Question

1.) x(4,2) y(1,-2) z(3,0) reflect across x - axis
2.) x(3,-4) y(-5,6) z(-7,8) reflect across y - axis
3.) x(3,1) y(-7,4) z(-3,3) reflect across x = - 2
4.) x(2,1) y(5,-1) z(4,-3) reflect across y = 1
5.) x(1,2) y(0,4) z(5,9) reflect across y = - x
6.) x(6,3) y(-1,-2) z(-10,6) reflect across y = x
7.) p(-3,8) q(7,5) r(2,3) s(-6,-1) reflect across x - axis
8.) p(5,3) q(1,-2) r(1,7) s(-2,0) reflect across y - axis
9.) p(4,2) q(2,-4) r(2,-5) s(5,0) reflect across x = 2
write coordinates after the reflection.

Explanation:

Step1: Recall reflection rules

• Reflection across x - axis: $(x,y)\to(x, - y)$
• Reflection across y - axis: $(x,y)\to(-x,y)$
• Reflection across $y = x$: $(x,y)\to(y,x)$
• Reflection across $y=-x$: $(x,y)\to(-y,-x)$
• Reflection across $x = a$: $(x,y)\to(2a - x,y)$
• Reflection across $y = b$: $(x,y)\to(x,2b - y)$

1. Reflect across x - axis

For $X(4,2)\to X'(4,-2)$
For $Y(1,-2)\to Y'(1,2)$
For $Z(3,0)\to Z'(3,0)$

2. Reflect across y - axis

For $X(3,-4)\to X'(-3,-4)$
For $Y(-5,6)\to Y'(5,6)$
For $Z(-7,8)\to Z'(7,8)$

3. Reflect across $x=-2$

The formula for reflection across $x = a$ is $(x,y)\to(2a - x,y)$. Here $a=-2$.
For $X(3,1)$: $2\times(-2)-3=-4 - 3=-7$, so $X'(-7,1)$
For $Y(-7,4)$: $2\times(-2)-(-7)=-4 + 7 = 3$, so $Y'(3,4)$
For $Z(-3,3)$: $2\times(-2)-(-3)=-4 + 3=-1$, so $Z'(-1,3)$

4. Reflect across $y = 1$

The formula for reflection across $y = b$ is $(x,y)\to(x,2b - y)$. Here $b = 1$.
For $X(2,1)$: $X'(2,2\times1 - 1)=X'(2,1)$
For $Y(5,-1)$: $Y'(5,2\times1-(-1))=Y'(5,3)$
For $Z(4,-3)$: $Z'(4,2\times1-(-3))=Z'(4,5)$

5. Reflect across $y=-x$

For $X(1,2)\to X'(-2,-1)$
For $Y(0,4)\to Y'(-4,0)$
For $Z(5,9)\to Z'(-9,-5)$

6. Reflect across $y = x$

For $X(6,3)\to X'(3,6)$
For $Y(-1,-2)\to Y'(-2,-1)$
For $Z(-10,6)\to Z'(6,-10)$

7. Reflect across x - axis

For $P(-3,8)\to P'(-3,-8)$
For $Q(7,5)\to Q'(7,-5)$
For $R(2,3)\to R'(2,-3)$
For $S(-6,-1)\to S'(-6,1)$

8. Reflect across y - axis

For $P(5,3)\to P'(-5,3)$
For $Q(1,-2)\to Q'(-1,-2)$
For $R(1,7)\to R'(-1,7)$
For $S(-2,0)\to S'(2,0)$

9. Reflect across $x = 2$

The formula for reflection across $x = a$ is $(x,y)\to(2a - x,y)$. Here $a = 2$.
For $P(4,2)$: $2\times2-4=0$, so $P'(0,2)$
For $Q(2,-4)$: $2\times2 - 2=2$, so $Q'(2,-4)$
For $R(2,-5)$: $2\times2-2 = 2$, so $R'(2,-5)$
For $S(5,0)$: $2\times2-5=-1$, so $S'(-1,0)$

Answer:

1. $X'(4,-2),Y'(1,2),Z'(3,0)$
2. $X'(-3,-4),Y'(5,6),Z'(7,8)$
3. $X'(-7,1),Y'(3,4),Z'(-1,3)$
4. $X'(2,1),Y'(5,3),Z'(4,5)$
5. $X'(-2,-1),Y'(-4,0),Z'(-9,-5)$
6. $X'(3,6),Y'(-2,-1),Z'(6,-10)$
7. $P'(-3,-8),Q'(7,-5),R'(2,-3),S'(-6,1)$
8. $P'(-5,3),Q'(-1,-2),R'(-1,7),S'(2,0)$
9. $P'(0,2),Q'(2,-4),R'(2,-5),S'(-1,0)$