Question

graph the image of the figure using the transformation given.

1. rotation $180^{\circ}$ about the origin
2. rotation $90^{\circ}$ counterclockwise about the origin
3. rotation $90^{\circ}$ clockwise about the origin
4. rotation $180^{\circ}$ about the origin
5. rotation $90^{\circ}$ clockwise about the origin

$u(1,-2), w(0,2), k(3,2), g(3,-3)$

1. rotation $180^{\circ}$ about the origin

$v(2,0), s(1,3), g(5,0)$

Explanation:

1) Rotation 180° about origin

Step1: Recall 180° rotation rule

A point $(x,y)$ becomes $(-x,-y)$.

Step2: Identify original points

$I(-3,1), H(-1,-2), Q(0,0)$

Step3: Calculate new points

$I(-3,1) \to (3,-1)$
$H(-1,-2) \to (1,2)$
$Q(0,0) \to (0,0)$

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2) Rotation 90° counterclockwise about origin

Step1: Recall 90° CCW rotation rule

A point $(x,y)$ becomes $(-y,x)$.

Step2: Identify original points

$S(2,3), D(4,5), L(5,1)$

Step3: Calculate new points

$S(2,3) \to (-3,2)$
$D(4,5) \to (-5,4)$
$L(5,1) \to (-1,5)$

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3) Rotation 90° clockwise about origin

Step1: Recall 90° CW rotation rule

A point $(x,y)$ becomes $(y,-x)$.

Step2: Identify original points

$B(-5,2), M(-5,-1), H(-1,-1), F(-1,3)$

Step3: Calculate new points

$B(-5,2) \to (2,5)$
$M(-5,-1) \to (-1,5)$
$H(-1,-1) \to (-1,1)$
$F(-1,3) \to (3,1)$

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4) Rotation 180° about origin

Step1: Recall 180° rotation rule

A point $(x,y)$ becomes $(-x,-y)$.

Step2: Identify original points

$U(-4,-2), H(-2,0), P(-1,-2)$

Step3: Calculate new points

$U(-4,-2) \to (4,2)$
$H(-2,0) \to (2,0)$
$P(-1,-2) \to (1,2)$

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5) Rotation 90° clockwise about origin

Step1: Recall 90° CW rotation rule

A point $(x,y)$ becomes $(y,-x)$.

Step2: Use given points

$U(1,-2), W(0,2), K(3,2), G(3,-3)$

Step3: Calculate new points

$U(1,-2) \to (-2,-1)$
$W(0,2) \to (2,0)$
$K(3,2) \to (2,-3)$
$G(3,-3) \to (-3,-3)$

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6) Rotation 180° about origin

Step1: Recall 180° rotation rule

A point $(x,y)$ becomes $(-x,-y)$.

Step2: Use given points

$V(2,0), S(1,3), G(5,0)$

Step3: Calculate new points

$V(2,0) \to (-2,0)$
$S(1,3) \to (-1,-3)$
$G(5,0) \to (-5,0)$

Answer:

1. Image points: $(3,-1), (1,2), (0,0)$
2. Image points: $(-3,2), (-5,4), (-1,5)$
3. Image points: $(2,5), (-1,5), (-1,1), (3,1)$
4. Image points: $(4,2), (2,0), (1,2)$
5. Image points: $(-2,-1), (2,0), (2,-3), (-3,-3)$
6. Image points: $(-2,0), (-1,-3), (-5,0)$

(To graph, plot each set of transformed points and connect them to match the original figure's shape)