FunctionsAI HL

Horizontal combinations from function notation

Read and animate transformations in the input of a function.

  • For , first stretch horizontally by . Reflect in the -axis if , then translate horizontally.
  • Horizontal operations change the -coordinates. The -coordinates stay the same.

Example 1

Sketch . First write .

−8−7−6−5−4−3−2−112345678−6−5−4−3−2−1123456xyA(−4, 0)B(0, 0)C(6, 0)A′(1, 0)B′(3, 0)C′(6, 0)Original y=f(x)Final image

  1. 1.Stretch parallel to the -axis by scale factor .

  2. 2.Translate by .

Halve the original -coordinates, then add 3. Follow each labelled intercept.

Example 2

Sketch . First write .

−14−12−10−8−6−4−22468101214−6−5−4−3−2−1123456xyA(−4, 0)B(0, 0)C(6, 0)A′(10, 0)B′(2, 0)C′(−10, 0)Original y=f(x)Final image

  1. 1.Stretch parallel to the -axis by scale factor .

  2. 2.Reflect in the -axis.

  3. 3.Translate by .

Double the horizontal distances, reflect in the -axis, then move 2 right.

Remember
Do not read the constant in an unfactored input as the translation.

Your turn

5 practice questions
Question 1

Fully describe one sequence mapping to . Then find the image of .

Check answer
Step 1: Stretch parallel to the -axis by scale factor . .Step 2: Translate by . .Image .

Horizontal operations act on the current -coordinates; vertical operations act on the current -coordinates.

Question 2

Fully describe one sequence mapping to . Then find the image of .

Check answer
Step 1: Stretch parallel to the -axis by scale factor . .Step 2: Reflect in the -axis. .Step 3: Translate by . .Image .

Horizontal operations act on the current -coordinates; vertical operations act on the current -coordinates.

Question 3

Fully describe one sequence mapping to . Then find the image of .

Check answer
Step 1: Stretch parallel to the -axis by scale factor . .Step 2: Translate by . .Image .

Horizontal operations act on the current -coordinates; vertical operations act on the current -coordinates.

Question 4

Fully describe one sequence mapping to . Then find the image of .

Check answer
Step 1: Stretch parallel to the -axis by scale factor . .Step 2: Translate by . .Image .

Horizontal operations act on the current -coordinates; vertical operations act on the current -coordinates.

Question 5

Fully describe one sequence mapping to . Then find the image of .

Check answer
Step 1: Stretch parallel to the -axis by scale factor . .Step 2: Reflect in the -axis. .Step 3: Translate by . .Image .

Horizontal operations act on the current -coordinates; vertical operations act on the current -coordinates.

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