Exercises: Represent and describe transformations

A

Warm-Up: Review What You Know

These problems review skills from earlier courses.

1.

A function $f$ takes inputs and produces exactly one output for each input. Which statement best describes how a geometric transformation is like a function?

A.

A transformation moves a figure, but leaves the rest of the plane unchanged.

B.

A transformation maps every point in the plane to exactly one image point, just as a function maps each input to one output.

C.

A transformation only works on points that are part of a named figure.

D.

A transformation can map one point to two different image points at the same time.

2.

Point $P$ has coordinates $(3, -1)$ . Which of the following correctly applies the rule $T(x, y) = (x + 2, y - 4)$ to find the image of $P$ ?

A.

$(5, -5)$

B.

$(1, 3)$

C.

$(3, -5)$

D.

$(5, -1)$

3.

Which of the following is NOT one of the four basic geometric transformations?

A.

Translation

B.

Reflection

C.

Projection

D.

Dilation

B

Fluency Practice

Apply the given coordinate rule to find image coordinates. Show your computation.

1.

Triangle $ABC$ has vertices $A(1, 2)$ , $B(4, 2)$ , and $C(2, 5)$ . A translation maps every point by the rule $T(x, y) = (x + 3, y - 1)$ .

What are the coordinates of $A'$ , the image of $A$ under $T$ ? Enter the $x$ -coordinate.

2.

Using the same triangle $ABC$ with $A(1, 2)$ , $B(4, 2)$ , $C(2, 5)$ and translation $T(x, y) = (x + 3, y - 1)$ .

What is the $y$ -coordinate of $C'$ , the image of $C(2, 5)$ under $T$ ?

3.

Triangle $PQR$ has vertices $P(2, 3)$ , $Q(5, 3)$ , $R(4, 6)$ . It is reflected over the $y$ -axis using the rule $M(x, y) = (-x, y)$ .

What is the $x$ -coordinate of $P'$ , the image of $P(2, 3)$ after reflection over the $y$ -axis?

4.

A $90^\circ$ counterclockwise rotation about the origin follows the rule $R(x, y) = (-y, x)$ . Which point is the image of $(3, 5)$ under this rotation?

A.

$(5, 3)$

B.

$(-5, 3)$

C.

$(3, -5)$

D.

$(-3, -5)$

5.

A dilation centered at the origin with scale factor $3$ maps triangle $DEF$ to $D'E'F'$ . Which statement is true about the distances and angles?

A.

All distances are preserved; all angles are preserved. The dilation is a rigid motion.

B.

All distances are tripled; all angles are preserved. The dilation is not a rigid motion.

C.

All distances and all angles are tripled.

D.

All angles are preserved; therefore the dilation is a rigid motion.

C

Mixed Practice

These problems test the same skills using different formats and representations.

D

Word Problems

Read each problem carefully, then apply the appropriate transformation reasoning.

1.

A graphic designer defines a coordinate rule to move every point in a logo design: $T(x, y) = (2x, y + 1)$ . She applies it to a triangle with vertices $A(1, 1)$ , $B(3, 1)$ , $C(2, 3)$ .

1.

What is the $x$ -coordinate of $B'$ , the image of $B(3, 1)$ under the rule $T(x, y) = (2x, y + 1)$ ?

2.

The original triangle has $AB = 2$ (horizontal distance). After applying $T(x, y) = (2x, y+1)$ , what is the length of $A'B'$ ?

A.

$2$ — the distance is preserved because $T$ is a rigid motion.

B.

$4$ — the horizontal distances are doubled by the factor of $2$ in $2x$ .

C.

$3$ — the rule shifts $y$ by $1$ , adding $1$ unit to the length.

D.

$2$ — vertical shifts do not affect horizontal lengths.

2.

An art teacher uses geometry software to rotate a star figure $60^\circ$ counterclockwise about the point $(0, 0)$ . One vertex of the star is at $(5, 0)$ . The rotation rule is $R(x, y) = (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)$ where $\theta = 60^\circ$ , $\cos 60^\circ = 0.5$ , and $\sin 60^\circ \approx 0.866$ .

To the nearest tenth, what is the $y$ -coordinate of the image of the vertex $(5, 0)$ ?

3.

Two triangles are drawn on a coordinate plane. After measurement, every pair of corresponding sides has the same length, and all corresponding angles are equal.

A classmate says the two triangles must be related by a dilation. Do you agree? Explain your reasoning using the concepts of rigid motions and what is preserved.

E

Find the Mistake

Each problem shows a student's incorrect claim. Identify the error and select the best explanation.

1.

Jordan reflected triangle $ABC$ over the $y$ -axis. Jordan says:

"When I flip the triangle over the $y$ -axis, the triangle leaves the plane and comes back on the other side — like turning a page in a book."

What is wrong with Jordan's description of reflection?

A.

Jordan is correct — reflection does involve rotating the figure through 3D space.

B.

Jordan should say the triangle is rotated, not flipped.

C.

Reflection is a function on the plane — each point maps to a mirror-image point within the same plane. No point leaves the plane.

D.

The error is that Jordan reflected over the wrong axis.

2.

Alex applied a dilation with scale factor $2$ to triangle $XYZ$ and got triangle $X'Y'Z'$ . Alex then wrote:

"All the angles in $X'Y'Z'$ are the same as in $XYZ$ . Since angles are preserved, this is a rigid motion."

What error did Alex make?

A.

Alex is correct — angle preservation is the definition of a rigid motion.

B.

Alex is wrong because dilation does not preserve angles.

C.

Alex is wrong because preserving angles is not sufficient for a rigid motion — distances must also be preserved, and dilation changes all distances by the scale factor.

D.

Alex is wrong because rigid motions are only translations.

F

Challenge Problems

These problems require multi-step reasoning. Show your work.

1.

A student claims: "Any transformation that preserves all angle measures must also preserve all distances."

Is this claim true or false? Either prove it or give a specific counterexample with numbers.

2.