Question

A triangular pyramid begins with these coordinates:

P(–1, 2, 0)
Y(2, 3, 0)
R(–3, 4, 0)
A(–2, 1, 5)
The pyramid is then reflected over the yz-plane. The reflected image is then translated 2 units forward, 4 units right, and 3 units down. Determine the coordinates of that final image, following these steps:
Write the general rule you use for the reflection.
Show the results of your calculations for the reflection.
Write the general rule you use for the translation. (Hint: Begin with the result of the rule for the reflection.)
Show the results of your calculations for the translation

Answer 1

1. The yz-plane is the plane x = 0.
So, The general rule of reflection over the yz- plane is (x,y,z) →(−x,y,z)

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2. Show the results of your calculations for the reflection

Applying the rule found in (1)
the new points of the shape after this reflection will be

P(–1, 2, 0) ⇒⇒⇒ P'(1, 2, 0)
Y(2, 3, 0) ⇒⇒⇒ Y'(-2, 3, 0)
R(–3, 4, 0) ⇒⇒⇒ R'(3, 4, 0)
A(–2, 1, 5) ⇒⇒⇒ A'(2, 1, 5)
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3. image is translated 2 units forward, 4 units right, and 3 units down

So, the transformation rule (x, y, z) → (x + 2, y + 4, z – 3)
Apply the transformation rule to the points from number 2

P'(1, 2, 0) ⇒⇒⇒ P" (3 , 6 , -3 )
Y'(-2, 3, 0) ⇒⇒⇒ Y" (0 , 7 , -3)
R'(3, 4, 0) ⇒⇒⇒ R" ( 5 , 8 , -3)
A'(2, 1, 5) ⇒⇒⇒ A" ( 4 , 5 , 2)

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4. Write the general rule you use for the translation.
The general rule for the translation from the basic shape to the final image.
(x , y , z ) → (-x + 2, y + 4, z – 3)
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5. the results of general the translation:
P(–1, 2, 0) ⇒⇒⇒ P'(1, 2, 0) ⇒⇒⇒ P" (3 , 6 , -3 )
Y(2, 3, 0) ⇒⇒⇒ Y'(-2, 3, 0) ⇒⇒⇒ Y" (0 , 7 , -3)
R(–3, 4, 0) ⇒⇒⇒ R'(3, 4, 0) ⇒⇒⇒R" ( 5 , 8 , -3)
A(–2, 1, 5) ⇒⇒⇒ A'(2, 1, 5) ⇒⇒⇒ A" ( 4 , 5 , 2)