The triangle JKL has vertices J(-3,-3), K(-4,2), and L(2,1)
1) This triangle was first translated using the rule:
(x,y) → (x+2,y-3)
This means it was translated 2 units to the right and 3 units down.
JKL → J'K'L'
J(-3,-3) → J'(-3+2,-3-3) = (-1,-6)
K(-4,2) → K'(-4+2,2-3) = (-2,-1)
L(2,1) → L'(2+2,1-3) = (4,-2)
The resulting triangle J'K'L' has vertices J'(-1,-6), K'(-2,-1), and L'(4,-2)
2) The triangle J'K'L' was translated using the algebraic rule
(x,y) → (x-2,y-1)
This indicates it was translated 2 units to the left and one unit down.
The translation is
J'K'L' → J''K''L''
J'(-1,-6) → J''(-1-2,-6-1) = (-3,-7)
K'(-2,-1) → K''(-2-2,-1-1) = (-4,-2)
L'(4,-2) → L''(4-2,-2-1) = (2,-3)
The triangle J''K''L'' has vertices J''(-3,-7), K''(-4,-2), and L''(2,-3)
To determine one algebraic rule that represents the translation from JKL to J''K''L'' directly, you have to compare the coordinates of the vertices of both triangles:
JKL → J''K''L''
J(-3,-3) → J''(-3,-7)
K(-4,2) → K''(-4,-2)
L(2,1) → L''(2,-3)
You have to compare the x-coordinates and the y-coordinates of both triangles.
If you compare the x-coordinates you will notice that they are the same, this means that to make a direct translation from JKL to J''K''L'' you don't have to make any horizontal movements.
If you compare the y-coordinates you will see that they are different, so between both triangles, there was a vertical movement done. To determine the length of said movement, you have to calculate the difference between the y-coordinate of one vertex of J''K''L'' and the corresponding y-coordinate of the vertex of JKL.
For example compare J(-3,-3) and J''(-3,-7)
[tex]y_{J^{\doubleprime}}-y_J=-7-(-3)=-4[/tex]→ So we can conclude that to move JKL to J''K''L'' you have to make a vertical translation 4 units down. The algebraic rule that represents this translation is
(x,y) → (x,y-4)
