Question

sketch △abc, △abc and △abc using the given compositions. then, reverse the order of the transformations and sketch △abc and △abc. determine if the order of the transformations affects the final image.

a (1,2) b (5,2) c (6,5)
translation: (x,y) → (x + 3, y)
reflection: in the x - axis

a (1,2) b (5,2) c (6,5)
reflection: in the x - axis
translation: (x,y) → (x + 3, y)

b.
a (-3,2) b (-1, 2) c (3,2)
translation: (x,y) → (x - 5, y + 2)
reflection: in the x - axis

a (-3,2) b (-1, 2) c (3,2)
reflection: in the x - axis
translation: (x,y) → (x - 5, y + 2)

Explanation:

---

7 (Top Left: Translation then Reflection)

Step1: Apply translation to ABC

Translation: $(x,y)\to(x+3,y)$

• $A(1,2)\to A'(1+3,2)=(4,2)$
• $B(5,2)\to B'(5+3,2)=(8,2)$
• $C(6,5)\to C'(6+3,5)=(9,5)$

Step2: Apply x-axis reflection to A'B'C'

Reflection: $(x,y)\to(x,-y)$

• $A'(4,2)\to A''(4,-2)$
• $B'(8,2)\to B''(8,-2)$
• $C'(9,5)\to C''(9,-5)$

---

7 (Top Right: Reflection then Translation)

Step1: Apply x-axis reflection to ABC

Reflection: $(x,y)\to(x,-y)$

• $A(1,2)\to A'(1,-2)$
• $B(5,2)\to B'(5,-2)$
• $C(6,5)\to C'(6,-5)$

Step2: Apply translation to A'B'C'

Translation: $(x,y)\to(x+3,y)$

• $A'(1,-2)\to A''(1+3,-2)=(4,-2)$
• $B'(5,-2)\to B''(5+3,-2)=(8,-2)$
• $C'(6,-5)\to C''(6+3,-5)=(9,-5)$

---

8 (Bottom Left: Translation then Reflection)

Step1: Apply translation to ABC

Translation: $(x,y)\to(x-5,y+2)$

• $A(-3,2)\to A'(-3-5,2+2)=(-8,4)$
• $B(-1,2)\to B'(-1-5,2+2)=(-6,4)$
• $C(3,2)\to C'(3-5,2+2)=(-2,4)$

Step2: Apply x-axis reflection to A'B'C'

Reflection: $(x,y)\to(x,-y)$

• $A'(-8,4)\to A''(-8,-4)$
• $B'(-6,4)\to B''(-6,-4)$
• $C'(-2,4)\to C''(-2,-4)$

---

8 (Bottom Right: Reflection then Translation)

Step1: Apply x-axis reflection to ABC

Reflection: $(x,y)\to(x,-y)$

• $A(-3,2)\to A'(-3,-2)$
• $B(-1,2)\to B'(-1,-2)$
• $C(3,2)\to C'(3,-2)$

Step2: Apply translation to A'B'C'

Translation: $(x,y)\to(x-5,y+2)$

• $A'(-3,-2)\to A''(-3-5,-2+2)=(-8,0)$
• $B'(-1,-2)\to B''(-1-5,-2+2)=(-6,0)$
• $C'(3,-2)\to C''(3-5,-2+2)=(-2,0)$

---

Answer:

7 (Top Left):

$A'(4,2)$, $B'(8,2)$, $C'(9,5)$
$A''(4,-2)$, $B''(8,-2)$, $C''(9,-5)$

7 (Top Right):

$A'(1,-2)$, $B'(5,-2)$, $C'(6,-5)$
$A''(4,-2)$, $B''(8,-2)$, $C''(9,-5)$

8 (Bottom Left):

$A'(-8,4)$, $B'(-6,4)$, $C'(-2,4)$
$A''(-8,-4)$, $B''(-6,-4)$, $C''(-2,-4)$

8 (Bottom Right):

$A'(-3,-2)$, $B'(-1,-2)$, $C'(3,-2)$
$A''(-8,0)$, $B''(-6,0)$, $C''(-2,0)$

Order of Transformation Conclusion:

For problem 7, the final image is identical regardless of order. For problem 8, the final images are different. The order of transformations can affect the final image, depending on the specific transformations used.