Question

in exercises 15 - 18, graph (overline{xy}) with endpoints (x(-3,1)) and (y(4,-5)) and its image after the composition. 15. translation: ((x,y)\to(x,y + 2)) rotation: (90^{circ}) counterclockwise about the origin 16. rotation: (180^{circ}) about the origin translation: ((x,y)\to(x - 1,y+1)) 17. rotation: (270^{circ}) counterclockwise about the origin reflection: in the (y) - axis 18. reflection: in the line (y = x) rotation: (90^{circ}) clockwise about the origin

Explanation:

Step1: Recall transformation rules

For a point $(x,y)$:

• A $180^{\circ}$ rotation about the origin changes $(x,y)$ to $(-x,-y)$.
• A translation $(x,y)\to(x - 1,y + 1)$ moves the point 1 unit left and 1 unit up.
• A $90^{\circ}$ counter - clockwise rotation about the origin changes $(x,y)$ to $(-y,x)$.
• A reflection in the $y$ - axis changes $(x,y)$ to $(-x,y)$.
• A $270^{\circ}$ counter - clockwise rotation about the origin is equivalent to a $90^{\circ}$ clockwise rotation, which changes $(x,y)$ to $(y,-x)$.
• A reflection in the line $y = x$ changes $(x,y)$ to $(y,x)$.

Let's consider point $X(-3,1)$ and $Y(4,-5)$ for each transformation.

Step2: Solve problem 15

Rotation and Translation for point $X(-3,1)$

First, $90^{\circ}$ counter - clockwise rotation about the origin: $(x,y)=(-3,1)\to(-1,-3)$.
Then translation: $(-1,-3)\to(-1 - 1,-3 + 1)=(-2,-2)$.

Rotation and Translation for point $Y(4,-5)$

First, $90^{\circ}$ counter - clockwise rotation about the origin: $(x,y)=(4,-5)\to(5,4)$.
Then translation: $(5,4)\to(5 - 1,4 + 1)=(4,5)$.

Step3: Solve problem 16

Rotation and Translation for point $X(-3,1)$

First, $180^{\circ}$ rotation about the origin: $(x,y)=(-3,1)\to(3,-1)$.
Then translation: $(3,-1)\to(3 - 1,-1 + 1)=(2,0)$.

Rotation and Translation for point $Y(4,-5)$

First, $180^{\circ}$ rotation about the origin: $(x,y)=(4,-5)\to(-4,5)$.
Then translation: $(-4,5)\to(-4 - 1,5 + 1)=(-5,6)$.

Step4: Solve problem 17

Rotation and Reflection for point $X(-3,1)$

First, $270^{\circ}$ counter - clockwise rotation about the origin: $(x,y)=(-3,1)\to(1,3)$.
Then reflection in the $y$ - axis: $(1,3)\to(-1,3)$.

Rotation and Reflection for point $Y(4,-5)$

First, $270^{\circ}$ counter - clockwise rotation about the origin: $(x,y)=(4,-5)\to(-5,4)$.
Then reflection in the $y$ - axis: $(-5,4)\to(5,4)$.

Step5: Solve problem 18

Rotation and Reflection for point $X(-3,1)$

First, reflection in the line $y = x$: $(x,y)=(-3,1)\to(1,-3)$.
Then $90^{\circ}$ clockwise rotation about the origin: $(1,-3)\to(-3,-1)$.

Rotation and Reflection for point $Y(4,-5)$

First, reflection in the line $y = x$: $(x,y)=(4,-5)\to(-5,4)$.
Then $90^{\circ}$ clockwise rotation about the origin: $(-5,4)\to(4,5)$.

Answer:

For problem 15: $X'(-2,-2)$ and $Y'(4,5)$
For problem 16: $X'(2,0)$ and $Y'(-5,6)$
For problem 17: $X'(-1,3)$ and $Y'(5,4)$
For problem 18: $X'(-3,-1)$ and $Y'(4,5)$