Functions and graphsUnit 1

Transformations of - notation to graphs

Read stretches, reflections and translations from function notation.

  • Write the equation in the form .
  • One reliable order is: horizontal stretch and reflection; vertical stretch and reflection; then translation by .
  • Negative multipliers also reflect; use positive scale factors and .

Example 1

Describe an ordered sequence of transformations from to . Sketch the result.

  1. Factor the input

    Read the horizontal shift after factoring the coefficient of .

  2. Stretch first

    Horizontal scale factor and vertical scale factor .

  3. Translate last

    Move to the left.

Original:
Original graph y = f(x).
Image:
Original dashed graph and its transformed solid green image.
Original graphTransformed graph

Example 2

Describe an ordered sequence of transformations from to . Sketch the result.

  1. Stretch and reflect

    Horizontal scale factor ; vertical scale factor ; reflect in the -axis.

  2. Translate last

    Move 2 left and 1 up.

  3. Check the image

    Apply each transformation in order to the labeled points.

Original:
Original graph y = g(x).
Image:
Original dashed graph and its transformed solid green image.
Original graphTransformed graph

Remember
Factor the entire input before reading the horizontal translation. Other orders can work if the translation vector is adjusted.

Your turn

5 practice questions
Question 1

Describe a sequence of transformations from to . Find the image of .

Check answer
Equation Image

One valid order: horizontal stretch, scale factor ; vertical stretch, scale factor ; translation by .

Question 2

Describe a sequence of transformations from to . Find the image of .

Check answer
Equation Image

One valid order: horizontal stretch, scale factor ; vertical stretch, scale factor and reflection in the -axis; translation by .

Question 3

Describe a sequence of transformations from to . Find the image of .

Check answer
Equation Image

One valid order: horizontal stretch, scale factor ; vertical stretch, scale factor and reflection in the -axis; translation by .

Question 4

Describe a sequence of transformations from to . Find the image of .

Check answer
Equation Image

One valid order: horizontal stretch, scale factor ; vertical stretch, scale factor ; translation by .

Question 5

Describe a sequence of transformations from to . Find the image of .

Check answer
Equation Image

One valid order: horizontal stretch, scale factor ; vertical stretch, scale factor ; translation by .

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