Question

Which of the following represents the translation of D(−5,4) along vector <6,−8> and its reflection across the y-axis?

Answer 1

D (−5, 4) → D ′(1, −4) → D ″(−1, −4)

Explanation:

Use the translation vector <6,−8> to determine the rule for translation of the coordinates: (x,y)→(x+6,y+(−8)).

Apply the rule to translate point D(−5,4).

D(−5,4)→(−5+6,4+(−8))→D'(1,−4).

To apply the reflection across y-axis use the rule for reflection: (x,y)→(−x,y).

Apply the reflection rule to point D'(1,−4).

D'(1,−4)→D''(−1,−4).

Therefore, D(−5,4)→D'(1,−4)→D''(−1,−4) represents the translation of D(−5,4) along vector <6,−8> and its reflection across the y-axis.

Answer 2

D (-5 , 4) → D' (1 , -4) → D" (-1 , -4) ⇒ 2nd answer

Explanation:

* Lets revise some transformation

- If the point (x , y) translated horizontally to the right by h units

∴ Its image is (x + h , y)

- If the point (x , y) translated horizontally to the left by h units

∴ Its image is (x - h , y)

- If the point (x , y) translated vertically up by k units

∴ Its image is (x , y + k)

- If the point (x , y) translated vertically down by k units

∴ Its image is(x , y - k)

- If point (x , y) reflected across the x-axis

∴ Its image is (x , -y)

- If point (x , y) reflected across the y-axis

∴ Its image is (-x , y)

* Now lets solve the problem

- The point D is (-5 , 4)

- The vector of the translation is <6 , -8>

∵ 6 is positive number

∴ Point D will translate horizontally 6 units to the right

∵ x-coordinate of D = -5

- Add the x-coordinate of D by 6 to find the x-coordinate of D'

∴ The x-coordinate of D' = -5 + 6 = 1

∴ The x-coordinate of D' = 1

∵ -8 is negative number

∴ Point D will translate vertically 8 units down

∵ y-coordinate of D = 4

- Add the y-coordinate of D by -8 to find the y-coordinate of D'

∴ The y-coordinate of D' = 4 + -8 = -4

∴ The y-coordinate of D' = -4

∴ The coordinates of D' are (1 , -4)

- If point (x , y) reflected across the y-axis then its image is (-x , y)

∵ D' is reflected across the y-axis

∵ D' = (1 , -4)

- Change the sign of its x-coordinate

∴ D" = (-1 , -4)

∴ The coordinates of D" are (-1 , -4)

* D (-5 , 4) → D' (1 , -4) → D" (-1 , -4)