Question

translations on the coordinate plane
use the translations shown on graphs a - c to answer 1 - 4.

1. which graph(s) show the image in quadrant iv?
2. which graph(s) show a translation left and up?
3. which translation(s) preserved orientation of the figure and its vertices?
4. give a verbal description and algebraic representation of each translation in the table below.

	verbal description	algebraic representation
graph b
graph c

the table represents the coordinates of triangle xyz after a translation. mark each statement as true or false and correct any false statements.

1. the triangle was translated from quadrant ii to quadrant i.
2. the translation can be represented by (x - 4, y + 9).
3. z will be located at (-10, 1).

pre - image	image
y (-6, 10)	y (3, 6)
z (-1, 5)	?

Explanation:

Graph A

Verbal Description

Move \(7\) units right and \(4\) units down.

Algebraic Representation

For a point \((x,y)\) in the pre - image, the image point \((x',y')\) is given by the rule \((x + 7,y-4)\).

Graph B

Verbal Description

Move \(3\) units down.

Algebraic Representation

For a point \((x,y)\) in the pre - image, the image point \((x',y')\) is given by the rule \((x,y - 3)\).

Graph C

Verbal Description

Move \(3\) units up.

Algebraic Representation

For a point \((x,y)\) in the pre - image, the image point \((x',y')\) is given by the rule \((x,y+3)\).

Statement 5

Check Quadrant

The pre - image \(X(-3,9)\) and \(Y(-6,10)\) are in quadrant II (where \(x<0,y > 0\)). The image \(X'(6,5)\) and \(Y'(3,6)\) are in quadrant I (where \(x>0,y>0\)). So, the statement “The triangle was translated from quadrant II to quadrant I” is True.

Statement 6

Find Translation Rule

For \(X(-3,9)\) to \(X'(6,5)\):
\(x\) - coordinate: \(6-(-3)=9\), \(y\) - coordinate: \(5 - 9=-4\). The translation rule is \((x + 9,y-4)\), not \((x - 4,y + 9)\). So, the statement “The translation can be represented by \((x - 4,y + 9)\)” is False. The correct rule is \((x+9,y - 4)\).

Statement 7

Find \(Z'\)

Using the translation rule \((x + 9,y-4)\) for \(Z(-1,5)\):
\(x\) - coordinate of \(Z'\): \(-1+9 = 8\), \(y\) - coordinate of \(Z'\): \(5-4 = 1\). So, \(Z'(8,1)\), not \((-10,1)\). The statement “\(Z'\) will be located at \((-10,1)\)” is False. The correct location is \(Z'(8,1)\).

Answer:

• Graph A: Verbal - Move \(7\) units right and \(4\) units down; Algebraic - \((x + 7,y-4)\)
• Graph B: Verbal - Move \(3\) units down; Algebraic - \((x,y - 3)\)
• Graph C: Verbal - Move \(3\) units up; Algebraic - \((x,y+3)\)
• Statement 5: True
• Statement 6: False. The correct translation rule is \((x + 9,y-4)\)
• Statement 7: False. \(Z'\) is located at \((8,1)\)