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Geometry Help: Representing Transformations As Functions

Review real example questions for Representing Transformations As Functions in Geometry.

Question 1 / 10

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A transformation $T$ maps point $(2, 3)$ to $(5, 7)$ and point $(4, 1)$ to $(7, 5)$ . If $T$ is a translation, what is the image of point $(0, -2)$ under this transformation?

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Question 1

A transformation $T$ maps point $(2, 3)$ to $(5, 7)$ and point $(4, 1)$ to $(7, 5)$ . If $T$ is a translation, what is the image of point $(0, -2)$ under this transformation?

$(3, 2)$ (correct answer)
$(3, 4)$
$(2, 3)$
$(-3, -6)$

Explanation: For a translation, the transformation function is $T(x,y) = (x+a, y+b)$ for some constants $a$ and $b$ . From $(2,3) \to (5,7)$ , we get $a = 3$ and $b = 4$ . We can verify with the second point: $(4,1) \to (4+3, 1+4) = (7,5)$ . Therefore, $T(0,-2) = (0+3, -2+4) = (3,2)$ .

Question 2

A transformation $g$ maps triangle $ABC$ with vertices $A(1,2)$ , $B(3,2)$ , and $C(2,4)$ to triangle $A'B'C'$ with vertices $A'(2,4)$ , $B'(6,4)$ , and $C'(4,8)$ . What type of transformation function does $g$ represent?

A translation function that shifts all points by the same vector displacement.
A dilation function that scales all distances from the origin by factor 2. (correct answer)
A rotation function that turns all points $90°$ about a fixed center point.
A reflection function that mirrors all points across a specific line of symmetry.

Explanation: Comparing corresponding points: $A(1,2) \to A'(2,4)$ , $B(3,2) \to B'(6,4)$ , $C(2,4) \to C'(4,8)$ . Each coordinate is multiplied by 2, indicating $g(x,y) = (2x, 2y)$ . This is a dilation with scale factor 2 centered at the origin. Dilations preserve angle measures but change distances by the scale factor.

Question 3

The transformation $h(x,y) = (3x, y)$ is applied to a square with vertices at $(-2,-1)$ , $(2,-1)$ , $(2,3)$ , and $(-2,3)$ . Which property is NOT preserved under this transformation?

The parallel relationships between opposite sides of the quadrilateral remain unchanged.
The right angle measures at each vertex are maintained at $90°$ .
The equal side lengths that define the original square are preserved. (correct answer)
The collinearity of points along each edge of the figure is maintained.

Explanation: The transformation $h(x,y) = (3x, y)$ is a horizontal stretch by factor 3. This preserves parallelism, collinearity, and angle measures, but not distances. The original square has side length 4, but after transformation, horizontal sides have length 12 while vertical sides remain length 4, creating a rectangle.

Question 4

Consider the transformation $f(x,y) = (-y, x)$ . Which statement best describes the properties of this transformation function?

It preserves both distance and angle measures, representing a $90°$ counterclockwise rotation about the origin. (correct answer)
It preserves distance but not angle measures, representing a reflection across the line $y = -x$ .
It preserves angle measures but not distances, representing a uniform scaling by factor $\sqrt{2}$ .
It preserves neither distance nor angle measures, representing a horizontal stretch combined with vertical compression.

Explanation: The transformation $f(x,y) = (-y, x)$ is a $90°$ counterclockwise rotation about the origin. Rotations are rigid transformations that preserve both distances and angle measures. We can verify: $(1,0) \to (0,1)$ and $(0,1) \to (-1,0)$ , which represents a $90°$ counterclockwise rotation.

Question 5

A composition of transformations is defined as $F(G(x,y))$ where $G(x,y) = (x+4, y-2)$ and $F(x,y) = (2x, 2y)$ . If this composite function maps point $A$ to point $A'(6,8)$ , what were the coordinates of the original point $A$ ?

$A(-1, 2)$
$A(7, 2)$
$A(1, 2)$
$A(-1, 6)$ (correct answer)

Explanation: When you encounter composition of transformations, you're working backwards from the final result through each transformation in reverse order. Think of it like undoing a series of steps to find where you started. Given that $F(G(x,y))$ maps point $A$ to $A'(6,8)$ , you need to work backwards through the transformations. Since $F(x,y) = (2x, 2y)$ is applied last, you first undo this dilation by dividing the coordinates of $A'$ by 2. This gives you $(6÷2, 8÷2) = (3, 4)$ , which represents the result after applying only transformation $G$ . Next, you undo transformation $G(x,y) = (x+4, y-2)$ by reversing its operations. Since $G$ adds 4 to the x-coordinate and subtracts 2 from the y-coordinate, you subtract 4 from the x-coordinate and add 2 to the y-coordinate: $(3-4, 4+2) = (-1, 6)$ . This is your original point $A$ . Choice A gives $(-1, 2)$ , which incorrectly subtracts 2 from the y-coordinate instead of adding 2 when undoing $G$ . Choice B gives $(7, 2)$ , which appears to add 4 instead of subtracting when undoing $G$ 's x-transformation. Choice C gives $(1, 2)$ , which makes errors in undoing both the dilation and translation components. The key strategy for composition problems is always to work backwards through the transformations in reverse order, undoing each operation step by step. Write out each intermediate step to avoid calculation errors.

Question 6

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 exactly one point by shifting 3 units left.
Each point maps to exactly one point by shifting 3 units up. (correct answer)
Each point maps to two different image points, so it is not a function.
Each image point maps to one original point by shifting 3 units up.

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.

Question 7

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?

Reflect across the $y$ -axis: $(x,y)\mapsto(-x,y)$ .
Translate 3 units right: $(x,y)\mapsto(x+3,y)$ . (correct answer)
Translate 3 units up: $(x,y)\mapsto(x,y+3)$ .
Each input point maps to two outputs: $(x,y)\mapsto(x+3,y)$ and $(x-3,y)$ .

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.

Question 8

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?

Distances and angles are preserved because each point shifts the same way. (correct answer)
Distances are not preserved because the $x$ -coordinates increase.
Angles are not preserved because the figure moves to the right.
The mapping is not a function because $J$ and $K$ have different images.

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.

Question 9

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?

Distances and angles are preserved because the mapping is a reflection. (correct answer)
Distances are doubled because the $y$ -values change sign.
Angles are not preserved because the points move to different quadrants.
One input point maps to two outputs, so distances cannot be preserved.

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.

Question 10

The figure shows quadrilateral $ABCD$ and its image $A'B'C'D'$ under transformation $R$ . What function rule best represents transformation $R$ ?

$R(x,y) = (y, -x)$ , representing a $90¬∞$ clockwise rotation about the origin. (correct answer)
$R(x,y) = (-y, x)$ , representing a $90¬∞$ counterclockwise rotation about the origin.
$R(x,y) = (-x, y)$ , representing a reflection across the $y$ -axis line.
$R(x,y) = (x, -y)$ , representing a reflection across the $x$ -axis line.

Explanation: From the figure, $A(1,2) \to A'(2,-1)$ , $B(3,1) \to B'(1,-3)$ , $C(4,3) \to C'(3,-4)$ , and $D(2,4) \to D'(4,-2)$ . The pattern shows $(x,y) \to (y,-x)$ , which represents a $90¬∞$ clockwise rotation about the origin.