HSG.CO.A.2: Transformations as Functions, Rigid and Non-Rigid
In plain English: HSG.CO.A.2 is the Common Core geometry standard that asks students to represent plane transformations with transparencies and geometry software and to describe them as functions that take points as inputs and give points as outputs. Students compare transformations that preserve distance and angle, like a translation, with ones that do not, like a horizontal stretch. It is usually taught in high school Geometry.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Common Core State Standards for Mathematics · Domain: Congruence (CO) · Cluster: Experiment with transformations in the plane Also written as HSG-CO.A.2 or G-CO.2 · Official standard
Students move from thinking of a transformation as "sliding" or "flipping" a picture to treating it as a function: a rule that takes every point of the plane as an input and assigns exactly one point as its output. They model transformations physically with tracing paper and digitally with dynamic geometry software, then write coordinate rules such as T(x, y) = (x + 6, y + 1) and use them in both directions, finding outputs from inputs and inputs from outputs.
The second half of the lesson compares transformations that preserve distance and angle measure (translations, reflections, rotations) with ones that do not, following the standard's own example: a translation versus a horizontal stretch. Students measure before and after, and they find that a stretch can keep some lengths and angles while changing others, so one unchanged measurement is not enough to call a transformation rigid.
Learning Objectives
By the end of this lesson, students will be able to:
Represent translations, reflections and rotations with tracing paper and with geometry software
Describe a transformation as a function whose inputs and outputs are points, and write and use coordinate rules
Find the output for a given input point and the input for a given output point
Test whether a transformation preserves distance and angle measure by computing and measuring
Explain why a translation is a rigid motion and a horizontal stretch is not
Prior Knowledge Required
Students should already be comfortable with:
Properties of rotations, reflections and translations verified with hands-on tools 8.G.A.1
Describing translations, rotations, reflections and dilations with coordinates 8.G.A.3
The distance between two points using the Pythagorean Theorem 8.G.B.8
Function notation: each input has exactly one output HSF.IF.A.1
Project a coordinate grid with the point K(2, -3) and the rule "add 4 to x, keep y". Ask students to apply it before discussing:
Warm-Up Prompt
"Apply the rule to K(2, -3), to (0, 0) and to (-4, 5). Which point has the output (1, 1)? Is there any point the rule cannot be applied to, or any point that gets two different outputs?"
Answers: (6, -3), (4, 0), (0, 5), and the input (-3, 1). Every point has exactly one output, which is what makes the rule a function. Tell students this is the view of transformations they will use for the rest of geometry: a transformation moves every point of the plane, not only the corners of a picture.
Direct Instruction20 minutes
Part 1: Transformations as functions. Define a transformation of the plane as a function that assigns to each input point P exactly one output point P', its image. Write rules in the form T(x, y) = (new x, new y). Show three ways to represent the same transformation: a coordinate rule, a traced copy on a transparency moved over the grid, and a software construction (translate by a vector, reflect across a line, rotate about a point). Work through these examples:
Translation as a function
Apply T(x, y) = (x + 5, y - 2) to the inputs (1, 3) and (-4, 0).
Equation: T(1, 3) = (6, 1) and T(-4, 0) = (1, -2)
Rotation as a function
The rotation of 90° counterclockwise about the origin is R(x, y) = (-y, x). Apply it to P(4, 1) and Q(2, 3) and compare PQ with P'Q'.
Equation: P' = (-1, 4), Q' = (-3, 2), and PQ = P'Q' = √8
Finding the input
For T(x, y) = (x - 3, y + 4), which input point has the output (2, 7)?
Equation: x - 3 = 2 and y + 4 = 7, so the input is (5, 3)
Representing a reflection with a transparency
Trace A(2, 5) and the y-axis on a transparency, flip the sheet over so the traced axis lands on the y-axis, and read where A lands.
Equation: A' = (-2, 5): the rule is M(x, y) = (-x, y)
Horizontal stretch
Apply S(x, y) = (3x, y) to D(1, 2), E(3, 2) and F(1, 4). Compare side lengths and the angle at E.
Equation: DE = 2 becomes 6, DF = 2 stays 2, EF = √8 becomes √40, and the angle at E changes from 45° to about 18.4°
Part 2: Comparing transformations. Show Diagram 1 (a translation) and Diagram 2 (a horizontal stretch of the same triangle). Define a transformation as preserving distance if every segment has the same length as its image, and as preserving angle if every angle has the same measure as its image. Translations, reflections and rotations do both; they are called rigid motions. A horizontal stretch does neither in general, even though it keeps vertical lengths and the right angle in Diagram 2. Ask: "Why can't a transparency show a stretch?" (The sheet is rigid: it can slide, flip and turn, but it cannot stretch.)
Guided Practice15-20 minutes
Pairs apply three rules to the triangle J(0, 0), K(2, 0), L(0, 3), first with tracing paper or software and then with coordinates: U(x, y) = (x + 1, y + 4), M(x, y) = (-x, y) and V(x, y) = (x, 2y). They record each image and the length of the image of KL. Answers: U gives (1, 4), (3, 4), (1, 7); M gives (0, 0), (-2, 0), (0, 3); V gives (0, 0), (2, 0), (0, 6). KL = √13 is preserved by U and M, but V gives √40. Ask pairs which segment V leaves unchanged (JK, which is horizontal) and why that does not make V rigid.
Independent Practice15 minutes
Students work alone: (1) Find the image of (5, 1) under (x - 2, y - 6). (Answer: (3, -5).) (2) Find the image of (3, 7) under (y, -x). (Answer: (7, -3).) (3) Find the image of (4, 8) under (x, y/2). (Answer: (4, 4).) (4) Which input has the output (0, 0) under (x + 4, y)? (Answer: (-4, 0).) (5) The segment from (0, 0) to (4, 3) has length 5. Find the length of its image under (x, y/2) and decide whether that rule preserves distance. (Answer: √18.25 ≈ 4.27, so no.)
Closure5 minutes
Exit ticket: (1) Write the rule for a translation 2 units left and 5 units up, and find the image of (3, -1). (Answer: (x - 2, y + 5), and (1, 4).) (2) Use the segment from (0, 0) to (3, 0) to show that (x/3, y) does not preserve distance. (Its image has length 1, not 3.) (3) In one sentence, explain what it means to call a transformation a function.
Differentiation Strategies
For Struggling Students
Give an input-output table with columns "input point", "rule applied" and "output point" so students apply the rule one coordinate at a time
Let students check every coordinate answer by moving tracing paper over the grid
Start with translations only, then add reflections across the axes before rotations
For Advanced Students
Ask students to prove that T(x, y) = (x + h, y + k) preserves the distance between any two points (a, b) and (c, d)
Ask for a transformation that preserves angle measure but not distance, and one that preserves neither, with coordinate evidence
Ask students to find every point that the reflection (x, y) → (y, x) leaves in place, and to describe the set geometrically
Assessment Guidance
What to Look For
Check that students treat a transformation as acting on every point of the plane and can work backward from an output to an input. When they compare transformations, look for evidence from more than one segment and angle: a student who checks only a vertical segment under a horizontal stretch will wrongly conclude that it preserves distance. Strong explanations connect the physical model to the property: a transparency cannot stretch, so everything it models preserves distance and angle.
02
Classroom Activities
3 Activities
1
Transparency Transformations
20 minPairs
Students model three transformations with tracing paper on a coordinate grid, record input and output points in a table, and then write each transformation as a coordinate rule.
Setup
Each pair gets a grid with quadrilateral WXYZ, W(1, 1), X(3, 1), Y(4, 3), Z(1, 4), a sheet of tracing paper and a pushpin.
Procedure
Translation: trace WXYZ and the axes, slide the sheet 4 units right and 2 units down without turning it, and record the new vertices: (5, -1), (7, -1), (8, 1), (5, 2)
Reflection: trace again, flip the sheet over so the traced y-axis lands on the y-axis, and record (-1, 1), (-3, 1), (-4, 3), (-1, 4)
Rotation: pin the sheet at the origin, turn it 90° counterclockwise until the traced x-axis lies on the y-axis, and record (-1, 1), (-1, 3), (-3, 4), (-4, 1)
For each table, write the coordinate rule and test it on a point that is not a vertex, such as the midpoint of WX
Discussion Questions
Which move of the sheet corresponds to which rule?
The traced copy always fits exactly on the image. What does that tell you about lengths and angles?
2
Function Machine Cards
15 minGroups of 3
One student secretly draws a rule card and acts as the "function machine". The others feed it input points and try to name the rule from the outputs, which reinforces that a transformation is a function on points.
The 6 Rule Cards
(x + 3, y)
(x, y - 4)
(-x, y)
(x, -y)
(-y, x)
(2x, y)
Procedure
The machine answers each input with its output, for example the input (2, 5) gives (5, 5), (2, 1), (-2, 5), (2, -5), (-5, 2) or (4, 5), depending on the card
Guessers may ask for at most three inputs before naming the rule and the type of transformation
After each round, the group decides whether the rule preserves distance, and tests one segment to confirm
Challenge Variation
Guessers may only ask for the input whose output is a given point, so they must think about the rule in reverse.
3
Rigid or Not? Software Lab
20 minPairs
In dynamic geometry software (for example GeoGebra), students build a triangle, apply a translation and a horizontal stretch to it, and measure every side and angle before and after.
Procedure
Construct any triangle PQR and measure its sides and angles
Translate it by a vector with the translation tool; enter the stretched image by defining each point as (2x(P), y(P)) and so on
Record all six measurements for both images in a table, then drag a vertex of PQR and watch which measurements stay equal
Drag PQR until one side is vertical. Which side of the stretched image keeps its length now, and do the angles still change?
Discussion Questions
How many measurements must match before you can call a transformation rigid?
Why would a transparency be enough to show the translation but not the stretch?
Modification for Distance Learning
Pairs work in a shared online geometry file and post a screenshot of their measurement table with one sentence of conclusion.
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Diagrams & Visual Aids
2 diagrams
Diagram 1: A Translation as a Function on Points
Triangle ABC with A(1, 1), B(4, 1), C(1, 5) and its image under T(x, y) = (x + 6, y + 1), drawn to scale. Each input point has exactly one output point, and every side length and angle of the image matches the original.
Diagram 2: A Horizontal Stretch Changes Distance and Angle
The same triangle under S(x, y) = (2x, y), drawn to scale. The vertical side keeps its length and the right angle stays 90°, but AB doubles, BC changes from 5 to √52, and the angles at B and C change, so the stretch is not a rigid motion.
04
Homework Assignment
~30 min
HSG.CO.A.2 Homework: Transformations as Functions
Directions: Show all work. Write each transformation in function notation, and justify every claim about distance with the distance formula or about angles with a measurement or a calculation. Use tracing paper or software to check your answers where you can.
Part 1: Inputs and Outputs (Problems 1-3)
Let T(x, y) = (x - 7, y + 2). (a) Find the images of A(3, 4), B(-1, 0) and C(0, -6). (b) Find the input point whose image is (-2, -2). (c) Compute AB and A'B' and compare.
The rotation of 90° counterclockwise about the origin is R(x, y) = (-y, x). (a) Find R(6, -1), R(0, 3) and R(-2, -5). (b) Find the point whose image is (4, 0). (c) Explain why R is a function.
Describe step by step how to represent each transformation (a) with tracing paper and (b) with geometry software: a translation 3 units right and 2 units down, a reflection across the x-axis, and a rotation of 90° about the point (1, 1).
Part 2: Comparing Transformations (Problems 4-6)
Triangle DEF has vertices D(0, 0), E(4, 3) and F(4, 0). (a) Find its image under V(x, y) = (2x, y). (b) Compare DE, EF and DF with the image lengths. (c) Compare the angle at D (about 36.9°) and the angle at F with the image angles. (d) Does V preserve distance? Angle?
For each rule, decide whether it preserves distance and whether it preserves angle measure, and test it on the segment from (0, 0) to (2, 2): (a) (x - 2, y + 5) (b) (x, -y) (c) (3x, 3y) (d) (x, y/2).
Find two different transformations, one reflection and one translation, that both send (1, 1) to (1, -1). Find the image of (4, 2) under each. Explain why both preserve distance, and why knowing the image of one point is not enough to identify a transformation.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Function Notation
Rules written and applied correctly, including in reverse
Outputs correct but a reverse question missed
Rules missing or misapplied
Representation
Clear tracing paper and software steps for all three transformations
Steps for some transformations
No usable steps
Distance and Angle Evidence
Correct computations support each claim
Correct claims with incomplete evidence
Claims without evidence
Comparison
Rigid and non-rigid transformations classified and explained
Classified but not explained
Misclassified
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Click an option to check it and read the explanation. Try the short answers on paper first, then open the answer. Reset quiz starts the quiz over.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Let T(x, y) = (x - 4, y + 6). What is the image of (3, -2)?
Answer: B
Subtract 4 from x and add 6 to y: (3 - 4, -2 + 6) = (-1, 4). Choice A adds 4 to x instead of subtracting it. Choice C subtracts 6 from y. Choice D reverses both operations.
Question 2 of 20 · Multiple Choice
The rotation of 90° clockwise about the origin is R(x, y) = (y, -x). What is R(5, 2)?
Answer: C
Swap the coordinates and change the sign of the new second coordinate: (2, -5). Choice A is the counterclockwise rotation (-y, x). Choice B is the reflection across the x-axis. Choice D is (-x, -y), the rotation of 180°.
Question 3 of 20 · Multiple Choice
The reflection across the x-axis is M(x, y) = (x, -y). What is M(-3, 7)?
Answer: A
Keep x and change the sign of y: (-3, -7). Choice B reflects across the y-axis instead. Choice C swaps the coordinates, and choice D is the rotation of 180° about the origin.
Question 4 of 20 · Multiple Choice
Let T(x, y) = (x + 2, y - 5). Which input point has the output (6, -1)?
Answer: D
Solve x + 2 = 6 and y - 5 = -1 to get (4, 4). Check: T(4, 4) = (6, -1). Choice A applies T to the output instead of undoing it. Choices B and C undo only one of the two coordinates.
Question 5 of 20 · Multiple Choice
Which transformation does NOT preserve distance?
Answer: D
The rule (x, 4y) is a vertical stretch: the segment from (0, 0) to (0, 1) becomes a segment of length 4. Choice A is a translation, choice B a reflection and choice C a rotation of 180°, and all three preserve distance.
Question 6 of 20 · Multiple Choice
A segment has endpoints (1, 3) and (4, 3). What is the length of its image under S(x, y) = (3x, y)?
Answer: B
The image endpoints are (3, 3) and (12, 3), so the image has length 9, three times the original length 3. Choice A assumes the stretch preserves length. Choice C adds 3 to the length instead of multiplying by 3, and choice D multiplies by 9.
Question 7 of 20 · Multiple Choice
Which statement about S(x, y) = (2x, y) is true?
Answer: C
Horizontal segments double in length and vertical segments keep their length, while slanted segments change by other factors. Choice B is false because vertical segments are unchanged. Choice D is false: a 45° angle between a horizontal and a slanted side does not stay 45°.
Question 8 of 20 · Multiple Choice
A student traces a figure on tracing paper, pins the paper at point P and turns it halfway around. Which transformation does this model?
Answer: A
Turning the sheet about a fixed pin is a rotation about that point, and halfway around is 180°. A reflection needs the sheet flipped over, so choice B is wrong. A translation slides the sheet without turning it.
Question 9 of 20 · Multiple Choice
Which move of a transparency models a reflection across a line?
Answer: C
Flipping the sheet over the line reverses the figure and keeps every point of the line in place, which is what a reflection does. Choice A is a translation and choice B a rotation. Choice D is impossible with a rigid sheet.
Question 10 of 20 · Multiple Choice
Why can a transparency not be used to model the transformation (x, y) → (2x, y)?
Answer: B
A transparency can slide, turn and flip, and all of these preserve distance and angle. The stretch changes horizontal lengths, so no move of the sheet can produce it. Choice D is false: the rule is a function, since each point has exactly one image.
Question 11 of 20 · Multiple Choice
The rotation R(x, y) = (-y, x) turns points 90° counterclockwise about the origin. Which point has the image (3, -4)?
Answer: D
Solve -y = 3 and x = -4, so the input is (-4, -3). Check: R(-4, -3) = (3, -4). Choice A is the image of (3, -4) itself, not its input. Choices B and C give other outputs: R(-4, 3) = (-3, -4) and R(3, -4) = (4, 3).
Question 12 of 20 · Multiple Choice
Points A(1, 1) and B(6, 13) are 13 units apart. What is the distance between their images under S(x, y) = (2x, y)?
Answer: C
The images are (2, 1) and (12, 13), so the distance is √(10² + 12²) = √244 ≈ 15.62. Choice B doubles the whole distance, but only the horizontal part doubles. Choice D doubles y instead of x.
Question 13 of 20 · Multiple Choice
Triangle with vertices (0, 0), (2, 0) and (0, 2) has a 45° angle at (2, 0). What is the angle at the image of (2, 0) under S(x, y) = (2x, y)?
Answer: C
The image triangle has vertices (0, 0), (4, 0) and (0, 2), so the angle at (4, 0) has tangent 2/4, about 26.6°. Choice A assumes the stretch preserves angles. Choice D halves the angle, but a stretch does not change angles by a simple factor.
Question 14 of 20 · Multiple Choice
For which rule is the image of every segment congruent to the segment?
Answer: C
The rule (-x, y + 3) is a reflection across the y-axis followed by a translation up 3; both preserve every distance. Choice B is a dilation, which doubles every length. Choices A and D stretch in one direction.
Question 15 of 20 · Short Answer
Write the rule of the translation that moves (2, -3) to (-4, 2). Then find the image of the origin.
The x-coordinate changes by -4 - 2 = -6 and the y-coordinate by 2 - (-3) = 5, so T(x, y) = (x - 6, y + 5). The image of the origin is (-6, 5).
Question 16 of 20 · Short Answer
Show that T(x, y) = (x + 7, y - 1) preserves the distance between P(-2, 1) and Q(1, 5), and that S(x, y) = (x, 3y) does not.
PQ = √(3² + 4²) = 5. T gives P'(5, 0) and Q'(8, 4), and P'Q' = √(3² + 4²) = 5, the same. S gives (-2, 3) and (1, 15), and the distance is √(3² + 12²) = √153 ≈ 12.37, so S does not preserve distance.
Question 17 of 20 · Short Answer
Describe how to use geometry software to show that a reflection preserves angle measure. What would convince you that it works for every triangle, not just one?
Construct a triangle and a line, use the reflect tool to create the image, and measure the three angles of both triangles: each image angle equals the matching original angle. Then drag the vertices and the line. If the measurements stay equal for every position, the evidence covers many triangles, not just one. (A proof comes later, in HSG.CO.A.4 and HSG.CO.B.6.)
Question 18 of 20 · Short Answer
The rotation of 180° about the origin is H(x, y) = (-x, -y). Which input has the output (2, -9)? Which points are their own images? Explain.
Solve -x = 2 and -y = -9: the input is (-2, 9). A point is its own image when (-x, -y) = (x, y), which means x = 0 and y = 0. So only the origin, the center of the rotation, stays in place.
Question 19 of 20 · Short Answer
For each rule, decide whether it preserves distance and whether it preserves angle measure: (a) (x + 2, y) (b) (y, x) (c) (x/2, y) (d) (2x, 2y).
(a) A translation: preserves both. (b) A reflection across the line y = x: preserves both. (c) A horizontal compression: preserves neither in general; for example, the triangle (0, 0), (2, 0), (0, 2) loses its 45° angles. (d) A dilation by 2: preserves angles but not distances, since every length doubles.
Question 20 of 20 · Short Answer
Find the image of the square with vertices (0, 0), (1, 0), (1, 1) and (0, 1) under S(x, y) = (4x, y). Which measurements are preserved and which change?
The image is the rectangle with vertices (0, 0), (4, 0), (4, 1), (0, 1). The four right angles and the vertical sides (length 1) are preserved. The horizontal sides change from 1 to 4, the diagonal from √2 to √17, and the angle between the diagonal and the base from 45° to about 14.0°. So S is not a rigid motion, even though it keeps some measurements.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.CO.A.2 mean?
HSG.CO.A.2 means students treat transformations as functions on points and compare those that keep distances and angles with those that do not. They represent transformations with tools such as tracing paper and geometry software, write rules like (x, y) → (x + 6, y + 1), and use measurements to tell a translation from a horizontal stretch.
Why call a transformation a function?
Because it assigns to each input point exactly one output point, in the same way that a function of numbers assigns each input a single output. This view prepares students for combining transformations, for undoing them, and for defining congruence and similarity in terms of transformations later in the course.
What is the difference between a rigid motion and a non-rigid transformation?
A rigid motion preserves distance, and therefore angle measure, while a non-rigid transformation changes at least some distances. Translations, reflections and rotations are rigid. Stretches and dilations are not, although a dilation still preserves angles.
Does a horizontal stretch change every length?
No. It leaves vertical segments unchanged, multiplies horizontal lengths by the stretch factor, and changes slanted lengths by other amounts. That is why testing only one segment can give the wrong answer. To show a transformation is not rigid, one changed length is enough; to argue that it is rigid, you need a reason that works for every segment.
Is HSG.CO.A.2 the same as the grade 8 transformation standards?
It builds on them but asks for more. In grade 8 (8.G.A.1 and 8.G.A.3) students verify properties of rigid motions and describe their effects with coordinates. In HSG.CO.A.2 they describe transformations as functions on the whole plane and compare rigid and non-rigid ones in general, which sets up the formal definitions in HSG.CO.A.4.
What is a common mistake with coordinate rules?
Mixing up the direction of a rule. For example, students often apply a rule to an output when asked for the input: if T(x, y) = (x - 1, y + 3) and the output is (2, 2), the input is (3, -1), not (1, 5). Another frequent error is confusing (-y, x), the 90° counterclockwise rotation, with (y, -x), the clockwise one.
Do students need geometry software for this standard?
No, but the standard names transparencies and geometry software as examples of tools, so students should use at least one physical and one digital representation. Tracing paper or transparency sheets work well for rigid motions. Software makes it easy to measure many cases at once and to define non-rigid rules.
How can parents help with transformations?
Ask your student to explain a rule in words and act it out. For example, "(x, y) → (x + 3, y)" means every point slides 3 units right. Photo editing apps are also useful: resizing a photo in one direction only is a stretch, and faces in it visibly change shape, while rotating the photo does not.
How is HSG.CO.A.2 assessed?
Typical items ask students to apply a coordinate rule, find an input from an output, identify which transformations preserve distance or angle, and justify their answer with calculations. Some tasks give a figure and its image and ask students to describe the transformation as a function.
What comes next after this standard?
Students define rotations, reflections and translations precisely (HSG.CO.A.4), draw images and sequences of transformations (HSG.CO.A.5), and then use rigid motions to define congruence (HSG.CO.B.6). Non-rigid transformations return with dilations and similarity in HSG.SRT.A.1.
07
Related Standards
6 standards
These standards connect to HSG.CO.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.A.1Prerequisite
Verify experimentally the properties of rotations, reflections and translations
