[Pic] Describe a combination of transformations.

[Pic] Describe a combination of transformations.

Question
Vectors and spaces
asked 2020-10-28
[Pic] Describe a combination of transformations.

Answers (1)

2020-10-29
From the figure, rectangle DEFG has vertices of D(6,3), E(5,3), F(5,−3), and G(6,−3) and rectangle D''E''F''G' has vertices of D''(−12,6),E''(−10,6), F''(−10,−6),and G''(−12,−6). Since rectangle DEFGDEFG is on the right side of the yy-axis and rectangle D''E''F''G''′ is on the left side of the yy-axis, the first transformation is a reflection across the yy-axis. To reflect a point across the y-axis, use the rule (x,y)\rightarrow (−x,y):
(x,y)\rightarrow (−x,y)
D(6,3)\rightarrow D′(−6,3)
E(5,3)\rightarrow E′(−5,3)
F(5,−3)\rightarrow F′(−5,−3)
G(6,−3)\rightarrow F′(−6,−3)
Notice that the coordinates of the vertices of rectangle D''E''F''G''′ are twice the coordinates of the vertices of rectangle D′E′F′G. The second transformation is then a dilation by a scale factor of 2 centered at the origin. To dilate a point by a scale factor of k centered at the origin, use the rule \(\displaystyle{\left({x},{y}\right)}\rightarrow{\left({k}{x},{k}{y}\right)}:\)
\(\displaystyle{\left({x},{y}\right)}\rightarrow{\left({2}{x},{2}{y}\right)}\)
\(\displaystyle{D}′{\left(−{6},{3}\right)}\rightarrow{D}′′{\left(−{6}⋅{2},{3}⋅{2}\right)}={D}′′{\left(−{12},{6}\right)}\)
\(\displaystyle{E}′{\left(−{5},{3}\right)}\rightarrow{E}′′{\left(−{5}⋅{2},{3}⋅{2}\right)}={E}′′{\left(−{10},{6}\right)}\)
\(\displaystyle{F}′{\left(−{5},−{3}\right)}\rightarrow{F}′′{\left(−{5}⋅{2},−{3}⋅{2}\right)}={F}′′{\left(−{10},−{6}\right)}\)
\(\displaystyle{G}′{\left(−{6},−{3}\right)}\rightarrow{G}′′{\left(−{6}⋅{2},−{3}⋅{2}\right)}={G}′′{\left(−{12},−{6}\right)}\) The combination of transformations used to obtain rectangle D''E''F''G''′ was then a: reflection across the y−axis and a dilation by a scale factor of 2 centered at the origin
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