Question

9.5 compositions of transformations
homework
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for each point, give the coordinates of its image under the translation.

1. given points ( a(1,2), b(5,4) ), and ( c(3,0) ), graph the image of ( \triangle abc )

under the composition of transformations in the order listed.
translation: ( (x,y)\to(x - 4,y) )
reflection: in the ( x )-axis
( a(_), b(_), c(___) )
( a(_), b(_), c(___) )

1. given points ( a(1,2), b(5,4) ), and ( c(3,0) ), graph the image of ( \triangle abc )

under the composition of transformations in the order listed.
translation: ( (0,-3) )
reflection: in the ( y )-axis
( a(_), b(_), c(___) )
( a(_), b(_), c(___) )

1. given points ( a(-3,-5), b(0,-5) ), and ( c(1,-2) ), graph the image of

( \triangle abc ) under the composition of transformations in the order listed.
translation: ( (4,5) )
reflection: in the line ( y = x )
( a(_), b(_), c(___) )
( a(_), b(_), c(___) )

1. given points ( a(-1,-3), b(-4,-2) ), and ( c(0,2) ), graph the image of

( \triangle abc ) under the composition of transformations in the order listed.
reflection: in the line ( x = 1 )
translation: ( (x,y)\to(x - 2,y + 3) )
( a(_), b(_), c(___) )
( a(_), b(_), c(___) )

Explanation:

Problem 1

Step - by - Step Format

Step1: Apply the translation \((x,y)\to(x - 4,y)\)

For point \(A(1,2)\): \(x=1,y = 2\), after translation \(x'=1-4=-3,y'=2\), so \(A'(-3,2)\)
For point \(B(5,4)\): \(x = 5,y = 4\), after translation \(x'=5 - 4=1,y'=4\), so \(B'(1,4)\)
For point \(C(3,0)\): \(x = 3,y = 0\), after translation \(x'=3-4=-1,y'=0\), so \(C'(-1,0)\)

Step2: Apply the reflection in the \(x\) - axis \((x,y)\to(x,-y)\)

For point \(A'(-3,2)\): after reflection \(A''(-3,- 2)\)
For point \(B'(1,4)\): after reflection \(B''(1,-4)\)
For point \(C'(-1,0)\): after reflection \(C''(-1,0)\)

Step1: Apply the translation \((x,y)\to(x,y - 3)\)

For point \(A(1,2)\): \(x = 1,y=2\), after translation \(x'=1,y'=2 - 3=-1\), so \(A'(1,-1)\)
For point \(B(5,4)\): \(x = 5,y = 4\), after translation \(x'=5,y'=4 - 3 = 1\), so \(B'(5,1)\)
For point \(C(3,0)\): \(x = 3,y = 0\), after translation \(x'=3,y'=0 - 3=-3\), so \(C'(3,-3)\)

Step2: Apply the reflection in the \(y\) - axis \((x,y)\to(-x,y)\)

For point \(A'(1,-1)\): after reflection \(A''(-1,-1)\)
For point \(B'(5,1)\): after reflection \(B''(-5,1)\)
For point \(C'(3,-3)\): after reflection \(C''(-3,-3)\)

Step1: Apply the translation \((x,y)\to(x + 4,y + 5)\)

For point \(A(-3,-5)\): \(x=-3,y=-5\), after translation \(x'=-3 + 4=1,y'=-5 + 5=0\), so \(A'(1,0)\)
For point \(B(0,-5)\): \(x = 0,y=-5\), after translation \(x'=0+4 = 4,y'=-5 + 5=0\), so \(B'(4,0)\)
For point \(C(1,-2)\): \(x = 1,y=-2\), after translation \(x'=1+4=5,y'=-2 + 5=3\), so \(C'(5,3)\)

Step2: Apply the reflection in the line \(y = x\) \((x,y)\to(y,x)\)

For point \(A'(1,0)\): after reflection \(A''(0,1)\)
For point \(B'(4,0)\): after reflection \(B''(0,4)\)
For point \(C'(5,3)\): after reflection \(C''(3,5)\)

Answer:

\(A'(-3,2),B'(1,4),C'(-1,0)\)
\(A''(-3,-2),B''(1,-4),C''(-1,0)\)

Problem 2

Step - by - Step Format