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A) Point P is at (-4,-5). The point was translated by the rule (x+3,y-6). Then the point was translated again by the rule (x-4,y+2). What quadrant will point P' be in?

(1 point)
• |I_
• IV
B) Write the rule for the composition transformation (4 points)
(x-[,y-|)

1 answer

To find the final position of point P after the two translations, let's go through the transformations step by step.

Step 1: Initial Point P

Point P is at (-4, -5).

Step 2: First Translation (x + 3, y - 6)

Using the first translation rule:

  • New x-coordinate: -4 + 3 = -1
  • New y-coordinate: -5 - 6 = -11

So after the first translation, point P' is at (-1, -11).

Step 3: Second Translation (x - 4, y + 2)

Using the second translation rule:

  • New x-coordinate: -1 - 4 = -5
  • New y-coordinate: -11 + 2 = -9

So after the second translation, point P'' is at (-5, -9).

Step 4: Determine the Quadrant

Since point P'' is at (-5, -9):

  • The x-coordinate is negative.
  • The y-coordinate is negative.

This places point P'' in Quadrant III.

Answer to A: III

B) Write the rule for the composition transformation

Now let's find the overall transformation rule from the original point P to P''.

  1. The first transformation can be expressed as: \[ (x, y) \rightarrow (x + 3, y - 6) \]

  2. The second transformation can be expressed as: \[ (x, y) \rightarrow (x - 4, y + 2) \]

Combining the transformations:

  • For the x-coordinate: \[ x' = (x + 3) - 4 = x - 1 \]

  • For the y-coordinate: \[ y' = (y - 6) + 2 = y - 4 \]

So the overall transformation can be expressed as: \[ (x, y) \rightarrow (x - 1, y - 4) \]

Answer to B: (x - 1, y - 4)