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′

Question

be in ?(1 point)
Responses

I
I

II
II

III
III

IV
IV
Question 2
Write the rule for the composition transformation (4 points)

Answer 1

To determine the new position of point P after the translations, we start with the original coordinates of point P, which are \((-4,5)\).

First Translation:
The rule for the first translation is \((x + 3, y - 6)\).

1. Calculate the new x-coordinate: \[ -4 + 3 = -1 \]
2. Calculate the new y-coordinate: \[ 5 - 6 = -1 \] So after the first translation, point P becomes: \[ P' = (-1, -1) \]

Second Translation:
The rule for the second translation is \((x - 4, y + 2)\).

1. Calculate the new x-coordinate: \[ -1 - 4 = -5 \]
2. Calculate the new y-coordinate: \[ -1 + 2 = 1 \] So after the second translation, point P' becomes: \[ P'' = (-5, 1) \]

Determining the Quadrant: The coordinates \((-5, 1)\) place point P'' in the second quadrant (II), where x is negative and y is positive.

Answer to Question 1:

• Point P' is in Quadrant II.

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Question 2: Write the rule for the composition transformation.

To find the rule for the composition, we need to combine both transformations systematically:

1. First translation: \((x + 3, y - 6)\)
2. Second translation: For the output of the first translation, we can plug in: \[ \text{New } x = x' - 4 \Rightarrow (x + 3) - 4 = x - 1 \] \[ \text{New } y = y' + 2 \Rightarrow (y - 6) + 2 = y - 4 \]

Putting it all together, the composition transformation rule from point \((x, y)\) is: \[ (x, y) \rightarrow (x - 1, y - 4) \]

Answer to Question 2:

• The rule for the composition transformation is \((x - 1, y - 4)\).