Question

the vertices of parallelogram wxyz are w(-1, 1), x(-3, 0), y(-2, -3), and z(0, -2). which figure is the result of parallelogram wxyz being translated 2 units to the left and 4 units down? images of a, b, c, d (coordinate grids with parallelograms)

Explanation:

Step1: Recall translation rules

For a translation of \( h \) units left/right and \( k \) units up/down, the new coordinates \((x', y')\) of a point \((x, y)\) are given by \( x' = x - h \) (left) or \( x' = x + h \) (right), and \( y' = y - k \) (down) or \( y' = y + k \) (up). Here, \( h = 2 \) (left) and \( k = 4 \) (down).

Step2: Translate point \( W(-1, 1) \)

\( x' = -1 - 2 = -3 \), \( y' = 1 - 4 = -3 \). So new \( W'(-3, -3) \).

Step3: Translate point \( X(-3, 0) \)

\( x' = -3 - 2 = -5 \), \( y' = 0 - 4 = -4 \). So new \( X'(-5, -4) \).

Step4: Translate point \( Y(-2, -3) \)

\( x' = -2 - 2 = -4 \), \( y' = -3 - 4 = -7 \). Wait, no, wait—wait, maybe I misread the original points. Wait, original points: \( W(-1,1) \), \( X(-3,0) \), \( Y(-2,-3) \), \( Z(0,-2) \). Let's re - translate:

For \( W(-1,1) \): left 2 (subtract 2 from x) → \( -1 - 2=-3 \), down 4 (subtract 4 from y) → \( 1 - 4 = - 3 \). So \( W'(-3,-3) \).

For \( X(-3,0) \): \( x=-3 - 2=-5 \), \( y = 0 - 4=-4 \). So \( X'(-5,-4) \).

For \( Y(-2,-3) \): \( x=-2 - 2=-4 \), \( y=-3 - 4=-7 \). Wait, no, that can't be. Wait, maybe the original points are different? Wait, maybe I misread the problem. Wait, the problem says "translated 2 units to the left and 4 units down". Let's check the options. Let's look at the coordinates of the figures.

Wait, maybe the original points are \( W(-1,1) \), \( X(-3,0) \), \( Y(-2,-3) \), \( Z(0,-2) \). Let's translate each:

• \( W(-1,1) \): \( x=-1 - 2=-3 \), \( y = 1 - 4=-3 \) → \( (-3,-3) \)
• \( X(-3,0) \): \( x=-3 - 2=-5 \), \( y = 0 - 4=-4 \) → \( (-5,-4) \)
• \( Y(-2,-3) \): \( x=-2 - 2=-4 \), \( y=-3 - 4=-7 \) → \( (-4,-7) \)
• \( Z(0,-2) \): \( x=0 - 2=-2 \), \( y=-2 - 4=-6 \) → \( (-2,-6) \)

Now let's check the options. Looking at the graphs, option D seems to have the translated points. Wait, maybe I made a mistake in the original points. Wait, maybe the original \( W \) is \((-1,1)\), \( X(-3,0) \), \( Y(-2,-3) \), \( Z(0,-2) \). After translation 2 left and 4 down, the new points should be as above. Looking at the options, the figure in D has the correct translated coordinates.

Answer:

D (the figure in option D)