Reflection in the -axis and translation - order matters

See how a vertical reflection changes an earlier vertical translation.

  • Reflecting in the -axis negates the whole current output.
  • A vertical translation made before the reflection changes direction.

Example 1

Reflect, then translate

Reflect in the -axis, then translate 3 up.

−5−4−3−2−112345−8−6−4−22468xyA(−2, 0)B(0, −4)C(2, 0)A′(−2, 3)B′(0, 7)C′(2, 3)Original y=f(x)Final image
  1. 1.Reflect in the -axis.

  2. 2.Translate by .

The result is . The vertex finishes at (0, 7).

Example 2

Translate, then reflect

Translate 3 up, then reflect in the -axis.

−5−4−3−2−112345−8−6−4−22468xyA(−2, 0)B(0, −4)C(2, 0)A′(−2, −3)B′(0, 1)C′(2, −3)Original y=f(x)Final image
  1. 1.Translate by .

  2. 2.Reflect in the -axis.

The result is . The vertex finishes at (0, 1).

Remember
Keep brackets until you have multiplied or negated every term.

Your turn

5 practice questions
Question 1

Starting with , Reflect in the -axis. Then translate by . Write the resulting function and the image of . Repeat with the order reversed.

Check answer
Given order: , .Reversed order: , .

Apply every new operation to the whole current graph. A stretch or reflection also changes any translation already made in that direction.

Question 2

Starting with , Reflect in the -axis. Then translate by . Write the resulting function and the image of . Repeat with the order reversed.

Check answer
Given order: , .Reversed order: , .

Apply every new operation to the whole current graph. A stretch or reflection also changes any translation already made in that direction.

Question 3

Starting with , Reflect in the -axis. Then translate by . Write the resulting function and the image of . Repeat with the order reversed.

Check answer
Given order: , .Reversed order: , .

Apply every new operation to the whole current graph. A stretch or reflection also changes any translation already made in that direction.

Question 4

Starting with , Reflect in the -axis. Then translate by . Write the resulting function and the image of . Repeat with the order reversed.

Check answer
Given order: , .Reversed order: , .

Apply every new operation to the whole current graph. A stretch or reflection also changes any translation already made in that direction.

Question 5

Starting with , Reflect in the -axis. Then translate by . Write the resulting function and the image of . Repeat with the order reversed.

Check answer
Given order: , .Reversed order: , .

Apply every new operation to the whole current graph. A stretch or reflection also changes any translation already made in that direction.