Representing Transformations as Functions
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Geometry › Representing Transformations as Functions
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 reflection function that mirrors all points across a specific line of symmetry.
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.
A rotation function that turns all points $$90°$$ about a fixed center point.
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.
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 right angle measures at each vertex are maintained at $$90°$$.
The collinearity of points along each edge of the figure is maintained.
The parallel relationships between opposite sides of the quadrilateral remain unchanged.
The equal side lengths that define the original square are preserved.
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.
Two transformations are given: $$M(x,y) = (x \cos 45° - y \sin 45°, x \sin 45° + y \cos 45°)$$ and $$N(x,y) = (2x, \frac{y}{2})$$. Which statement correctly compares these transformation functions?
Both $$M$$ and $$N$$ preserve angles; $$M$$ preserves distances while $$N$$ does not preserve distances.
$$M$$ preserves both distances and angles; $$N$$ preserves neither distances nor angles between points.
Both $$M$$ and $$N$$ preserve distances; $$M$$ preserves angles while $$N$$ does not preserve angles.
$$M$$ preserves both distances and angles; $$N$$ preserves angles but not distances between points.
Explanation
$$M$$ is a $$45°$$ rotation (using rotation matrix formulas), which preserves both distances and angles. $$N$$ is a non-uniform scaling that doubles $$x$$-coordinates and halves $$y$$-coordinates, which changes both distances and angles. For example, a right angle would become non-perpendicular under $$N$$.
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 point $(x,y)$ maps to exactly one point $(y,x)$.
Each image point maps to one original point by swapping coordinates.
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.
Consider the transformation $$f(x,y) = (-y, x)$$. Which statement best describes the properties of this transformation function?
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.
It preserves distance but not angle measures, representing a reflection across the line $$y = -x$$.
It preserves both distance and angle measures, representing a $$90°$$ counterclockwise rotation about the origin.
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.
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 $$A$$?
$$A(-1, 2)$$
$$A(-1, 6)$$
$$A(7, 2)$$
$$A(1, 2)$$
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.
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, 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 image point maps to one original point by shifting 3 units up.
Each point maps to exactly one point by shifting 3 units up.
Each point maps to two different image points, so it is not a function.
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.
A transformation $$S$$ has the property that $$S(S(x,y)) = (x,y)$$ for all points $$(x,y)$$. Additionally, $$S(3,0) = (-3,0)$$ and $$S(0,5) = (0,-5)$$. Which function represents transformation $$S$$?
$$S(x,y) = (y, x)$$, which represents a reflection across the line $$y = x$$.
$$S(x,y) = (x, -y)$$, which represents a reflection across the $$x$$-axis line.
$$S(x,y) = (-x, y)$$, which represents a reflection across the $$y$$-axis line.
$$S(x,y) = (-x, -y)$$, which represents a $$180°$$ rotation about the origin point.
Explanation
When you encounter a transformation that satisfies $$S(S(x,y)) = (x,y)$$, you're looking at an involution — a transformation that is its own inverse. This means applying the transformation twice returns you to the original point.
To find the correct transformation, test each option against the given conditions. First, check which transformations satisfy $$S(3,0) = (-3,0)$$ and $$S(0,5) = (0,-5)$$.
Option D, $$S(x,y) = (-x, -y)$$, works perfectly: $$S(3,0) = (-3,0)$$ ✓ and $$S(0,5) = (0,-5)$$ ✓. Let's verify the involution property: $$S(S(x,y)) = S(-x,-y) = (-(-x), -(-y)) = (x,y)$$ ✓.
Now let's see why the other options fail:
Option A $$S(x,y) = (x,-y)$$ gives $$S(3,0) = (3,0)$$, not $$(-3,0)$$.
Option B $$S(x,y) = (y,x)$$ gives $$S(3,0) = (0,3)$$, not $$(-3,0)$$.
Option C $$S(x,y) = (-x,y)$$ gives $$S(0,5) = (0,5)$$, not $$(0,-5)$$.
While options A, B, and C are all valid involutions (reflections are their own inverses), they don't satisfy the specific point conditions given in the problem.
Study tip: When solving transformation problems, always test the given specific points first to eliminate incorrect options quickly. Then verify that your remaining candidate satisfies any additional properties like the involution condition.
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?
Translate 3 units right: $(x,y)\mapsto(x+3,y)$.
Each input point maps to two outputs: $(x,y)\mapsto(x+3,y)$ and $(x-3,y)$.
Translate 3 units up: $(x,y)\mapsto(x,y+3)$.
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.