FunctionsAI HL

Vertical combinations from function notation

Read and animate transformations outside a function.

  • For , stretch vertically by , reflect in the -axis if , then translate vertically by .
  • Vertical operations change the -coordinates. The -coordinates stay the same.

Example 1

Sketch .

−4−3−2−11234−12−10−8−6−4−224681012xyA(−2, 0)B(0, −4)C(2, 0)A′(−2, 3)B′(0, −5)C′(2, 3)Original y=f(x)Final image

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

  2. 2.Translate by .

Double each original -coordinate, then add 3. The vertex moves from (0, −4) to (0, −8), then to (0, −5).

Example 2

Sketch .

−4−3−2−11234−10−8−6−4−2246810xyA(−2, 0)B(0, −4)C(2, 0)A′(−2, 4)B′(0, 6)C′(2, 4)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 .

Halve the vertical distances, reflect in the -axis, then move 4 up.

Remember
An outside multiplier acts on the output. An outside addition is a vertical 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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