) ). graph the image of ( \triangle abc ) under ( r_{x - axis}circ t_{-5,-3} )
6. graph ( \triangle pqr ) with ( p(2,-1), q(2,1) ) and ( r(-2,0) ). graph the image of ( \triangle pqr ) under ( r_{y - axis}circ t_{-4,4} ).

Explanation:

Step1: Understand the composition of transformations

For \(R_{90}\circ T_{1,5}\), first perform the translation \(T_{1,5}\) on the point \(A(-3,5)\).
The rule for translation \(T_{a,b}\) is \((x,y)\to(x + a,y + b)\).
For \(T_{1,5}\), if \(x=-3\) and \(y = 5\), then after translation:
\(x=-3+1=-2\), \(y=5 + 5=10\). The point becomes \(A'(-2,10)\).

Step2: Perform the rotation

The rule for a \(90^{\circ}\) counter - clockwise rotation \(R_{90}\) is \((x,y)\to(-y,x)\).
For the point \(A'(-2,10)\), after rotation:
\(x=-10\), \(y=-2\).

Step1: Reflect first

For \(T_{(2,-4)}\circ r_{y = 1}\), first perform the reflection \(r_{y=1}\) on the point \(P(-3,-9)\).
The rule for reflection over the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k = 1\), so \((x,y)\to(x,2\times1-y)=(x,2 - y)\).
For \(P(-3,-9)\), after reflection: \(x=-3\), \(y=2-(-9)=11\). The point becomes \(P'(-3,11)\).

Step2: Translate

The rule for translation \(T_{a,b}\) is \((x,y)\to(x + a,y + b)\). For \(T_{(2,-4)}\), \(x=-3+2=-1\), \(y=11-4 = 7\).

Step1: Translate first

For \(R_{180}\circ T_{(1,2)}\), first perform the translation \(T_{(1,2)}\) on the point \(S(3,6)\).
Using the rule \((x,y)\to(x + a,y + b)\) with \(a = 1\) and \(b=2\), we get \(x=3 + 1=4\), \(y=6+2 = 8\). The point becomes \(S'(4,8)\).

Step2: Rotate

The rule for a \(180^{\circ}\) rotation \(R_{180}\) is \((x,y)\to(-x,-y)\).
For the point \(S'(4,8)\), after rotation: \(x=-4\), \(y=-8\).

Step1: Reflect over \(y = 1\)

For \(r_{y - axis}\circ r_{y=1}\), first perform the reflection \(r_{y=1}\) on the point \(Q(-2,8)\).
Using the rule \((x,y)\to(x,2k - y)\) with \(k = 1\), we have \(x=-2\), \(y=2\times1 - 8=-6\). The point becomes \(Q'(-2,-6)\).

Step2: Reflect over \(y - axis\)

The rule for reflection over the \(y - axis\) is \((x,y)\to(-x,y)\).
For the point \(Q'(-2,-6)\), after reflection: \(x = 2\), \(y=-6\).

Answer:

\((-10,-2)\)