Question

write the coordinates of the vertices after a translation 8 units left and 10 units down.

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of each vertex. From the graph:

• \( P \): Let's assume the grid. Looking at the x and y. Let's say \( P \) is at \( ( -1, 6) \)? Wait, no, wait. Wait, the grid: each square is 1 unit. Let's check the positions. Let's see, \( P \): x-coordinate: let's count the grid. From the y-axis (x=0), moving left: so if \( P \) is 1 unit left of y-axis, x=-1? Wait, no, maybe I misread. Wait, looking at the graph, \( P \) is at ( -1, 6)? Wait, no, maybe: let's see, the x-axis: each grid line is 1 unit. Let's check \( P \): x is -1 (since it's 1 unit left of y-axis), y is 6. \( Q \): x is 5 (since 5 units right of y-axis), y is 6. \( R \): x is 1 (1 unit right of y-axis), y is 3? Wait, no, the graph shows \( R \) at (1, 3)? Wait, no, the point \( R \) is at (1, 3)? Wait, no, looking at the graph, \( R \) is at (1, 3)? Wait, no, the user's graph: \( P \) is at (-1, 6), \( Q \) is at (5, 6), \( R \) is at (1, 3). Wait, maybe I should re-express. Wait, let's do it properly.

Wait, the translation is 8 units left (so subtract 8 from x-coordinate) and 10 units down (subtract 10 from y-coordinate).

First, find original coordinates:

• \( P \): Let's look at the graph. The x-coordinate: from the y-axis (x=0), moving left: how many units? Let's see, the point \( P \) is at x = -1? Wait, no, maybe the grid: each square is 1 unit. Let's count the x for \( P \): if the y-axis is x=0, then \( P \) is at x = -1 (1 unit left), y = 6. \( Q \) is at x = 5 (5 units right), y = 6. \( R \) is at x = 1 (1 unit right), y = 3. Wait, but maybe I made a mistake. Wait, maybe \( P \) is at ( -1, 6), \( Q \) at (5, 6), \( R \) at (1, 3). Let's confirm.

So original coordinates:

• \( P \): \( (x_1, y_1) = (-1, 6) \)
• \( Q \): \( (x_2, y_2) = (5, 6) \)
• \( R \): \( (x_3, y_3) = (1, 3) \)

Now, translation: 8 units left (so \( x' = x - 8 \)) and 10 units down ( \( y' = y - 10 \) )

Step2: Translate \( P \)

For \( P(-1, 6) \):
New x: \( -1 - 8 = -9 \)
New y: \( 6 - 10 = -4 \)
So \( P' = (-9, -4) \)

Step3: Translate \( Q \)

For \( Q(5, 6) \):
New x: \( 5 - 8 = -3 \)
New y: \( 6 - 10 = -4 \)
So \( Q' = (-3, -4) \)

Step4: Translate \( R \)

For \( R(1, 3) \):
New x: \( 1 - 8 = -7 \)
New y: \( 3 - 10 = -7 \)
So \( R' = (-7, -7) \)

Wait, but maybe I made a mistake in original coordinates. Let's recheck the original coordinates. Wait, maybe \( P \) is at ( -1, 6), \( Q \) at (5, 6), \( R \) at (1, 3). Let's confirm with the graph. The line \( PQ \) is horizontal at y=6, from x=-1 to x=5. Then \( R \) is at (1, 3). So that's correct.

So applying translation: 8 left (x -8), 10 down (y -10).

So:

• \( P(-1, 6) \): \( x' = -1 -8 = -9 \), \( y' = 6 -10 = -4 \) → \( (-9, -4) \)
• \( Q(5, 6) \): \( x' = 5 -8 = -3 \), \( y' = 6 -10 = -4 \) → \( (-3, -4) \)
• \( R(1, 3) \): \( x' = 1 -8 = -7 \), \( y' = 3 -10 = -7 \) → \( (-7, -7) \)

Wait, but maybe the original coordinates are different. Wait, maybe \( P \) is at ( -1, 6), \( Q \) at (5, 6), \( R \) at (1, 3). Let's check the translation again.

Yes, translation 8 left: subtract 8 from x, 10 down: subtract 10 from y.

So that's the process.

Answer:

• \( P' \): \(\boldsymbol{(-9, -4)}\)
• \( Q' \): \(\boldsymbol{(-3, -4)}\)
• \( R' \): \(\boldsymbol{(-7, -7)}\)