Composition of transformations applies two or more transformations in sequence to a figure, where the output of one transformation becomes the input of the next. The order matters because transformation composition is generally not commutative.
Order matters, like doing rotate then reflect versus reflect then rotate.
Showing a random 20 of 50 problems.
Example 1
easy
Composing two translations gives what?
Example 2
medium
Composing two dilations centered at the same point with factors 2 and 3 gives what?
Example 3
easy
What is a composition of transformations?
Example 4
challenge
Prove that every rigid motion of the plane is a composition of at most three reflections.
Example 5
hard
Find the image of (4,1) under reflection in the line y=x followed by translation ⟨−2,3⟩, then explain why swapping the order generally fails.(4,1) reflected in line y=x then translated by ⟨−2,3⟩
Example 6
medium
Point (1,2) is reflected over the x-axis, then translated by (3,0). Find its image.(1,2) reflected over x-axis, then translated by (3,0)
Example 7
easy
Point Q(2,−3) is translated by ⟨1,5⟩ and then reflected over the x-axis. Find the final image.Q(2,-3) translated by ⟨1,5⟩ then reflected over the x-axis
Example 8
hard
Triangle ABC with A(1,2), B(3,2), C(2,5) is dilated by factor 2 about origin, then reflected over the x-axis. Find the image vertices.Triangle ABC dilated by factor 2 about origin then reflected over the x-axis
Example 9
medium
Triangle with vertices A(1,1), B(4,1), C(1,3) is translated by ⟨2,3⟩ then reflected over the x-axis. Find the new vertices.Triangle ABC translated by ⟨2,3⟩ then reflected over the x-axis
Example 10
medium
Apply to (2,−3): rotate 90° counterclockwise about origin, then rotate 180° about origin. What single rotation does this equal, and what is the image?
Example 11
hard
A point is dilated by factor 3 about (0,0), then dilated by factor 21 about (0,0). What single dilation does this equal, and what is the image of (4,−2)?
Example 12
medium
Show that reflecting over y=x then over the x-axis is NOT the same as doing them in the opposite order, using the point (2,5).
Example 13
medium
Point E(−3,2) is reflected over the line y=x, then rotated 90° clockwise about the origin. Find E′′.E(-3,2) reflected over y=x then rotated 90° CW about origin
Example 14
medium
What single transformation does the identity followed by a 90° rotation about the origin equal?
Example 15
medium
What is a glide reflection?
Example 16
easy
Point P(2,5) undergoes the translation ⟨−3,1⟩ followed by ⟨4,−2⟩. Find P′′.
Example 17
easy
Composing two rotations about the same center gives what?
Example 18
easy
A point is translated by (2,3) then by (4,−1). Find the single equivalent translation vector.
Example 19
medium
Two mirror lines intersect at 35°. A double reflection rotates a figure by how much?
Example 20
hard
Triangle ABC with A(1,0), B(3,0), C(2,2) is rotated 90° counterclockwise about the origin, then reflected over the y-axis. Find the final vertices.Triangle ABC → 90° CCW rotation → reflection over y-axis