Question

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

	verbal description	algebraic representation
graph b		(x - 2,y - 2)
graph c		(x - 4,y + 10)

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

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

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).

Explanation:

Step1: Analyze Graph A

For Graph A, to find the verbal description and algebraic representation of the translation. Observe the movement of points. If we assume a general point \((x,y)\) on the pre - image, and compare it to the image points, we can see that it moves right and down. The algebraic representation for moving right \(h\) units and down \(k\) units is \((x + h,y - k)\). By counting the grid squares, we find it moves 2 units right and 3 units down, so the algebraic representation is \((x + 2,y-3)\), and the verbal description is "translation 2 units right and 3 units down".

Step2: Analyze Graph B

For Graph B, by comparing pre - image and image points, we see that it moves 2 units left and 0 units up or down in the vertical direction (no vertical movement). The algebraic representation for moving left \(h\) units (where moving left means subtracting from the \(x\) - coordinate) and no vertical movement is \((x - 2,y)\), and the verbal description is "translation 2 units left".

Step3: Analyze Graph C

For Graph C, by comparing pre - image and image points, we find that it moves 4 units left and 10 units up. The algebraic representation for moving left \(h\) units and up \(k\) units is \((x - h,y + k)\), so it is \((x - 4,y + 10)\), and the verbal description is "translation 4 units left and 10 units up".

Step4: Analyze Statement 5

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

Step5: Analyze Statement 6

For the translation from \(X(-3,9)\) to \(X'(6,5)\): \(x\) changes from \(-3\) to \(6\) (\(\Delta x=6-(-3)=9\)) and \(y\) changes from \(9\) to \(5\) (\(\Delta y = 5 - 9=-4\)). 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.

Step6: Analyze Statement 7

The translation rule is \((x + 9,y-4)\). For point \(Z(-1,5)\), \(Z'\) has \(x\) - coordinate \(-1+9 = 8\) and \(y\) - coordinate \(5-4 = 1\). So \(Z'\) is located at \((8,1)\), not \((-10,1)\). The statement " \(Z'\) will be located at \((-10,1)\)" is False.

Answer:

Graph	Verbal Description	Algebraic Representation
Graph B	Translation 2 units left	\((x - 2,y)\)
Graph C	Translation 4 units left and 10 units up	\((x - 4,y + 10)\)

1. True
2. False (The translation rule is \((x + 9,y-4)\))
3. False (\(Z'\) is at \((8,1)\))