Composition of Transformations Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Point Q(2,−3)Q(2, -3) is translated by ⟨1,5⟩\langle 1, 5 \rangle and then reflected over the xx-axis. Find the final image.

Solution

  1. 1
    Translate: Q(2,−3)→Q′(2+1, −3+5)=Q′(3,2)Q(2,-3) \to Q'(2+1,\,-3+5) = Q'(3, 2).
  2. 2
    Reflect over xx-axis: (x,y)→(x,−y)(x,y)\to(x,-y), so Q′(3,2)→Q′′(3,−2)Q'(3,2) \to Q''(3,-2).

Answer

Q′′=(3,−2)Q'' = (3, -2)
Apply each transformation to the result of the previous one. Translation shifts both coordinates by the vector components; reflection over the xx-axis negates only the yy-coordinate.

About Composition of Transformations

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.

Learn more about Composition of Transformations →

More Composition of Transformations Examples

Example 1 medium

Point [formula] is first reflected over the [formula]-axis, then translated by vector [formula]. Fin

Example 2 hard

Triangle [formula] with [formula], [formula], [formula] is rotated [formula] counterclockwise about

Example 4 medium

Describe the single transformation equivalent to two successive reflections over parallel lines [for

← All Composition of Transformations Examples Composition of Transformations Concept