Representing Transformations as Functions
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Geometry › Representing Transformations as Functions
On the coordinate plane, triangle $ABC$ (solid) is mapped to triangle $A'B'C'$ (dashed). The labeled point-image pairs include $A(-2,1)\to A'(1,1)$ and $B(-1,4)\to B'(2,4)$. Which mapping rule matches the diagram?
Each input point maps to two outputs: $(x,y)\mapsto(x+3,y)$ and $(x-3,y)$.
Translate 3 units up: $(x,y)\mapsto(x,y+3)$.
Translate 3 units right: $(x,y)\mapsto(x+3,y)$.
Reflect across the $y$-axis: $(x,y)\mapsto(-x,y)$.
Explanation
The skill is representing geometric transformations as functions. A transformation is defined as an input-output rule that assigns to each point (x, y) in the plane a unique output point. In the diagram, points move horizontally to the right by 3 units while keeping the y-coordinate the same, as seen from A(-2,1) to A'(1,1) and B(-1,4) to B'(2,4). This mapping preserves distances, angles, and orientations since the figure remains congruent to the original. The correct answer is B because adding 3 to the x-coordinate matches both point pairs exactly. A distractor like D suggests multiple outputs, but transformations as functions map each input to exactly one output. To analyze similar problems, track one point at a time to identify the pattern in coordinates.
On the coordinate plane, point $G(1,2)$ maps to $G'(2,1)$ and point $H(4,-1)$ maps to $H'(-1,4)$ under transformation $T$. Which description correctly represents this transformation as a function?
Each point maps to two outputs by swapping or negating coordinates.
Each image point maps to one original point by swapping coordinates.
Each point $(x,y)$ maps to exactly one point $(y,x)$.
Each point $(x,y)$ maps to exactly one point $(x,-y)$.
Explanation
This question focuses on representing transformations as functions with unique outputs. A transformation function maps each input point to exactly one output point according to a specific rule. Looking at the mappings, G(1,2) maps to G'(2,1) and H(4,-1) maps to H'(-1,4), which shows coordinates are being swapped. This pattern matches T(x,y) = (y,x), representing reflection across the line y=x. Each point has exactly one image under this transformation, satisfying the function definition. The transformation preserves distances and angles as a rigid motion. Students might think swapping creates multiple outputs or confuse this with other coordinate manipulations. To verify transformation functions, ensure each input produces exactly one output.
On the coordinate plane, a transformation $T$ maps $R(1,2)$ to $R'(2,4)$ and $S(3,1)$ to $S'(6,2)$. Which description treats the transformation as a function?
Each image point maps back to the original by doubling coordinates.
Point $(x,y)$ can map to both $(2x,2y)$ and $(-2x,-2y)$.
Each point $(x,y)$ maps to exactly one point $(2x,2y)$.
Each point $(x,y)$ maps to exactly one point $(x+1,y+2)$.
Explanation
This question focuses on recognizing transformations as functions with unique outputs. A transformation is a function when each input point (x,y) maps to exactly one output point. Analyzing the given mappings, R(1,2) maps to R'(2,4) and S(3,1) maps to S'(6,2), which shows both coordinates are being doubled. The pattern T(x,y) = (2x,2y) represents a dilation centered at the origin with scale factor 2. This transformation stretches distances but preserves angles and ensures each point has exactly one image. The incorrect option suggesting multiple outputs violates the function definition. To identify transformation patterns, compare corresponding coordinates systematically.
On the coordinate plane, triangle $ABC$ is mapped to triangle $A'B'C'$ by a transformation $T$. Points $A(1,1)$ and $B(4,1)$ map to $A'(1,4)$ and $B'(4,4)$. Which statement correctly describes how points are mapped?
Each point maps to two different image points, so it is not a function.
Each point maps to exactly one point by shifting 3 units up.
Each image point maps to one original point by shifting 3 units up.
Each point maps to exactly one point by shifting 3 units left.
Explanation
This question tests understanding of transformations as functions that map points to new locations. A transformation is a function when each input point maps to exactly one output point. Looking at the given points, A(1,1) maps to A'(1,4) and B(4,1) maps to B'(4,4), which shows that the x-coordinates stay the same while the y-coordinates increase by 3. This pattern indicates that every point (x,y) maps to (x,y+3), meaning each point shifts 3 units up. The correct answer recognizes this vertical translation. A common misconception is confusing the direction of movement or thinking that transformations can map one point to multiple outputs, which would violate the function definition. To verify transformations, track individual coordinates systematically.
On the coordinate plane, a transformation $T$ maps triangle $JKL$ to $J'K'L'$. Point $J(0,1)$ maps to $J'(2,1)$ and point $K(1,4)$ maps to $K'(3,4)$. Which claim about distance or angle preservation is supported?
The mapping is not a function because $J$ and $K$ have different images.
Angles are not preserved because the figure moves to the right.
Distances are not preserved because the $x$-coordinates increase.
Distances and angles are preserved because each point shifts the same way.
Explanation
This question examines distance and angle preservation in transformations. A transformation is a function mapping each point to exactly one new location, and translations are rigid motions that preserve geometric properties. Analyzing the mappings, J(0,1) maps to J'(2,1) and K(1,4) maps to K'(3,4), showing both points shift 2 units right with no vertical change. This consistent pattern T(x,y) = (x+2,y) represents a horizontal translation. Since every point moves the same distance in the same direction, all distances and angles are preserved. Students might incorrectly think that coordinate changes affect distances or that different image points mean the transformation isn't a function. When analyzing translations, verify that all points shift by the same vector.
A transformation $T$ maps points in the plane. On the coordinate plane, $E(2,3)$ maps to $E'(-2,-3)$ and $F(-1,4)$ maps to $F'(1,-4)$. Which statement correctly describes how points are mapped?
Each point maps to exactly one point by reflecting through the origin.
Each image point maps to one original point by reflecting across the $x$-axis.
Each point maps to exactly one point by rotating $90^\circ$ counterclockwise.
Each point maps to exactly one point by reflecting across the $y$-axis.
Explanation
This question tests recognizing transformation patterns from point mappings. A transformation is a function where each input point maps to exactly one output point following a consistent rule. Examining the mappings, E(2,3) becomes E'(-2,-3) and F(-1,4) becomes F'(1,-4), showing that both coordinates change sign. This pattern indicates reflection through the origin, where T(x,y) = (-x,-y). This transformation is a 180° rotation about the origin, preserving distances and angles as a rigid motion. Students might confuse this with reflections across axes or think sign changes create multiple outputs. When identifying transformations, check that the rule works consistently for all given points.
On the coordinate plane, segment $\overline{MN}$ is mapped to $\overline{M'N'}$ by a transformation $T$ where $M(1,1)\to M'(1,-1)$ and $N(4,2)\to N'(4,-2)$. Which claim about distance or angle preservation is supported?
One input point maps to two outputs, so distances cannot be preserved.
Distances and angles are preserved because the mapping is a reflection.
Distances are doubled because the $y$-values change sign.
Angles are not preserved because the points move to different quadrants.
Explanation
This question examines properties preserved by transformations as functions. A transformation function maps each point to exactly one new location, and different transformations preserve different properties. Analyzing the mappings, M(1,1) becomes M'(1,-1) and N(4,2) becomes N'(4,-2), showing that x-coordinates stay constant while y-coordinates change sign. This pattern indicates reflection across the x-axis, which is a rigid motion that preserves both distances and angles. The segment length and orientation angles remain unchanged under this transformation. Students might incorrectly think that sign changes affect distances or that moving to different quadrants changes angles. When analyzing transformations, calculate distances before and after to verify preservation.