Question

graph and label the image of the figure using the transformation given.

1. reflection across $y = x$
2. reflection across the x-axis

find the coordinates of the vertices of each figure after the given transformation.

1. reflection across $x = 4$

$f(3, -5), c(3, -4), p(5, -4)$

1. reflection across $y = -x$

$x(-4, -3), m(-3, -2), i(-1, -5)$

1. reflection across the y-axis

$n(-3, 1), g(0, 4), b(-1, 1)$

1. reflection across the x-axis

$w(-4, 4), u(1, 5), k(0, 0)$

graph the image and the preimage of the figure using the transformation given.

1. reflection across $x = -1$

$z(0, 2), u(0, 5), b(3, 2)$

1. reflection across $y = x$

$c(-4, 2), v(-2, 5), t(-2, 1)$

Explanation:

Step1: Identify reflection rule for $y=x$

For a point $(x,y)$, reflection over $y=x$ is $(y,x)$.

Step2: Identify original vertices (1)

Assume vertices: $L(-4,1), K(-2,3), S(2,1), X(-2,-2)$

Step3: Apply rule to (1)

$L'(1,-4), K'(3,-2), S'(1,2), X'(-2,-2)$

Step4: Identify reflection rule for x-axis

For a point $(x,y)$, reflection over x-axis is $(x,-y)$.

Step5: Identify original vertices (2)

Assume vertices: $D(-2,-1), Z(-1,2), S(1,2)$

Step6: Apply rule to (2)

$D'(-2,1), Z'(-1,-2), S'(1,-2)$

Step7: Reflection over $x=4$ formula

For $(x,y)$, new $x=2*4-x=8-x$, $y$ stays.

Step8: Apply to (3) points

$F(3,-5)\to(8-3,-5)=(5,-5)$
$C(3,-4)\to(8-3,-4)=(5,-4)$
$P(5,-4)\to(8-5,-4)=(3,-4)$

Step9: Reflection over $y=-x$ formula

For $(x,y)$, new point is $(-y,-x)$.

Step10: Apply to (4) points

$X(-4,-3)\to(3,4)$
$M(-3,-2)\to(2,3)$
$I(-1,-5)\to(5,1)$

Step11: Reflection over y-axis formula

For $(x,y)$, new point is $(-x,y)$.

Step12: Apply to (5) points

$N(-3,1)\to(3,1)$
$G(0,4)\to(0,4)$
$B(-1,1)\to(1,1)$

Step13: Apply x-axis rule to (6)

$W(-4,4)\to(-4,-4)$
$U(1,5)\to(1,-5)$
$K(0,0)\to(0,0)$

Step14: Reflection over $x=-1$ formula

For $(x,y)$, new $x=2*(-1)-x=-2-x$, $y$ stays.

Step15: Apply to (7) points

$Z(0,2)\to(-2-0,2)=(-2,2)$
$U(0,5)\to(-2-0,5)=(-2,5)$
$B(3,2)\to(-2-3,2)=(-5,2)$

Step16: Apply $y=x$ rule to (8)

$C(-4,2)\to(2,-4)$
$V(-2,5)\to(5,-2)$
$T(-2,1)\to(1,-2)$

Answer:

1. Reflected vertices over $y=x$: $L'(1,-4), K'(3,-2), S'(1,2), X'(-2,-2)$ (graph by plotting these points)
2. Reflected vertices over x-axis: $D'(-2,1), Z'(-1,-2), S'(1,-2)$ (graph by plotting these points)
3. $F'(5,-5), C'(5,-4), P'(3,-4)$
4. $X'(3,4), M'(2,3), I'(5,1)$
5. $N'(3,1), G'(0,4), B'(1,1)$
6. $W'(-4,-4), U'(1,-5), K'(0,0)$
7. Reflected vertices over $x=-1$: $Z'(-2,2), U'(-2,5), B'(-5,2)$ (graph preimage and image)
8. Reflected vertices over $y=x$: $C'(2,-4), V'(5,-2), T'(1,-2)$ (graph preimage and image)