FunctionsAA SL

Transformations of - notation to graphs

Combine vertical operations with one horizontal translation or stretch.

  • A vertical operation changes outputs; a horizontal operation changes inputs.
  • For , move horizontally, multiply each output by , then add .
  • For , divide each input coordinate by , multiply each output by , then add .
  • Follow the stated order. A later vertical stretch or reflection also changes an earlier vertical translation.

Example 1

Translate 2 right and 1 up, then stretch vertically by scale factor 2.

−6−5−4−3−2−1123456−10−8−6−4−2246810xyA(−2, 0)B(0, −4)C(2, 0)A′(0, 2)B′(2, −6)C′(4, 2)Original y=f(x)Final image

  1. 1.Translate by .

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

Example 2

Stretch horizontally by scale factor one half, reflect in the -axis, then translate 3 up.

−4−3−2−11234−10−8−6−4−2246810xyA(−2, 0)B(0, −4)C(2, 0)A′(−1, 3)B′(0, 7)C′(1, 3)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 .

Remember
AA SL includes composite transformations. This card uses either a horizontal translation or a horizontal scale; transformations of the form with both are AA HL content.

Your turn

Question 1

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

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

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

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