Perform sequences of transformations

Detailed Explanation: Perform sequences of transformations

A sequence of transformations means applying transformations one at a time in the stated order. The result of the first transformation becomes the input for the second.

Example

Triangle $ABC$ has vertices

Perform these transformations in order:

1. Translate the triangle $3$ units left and $1$ unit up.
2. Reflect the image over the $x$-axis.

Step 1: Translate

Moving $3$ units left changes each $x$-coordinate by $-3$. Moving $1$ unit up changes each $y$-coordinate by $+1$.

Apply this to each vertex:

• $A(1,2) \rightarrow A'(-2,3)$
• $B(4,2) \rightarrow B'(1,3)$
• $C(1,5) \rightarrow C'(-2,6)$

The translated triangle is $A'B'C'$.

Step 2: Reflect over the $x$-axis

When a point is reflected over the $x$-axis, its $x$-coordinate stays the same and the sign of its $y$-coordinate changes.

Apply the reflection to the translated points:

• $A'(-2,3) \rightarrow A''(-2,-3)$
• $B'(1,3) \rightarrow B''(1,-3)$
• $C'(-2,6) \rightarrow C''(-2,-6)$

The final image has vertices

Remember to use the image from Step 1 as the input for Step 2. The order matters. For example, reflecting $A(1,2)$ first gives $(1,-2)$, and then translating gives $(-2,-1)$, which is different from the final point $A''(-2,-3)$.

The triangle’s size and shape stay the same because translations and reflections preserve lengths and angle measures.