Question

follow the guided instructions below to rotate the figure 270° counter - clockwise about the point (-3, -2). fill out all of the corresponding points of the image figure. animate rotation again pre - image image (-4,1) → () () (-1,2) → () () (-1,4) → () () (-4,3) → () () try values

Explanation:

Step1: Translate points

First, translate the pre - image points so that the center of rotation $(-3,-2)$ becomes the origin. Let $(x,y)$ be a pre - image point and $(x',y')$ be the translated point. Then $x'=x + 3$ and $y'=y + 2$.

Step2: Apply 270 - degree counter - clockwise rotation rule

The rule for a 270 - degree counter - clockwise rotation about the origin is $(x',y')\to(y',-x')$.

Step3: Translate back

After rotation, translate the points back to the original coordinate system. Let $(x'',y'')$ be the rotated and translated point. Then $x''=x'-3$ and $y''=y'-2$.

For the point $(-4,1)$:

• Translate: $x'=-4 + 3=-1$, $y'=1 + 2 = 3$.
• Rotate: $(x',y')=(-1,3)\to(3,1)$.
• Translate back: $x''=3-3 = 0$, $y''=1-2=-1$.

For the point $(-1,2)$:

• Translate: $x'=-1 + 3 = 2$, $y'=2+2 = 4$.
• Rotate: $(x',y')=(2,4)\to(4,-2)$.
• Translate back: $x''=4-3 = 1$, $y''=-2-2=-4$.

For the point $(-1,4)$:

• Translate: $x'=-1 + 3 = 2$, $y'=4 + 2=6$.
• Rotate: $(x',y')=(2,6)\to(6,-2)$.
• Translate back: $x''=6-3 = 3$, $y''=-2-2=-4$.

For the point $(-4,3)$:

• Translate: $x'=-4 + 3=-1$, $y'=3 + 2 = 5$.
• Rotate: $(x',y')=(-1,5)\to(5,1)$.
• Translate back: $x''=5-3 = 2$, $y''=1-2=-1$.

Answer:

$(-4,1)\to(0,-1)$
$(-1,2)\to(1,-4)$
$(-1,4)\to(3,-4)$
$(-4,3)\to(2,-1)$