Question

1. which of the following transformations always preserves the orientation of a figure?

a. translations
b. rotations
c. reflections
d. translations and reflections

1. which of the following is the algebraic representation for a translation 7 units left and 6 units up?

a. (x + 7,y - 6)
b. (x - 7,y + 6)
c. (x + 6,y - 7)
d. (x - 6,y + 7)
use the graph to answer questions 9 - 11.

1. triangle ijk has the coordinates listed. find j after a reflection over the y - axis.

i(5,-8) j(10,-8) k(7,-4)

1. the coordinates for triangle lmn are given below. if joel plans to transform lmn using the rule (x,y)→(-y,x), which is a true statement?

l(-1,9) m(-8,8) n(-3,5)
a. joel will reflect lmn over the x - axis.
b. l will be located at (9,1).
c. joel will rotate lmn 270° clockwise.
d. m will be located at (8,-8).

1. reflect figure opqr over the x - axis. record the coordinates for p.
2. translate figure stuv 7 units down and 5 units left. record the coordinates for t.
3. in which quadrant would figure stuv lie after a rotation 270° counterclockwise?
4. which is the algebraic representation for a rotation 180° clockwise?

(y,x)
(-y,x)
(y,-x)
(x,-y)

1. which transformation will always produce the same image as a rotation 90° counterclockwise?

a. a reflection over the y - axis.
b. a rotation 270° clockwise.
c. a rotation 90° clockwise.
d. a reflection over the x - axis.

1. point w(-6,7) is rotated 90° clockwise. where is w?

point j(-8,-12) is reflected over the x - axis. where is j?

Explanation:

Step1: Recall transformation - orientation rules

Translations slide a figure without changing its orientation. Rotations and reflections change the orientation. So the transformation that always preserves the orientation of a figure is translation.

Step2: Recall translation rule

For a translation \(a\) units left and \(b\) units up, the rule is \((x,y)\to(x - a,y + b)\). For 7 units left and 6 units up, \(a = 7\) and \(b=6\), so the rule is \((x,y)\to(x - 7,y + 6)\).

Step3: Recall rotation rule \((x,y)\to(-y,x)\)

The rule \((x,y)\to(-y,x)\) represents a 90 - degree counter - clockwise rotation. For point \(L(-1,9)\), applying the rule: \(L'=(-9,-1)\). For point \(M(-8,8)\), \(M'=(-8,-8)\). For point \(N(-3,5)\), \(N'=(-5,-3)\).

Step4: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) over the \(x\) - axis is \((x,y)\to(x,-y)\).

Step5: Recall translation rule for STUV

For a translation 7 units down and 5 units left, the rule is \((x,y)\to(x - 5,y - 7)\).

Step6: Recall rotation rules for quadrants

A 270 - degree counter - clockwise rotation is the same as a 90 - degree clockwise rotation. Analyze the original quadrant of STUV and the new quadrant after rotation.

Step7: Recall rotation and reflection equivalence

A 90 - degree counter - clockwise rotation and a 270 - degree clockwise rotation produce the same result.

Step8: Recall 90 - degree clockwise rotation rule

The rule for a 90 - degree clockwise rotation of a point \((x,y)\) is \((x,y)\to(y,-x)\). For point \(W(-6,7)\), \(W'=(7,6)\).

Answer:

1. A. Translations
2. B. \((x - 7,y + 6)\)
3. C. Joel will rotate LMN 270° clockwise.
4. (Coordinates of \(P\) are needed from the graph to calculate \(P'\) using \((x,y)\to(x,-y)\) for reflection over \(x\) - axis)
5. (Coordinates of \(T\) are needed from the graph to calculate \(T'\) using \((x,y)\to(x - 5,y - 7)\))
6. (Analysis based on original quadrant of STUV and 270° counter - clockwise rotation rule)
7. (Not enough information provided in the solution steps above to answer this one completely as no coordinates or general rules are applied yet in a full - fledged way)
8. B. A rotation 270° clockwise.
9. \((7,6)\)