Question

1. determine the coordinates of the image, plot the image and determine if it is an isometric transformation or not

pre - image
transformation
coordinates
plot the image
a) pre - image points
a (1,-4)
b (2,-1)
c (6,-4)
image points
a (_,_)
b (_,_)
c (_,_)
coordinate rule
(x,y)→(x - 7,y + 5)
isometry? yes or no
transformation type:
b) pre - image points
a (-1,-2)
b (0,1)
c (4,-2)
image points
a (_,_)
b (_,_)
c (_,_)
coordinate rule
(x,y)→(-2y,-x)
isometry? yes or no
transformation type:
c) pre - image points
a (-3,1)
b (-2,4)
c (2,1)
image points
a (_,_)
b (_,_)
c (_,_)
coordinate rule
(x,y)→(-y - 2,x + 3)
isometry? yes or no
transformation type(s):
d) pre - image points
a (-6,-4)
b (-3,2)
c (6,-4)
image points
a (_,_)
b (_,_)
c (_,_)
coordinate rule
(x,y)→(.5x,.5y)
isometry? yes or no
transformation type:
e) pre - image points
a (0,0)
b (1,3)
c (5,0)
image points
a (_,_)
b (_,_)
c (_,_)
coordinate rule
(x,y)→(-y,x)
isometry? yes or no
transformation type:
f) pre - image points
a (3,-2)
b (2,1)
c (-2,-2)
image points
a (_,_)
b (_,_)
c (_,_)
coordinate rule
(x,y)→(x + 2,y + 2)
isometry? yes or no
transformation type:

Explanation:

Step1: Calculate image points for part a

For point A(1, - 4) with rule $(x,y)\to(x - 7,y + 5)$, $A'=(1-7,-4 + 5)=(-6,1)$. For B(2,-1), $B'=(2-7,-1 + 5)=(-5,4)$. For C(6,-4), $C'=(6-7,-4 + 5)=(-1,1)$. An isometry preserves distances. Since it is a translation, it is an isometry. The transformation type is translation.

Step2: Calculate image points for part b

For point A(-1,-2) with rule $(x,y)\to(-2y,-x)$, $A'=(-2\times(-2),-(-1))=(4,1)$. For B(0,1), $B'=(-2\times1,-0)=(-2,0)$. For C(4,-2), $C'=(-2\times(-2),-4)=(4,-4)$. This is not an isometry as distances change. The transformation type is a non - isometric transformation (a type of linear transformation).

Step3: Calculate image points for part c

For point A(-3,1) with rule $(x,y)\to(-y - 2,x + 3)$, $A'=(-1-2,-3 + 3)=(-3,0)$. For B(-2,4), $B'=(-4-2,-2 + 3)=(-6,1)$. For C(2,1), $C'=(-1-2,2 + 3)=(-3,5)$. This is not an isometry. The transformation type is a non - isometric transformation.

Step4: Calculate image points for part d

For point A(-6,-4) with rule $(x,y)\to(0.5x,0.5y)$, $A'=(0.5\times(-6),0.5\times(-4))=(-3,-2)$. For B(-3,2), $B'=(0.5\times(-3),0.5\times2)=(-1.5,1)$. For C(6,-4), $C'=(0.5\times6,0.5\times(-4))=(3,-2)$. This is not an isometry as it is a dilation. The transformation type is dilation.

Step5: Calculate image points for part e

For point A(0,0) with rule $(x,y)\to(-y,x)$, $A'=(0,0)$. For B(1,3), $B'=(-3,1)$. For C(5,0), $C'=(0,5)$. This is an isometry (a rotation). The transformation type is rotation.

Step6: Calculate image points for part f

For point A(3,-2) with rule $(x,y)\to(x + 2,y + 2)$, $A'=(3+2,-2 + 2)=(5,0)$. For B(2,1), $B'=(2+2,1 + 2)=(4,3)$. For C(-2,-2), $C'=(-2+2,-2 + 2)=(0,0)$. This is an isometry as it is a translation. The transformation type is translation.

Answer:

a) $A'(-6,1),B'(-5,4),C'(-1,1)$; Isometry? Yes; Transformation Type: Translation
b) $A'(4,1),B'(-2,0),C'(4,-4)$; Isometry? No; Transformation Type: Non - isometric transformation
c) $A'(-3,0),B'(-6,1),C'(-3,5)$; Isometry? No; Transformation Type: Non - isometric transformation
d) $A'(-3,-2),B'(-1.5,1),C'(3,-2)$; Isometry? No; Transformation Type: Dilation
e) $A'(0,0),B'(-3,1),C'(0,5)$; Isometry? Yes; Transformation Type: Rotation
f) $A'(5,0),B'(4,3),C'(0,0)$; Isometry? Yes; Transformation Type: Translation