Perform sequences of transformations

A sequence of transformations consists of applying two or more translations, reflections, rotations, or dilations in a specified order, using coordinate rules or diagrams to track each point through successive images. The resulting correspondence preserves congruence under rigid motions and produces similarity under dilations; the order of transformations can affect the final image. Work is limited to standard plane transformations with familiar centers, angles, and scale factors, not abstract matrix composition or generalized transformation theory.

Detailed Explanation: Perform sequences of transformations

To perform a sequence of transformations, apply the transformations one at a time in the stated order. Use the coordinates after one transformation as the input for the next.

Example: Triangle $ABC$ has vertices

A(1,2), \quad B(3,2), \quad C(1,4).

Translate the triangle $2$ units right and $1$ unit down. Then reflect it across the $x$ -axis.

Step 1: Apply the translation

The coordinate rule for translating $2$ units right and $1$ unit down is

(x,y)\rightarrow (x+2,y-1).

Apply this rule to each vertex:

\begin{aligned} A(1,2)&\rightarrow A'(3,1),\\ B(3,2)&\rightarrow B'(5,1),\\ C(1,4)&\rightarrow C'(3,3). \end{aligned}

Step 2: Apply the reflection

The coordinate rule for reflecting across the $x$ -axis is

(x,y)\rightarrow (x,-y).

Use the translated coordinates as the starting points:

\begin{aligned} A'(3,1)&\rightarrow A''(3,-1),\\ B'(5,1)&\rightarrow B''(5,-1),\\ C'(3,3)&\rightarrow C''(3,-3). \end{aligned}

The final image has vertices

A''(3,-1), \quad B''(5,-1), \quad C''(3,-3).

Keep the vertex labels connected: $A$ becomes $A'$ , then $A''$ ; $B$ becomes $B'$ , then $B''$ ; and so on. Also, the order matters. You must complete the translation before reflecting because each transformation uses the coordinates produced by the previous step.

Learn by doing: Perform sequences of transformations

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Multiple Transformations - Grid Image & Transformations to Similar or Congruent

Earned ?