which rule describes a composition of transformations that maps pre-image pqrs to image p\q
\s\?

$r_{o, 270^{\circ}} \circ t_{-2, 0}(x, y)$
$t_{-2, 0} \circ r_{o, 270^{\circ}}(x, y)$
$r_{o, 270^{\circ}} \circ r_{y-\text{axis}}(x, y)$
$r_{y-\text{axis}} \circ r_{o, 270^{\circ}}(x, y)$

Question

which rule describes a composition of transformations that maps pre-image pqrs to image p\q
\s\?

$r_{o, 270^{\circ}} \circ t_{-2, 0}(x, y)$
$t_{-2, 0} \circ r_{o, 270^{\circ}}(x, y)$
$r_{o, 270^{\circ}} \circ r_{y-\text{axis}}(x, y)$
$r_{y-\text{axis}} \circ r_{o, 270^{\circ}}(x, y)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Coordinate Transformations", "Rotational Transformations", "Composition of Transformations" ], "new_concepts": [], "current_concepts": [ "Coordinate Transformations", "Rotational Transformations", "Composition of Transformations" ] } ## Identify coordinates of pre-image and image vertices $$ \begin{aligned} &P(1, 1), \quad Q(1, 5), \quad R(3, 5), \quad S(3, 1) \ &P''(-1, -1), \quad Q''(-5, -1), \quad R''(-5, -3), \quad S''(-1, -3) \end{aligned} $$ ## Test the composition options on a reference point $$ \begin{aligned} &\text{Option 1: } R_{O, 270^\circ} \circ T_{-2, 0}(x, y) \ &T_{-2, 0}(1, 1) = (1 - 2, 1) = (-1, 1) \ &R_{O, 270^\circ}(-1, 1) = (1, 1) eq P''(-1, -1) \\ &\text{Option 2: } T_{-2, 0} \circ R_{O, 270^\circ}(x, y) \ &R_{O, 270^\circ}(1, 1) = (1, -1) \ &T_{-2, 0}(1, -1) = (1 - 2, -1) = (-1, -1) = P'' \\ &R_{O, 270^\circ}(1, 5) = (5, -1) \ &T_{-2, 0}(5, -1) = (5 - 2, -1) = (3, -1) eq Q''(-5, -1) \\ &\text{Option 3: } R_{O, 270^\circ} \circ r_{y\text{-axis}}(x, y) \ &r_{y\text{-axis}}(1, 1) = (-1, 1) \ &R_{O, 270^\circ}(-1, 1) = (1, 1) eq P''(-1, -1) \\ &\text{Option 4: } r_{y\text{-axis}} \circ R_{O, 270^\circ}(x, y) \ &R_{O, 270^\circ}(1, 1) = (1, -1) \implies r_{y\text{-axis}}(1, -1) = (-1, -1) = P'' \ &R_{O, 270^\circ}(1, 5) = (5, -1) \implies r_{y\text{-axis}}(5, -1) = (-5, -1) = Q'' \ &R_{O, 270^\circ}(3, 5) = (5, -3) \implies r_{y\text{-axis}}(5, -3) = (-5, -3) = R'' \ &R_{O, 270^\circ}(3, 1) = (1, -3) \implies r_{y\text{-axis}}(1, -3) = (-1, -3) = S'' \end{aligned} $$ ## Confirm the correct composition rule $$ (r_{y\text{-axis}} \circ R_{O, 270^\circ})(x, y) = r_{y\text{-axis}}(y, -x) = (-y, -x) $$

(A) $R_{O, 270^\circ} \circ T_{-2, 0}(x, y)$

(B) $T_{-2, 0} \circ R_{O, 270^\circ}(x, y)$

(C) $R_{O, 270^\circ} \circ r_{y\text{-axis}}(x, y)$

(D) $r_{y\text{-axis}} \circ R_{O, 270^\circ}(x, y)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Composition of Transformations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify coordinates of pre-image and image vertices

Test the composition options on a reference point

Confirm the correct composition rule

Answer: