Question

suppose we perform a translation, then a reflection, then a rotation, then another translation on the point a. a gets translated to ____, which gets reflected to __, which gets rotated to __, which finally gets translated again to ____. pre - image: a(6, -1), t(-3, 5), r_x - axis, r_270, t(1, 3). notation: which rigid motion would be performed first? r_x - axis ∘ r_270, t(-5, 1) ∘ r_y = x, r_y=-x ∘ r_y - axis, r_x - axis ∘ r_x - axis. the ____________ is performed first. for the last box, where would the point image of the point (-4, 2) be?

Explanation:

Step1: First translation

For a translation $T_{(-3,5)}$ on point $A(6, - 1)$, we add the translation values to the coordinates of the point. The $x$-coordinate of the new - point is $x = 6+( - 3)=3$, and the $y$-coordinate is $y=-1 + 5 = 4$. So the point after the first translation is $(3,4)$.

Step2: Reflection over the x - axis

The rule for reflecting a point $(x,y)$ over the $x$-axis is $(x,-y)$. For the point $(3,4)$, after reflection over the $x$-axis, it becomes $(3,-4)$.

Step3: Rotation of 270 degrees counter - clockwise

The rule for rotating a point $(x,y)$ 270 degrees counter - clockwise about the origin is $(y,-x)$. For the point $(3,-4)$, after rotation, it becomes $(-4,-3)$.

Step4: Second translation

For a translation $T_{(1,3)}$ on the point $(-4,-3)$, we add the translation values to the coordinates of the point. The $x$-coordinate of the new - point is $x=-4 + 1=-3$, and the $y$-coordinate is $y=-3 + 3 = 0$. So the final point is $(-3,0)$.

For the rigid - motion notation part, in the notation $a\circ b$, the transformation $b$ is performed first.
For $r_{x - axis}\circ R_{270}$, $R_{270}$ is performed first.
For $T_{(-5,1)}\circ r_{y = x}$, $r_{y = x}$ is performed first.
For $r_{y=-x}\circ r_{y - axis}$, $r_{y - axis}$ is performed first.
For $r_{x - axis}\circ r_{x - axis}$, $r_{x - axis}$ is performed first.

For the last question, when we have $r_{x - axis}\circ r_{x - axis}$ on the point $(-4,2)$:

Step1: First reflection over the x - axis

The rule for reflecting a point $(x,y)$ over the $x$-axis is $(x,-y)$. For the point $(-4,2)$, after the first reflection over the $x$-axis, it becomes $(-4,-2)$.

Step2: Second reflection over the x - axis

Applying the reflection rule again to the point $(-4,-2)$, we get $(-4,2)$.

Answer:

The point $A(6,-1)$:

• After first translation: $(3,4)$
• After reflection: $(3,-4)$
• After rotation: $(-4,-3)$
• After second translation: $(-3,0)$

For rigid - motion notation:

• In $r_{x - axis}\circ R_{270}$, $R_{270}$ is performed first.
• In $T_{(-5,1)}\circ r_{y = x}$, $r_{y = x}$ is performed first.
• In $r_{y=-x}\circ r_{y - axis}$, $r_{y - axis}$ is performed first.
• In $r_{x - axis}\circ r_{x - axis}$, $r_{x - axis}$ is performed first.

For the point $(-4,2)$ under $r_{x - axis}\circ r_{x - axis}$: $(-4,2)$