Lesson 22 Matching a Transformation to a Graph

1

In the previous lesson we looked at graphical transformations, which

means taking the graph of a function and shifting it, stretching it,

compressing it, and/or reflecting it to get a new graph.

Vertical shifts: a number is added to or subtracted from the outputs of a

function, leaving the inputs of the function unaffected.

Examples: 𝑓(𝑥) + 2 is the graph of 𝑓(𝑥) shifted up 2 units

𝑓(𝑥) − 3 is the graph of 𝑓(𝑥) shifted down 3 units

Horizontal shifts: a number is added to or subtracted from the inputs of a

function, leaving the outputs of the function unaffected.

Examples: 𝑓(𝑥 + 2) is the graph of 𝑓(𝑥) shifted to the left 2 units

𝑓(𝑥 − 3) is the graph of 𝑓(𝑥) shifted to the right 3 units

Vertical stretching/compressing: the outputs of a function are multiplied

or divide by a number, leaving the inputs of the function unaffected.

Examples: 2 ∙ 𝑓(𝑥) is the graph of 𝑓(𝑥) stretched by a factor of 2

𝑓(𝑥)

3 is the graph of 𝑓(𝑥) compressed by a factor of 3

Horizontal stretching/compressing: the inputs of a function are

multiplied or divide by a number, leaving the outputs of the function

unaffected.

Examples: 𝑓(2 ∙ 𝑥) is the graph of 𝑓(𝑥) compressed by a factor of 2

𝑓 (𝑥

3) is the graph of 𝑓(𝑥) stretched by a factor of 3

Reflections: the inputs or outputs of a function are negated

Examples: −𝑓(𝑥) is the graph of 𝑓(𝑥) reflected vertically through

the 𝑥-axis (the outputs are negated)

𝑓(−𝑥) is the graph of 𝑓(𝑥) reflected horizontally through

the 𝑦-axis (the inputs are negated)

Remember that when changes take place INside the parentheses, those

changes only effect the INputs, and we do the INverse operation.

Lesson 22 Matching a Transformation to a Graph

2

The following examples contain only the transformations shifting and

reflecting. Stretching and compressing will be emphasized in the next two

sets of notes.

Example 1a: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 1. Use transformations (shifting and/or reflecting

only) to express the graph labeled 1 in terms of 𝑓.

Notice that only the outputs have changed when going from the original

function 𝑓(𝑥) to the transformation 1. Keep in mind that the tables of

inputs and outputs will not show up on homework, quizzes, or exams.

You are welcome to make your own, but they are not required.

𝑓(𝑥)

1

𝑓(𝑥) Inputs Outputs

0 1 1 2 2 4 3 8

1: 𝑦 = −𝒇(𝒙) Inputs Outputs

0 −1 1 −2 2 −4 3 −8

𝑥

Inputs

𝑓(𝑥)

Outputs

Lesson 22 Matching a Transformation to a Graph

3

Example 1b: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 2. Use transformations (shifting and/or reflecting

only) to express the graph labeled 2 in terms of 𝑓.

Think about what additional transformations have been made since the

previous example. After you determine the transformations, use ordered

pairs to check that the transformations you came up with are correct.

𝑓(𝑥) Inputs Outputs

0 1 1 2 2 4 3 8

2: 𝑦 = −𝒇(𝒙 − 𝟓) Inputs Outputs

5 −1 6 −2 7 −4 8 −8

𝑓(𝑥)

2

𝑥

Inputs

𝑓(𝑥)

Outputs

1

Lesson 22 Matching a Transformation to a Graph

4

Example 1c: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 3. Use transformations (shifting and/or reflecting

only) to express the graph labeled 3 in terms of 𝑓.

Think about what additional transformations have been made since the

previous example. After you determine the transformations, use ordered

pairs to check that the transformations you came up with are correct.

𝑓(𝑥)

3

𝑥

Inputs

𝑓(𝑥)

Outputs

𝑓(𝑥) Inputs Outputs

0 1 1 2 2 4 3 8

3: 𝑦 =

Inputs Outputs

5 −4 6 −5 7 −7 8 −11

2

Lesson 22 Matching a Transformation to a Graph

5

On Examples 1a, 1b, and 1c the transformations went in order, where each

successive transformation built on the previous one. There will be a

couple of homework problems like these, but there will also be homework

problems where each transformation is independent of the previous ones.

That is what we will see on the remaining examples, which will also

contain multiple transformations taking place at once.

Example 2: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 1. Use transformations (shifting and/or reflecting

only) to express the graph labeled 1 in terms of 𝑓.

𝑓(𝑥)

1

𝑓(𝑥) Inputs Outputs

0 0 1 1 4 2 9 3

1: 𝑦 =

Inputs Outputs

−2 4 −1 5 2 6 7 7

𝑥

Inputs

𝑓(𝑥)

Outputs

Lesson 22 Matching a Transformation to a Graph

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Example 3: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 2. Use transformations (shifting and/or reflecting

only) to express the graph labeled 2 in terms of 𝑓.

Once again, completing the tables of inputs and outputs is optional. These

tables will not show up anywhere else but in these notes. You are

welcome to make your own, but they are not required.

𝑓(𝑥)

2

𝑥

Inputs

𝑓(𝑥)

Outputs

𝑓(𝑥) Inputs Outputs

0 0 1 1 4 2 9 3

2: 𝑦 =

Inputs Outputs

0 1 −1 2 −4 3 −9 4

Lesson 22 Matching a Transformation to a Graph

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Example 4: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 3. Use transformations (shifting and/or reflecting

only) to express the graph labeled 3 in terms of 𝑓.

𝑓(𝑥)

3

𝑥

Inputs

𝑓(𝑥)

Outputs

𝑓(𝑥) Inputs Outputs

0 0 1 1 4 2 9 3

3: 𝑦 =

Inputs Outputs

2 −1 3 −2 6 −3

11 −4

Lesson 22 Matching a Transformation to a Graph

8

Example 5: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 1. Use transformations (shifting and/or reflecting

only) to express the graph labeled 1 in terms of 𝑓.

𝑓(𝑥)

1

𝑓(𝑥)

Inputs Outputs

−5 6 −2 3 1 6 4 3

7 6

1: 𝑦 =

Inputs Outputs

−9 −1 −6 −4 −3 −1 0 −4

3 −1

𝑓(𝑥)

Outputs

𝑥

Inputs

Lesson 22 Matching a Transformation to a Graph

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Example 6: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 2. Use transformations (shifting and/or reflecting

only) to express the graph labeled 2 in terms of 𝑓.

𝑓(𝑥)

2

𝑓(𝑥)

Inputs Outputs

−5 6 −2 3 1 6 4 3 7 6

2: 𝑦 =

Inputs Outputs

−9 1 −6 4 −3 1 0 4 3 1

𝑓(𝑥)

Outputs

𝑥

Inputs

Lesson 22 Matching a Transformation to a Graph

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Example 7: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 1. Use transformations (shifting and/or reflecting

only) to express the graph labeled 1 in terms of 𝑓. (there is more than one

possible answer for graph 𝟏)

𝑓(𝑥)

1

1: 𝑦 =

Inputs Outputs

−4 −4 −2 −2 0 −4 2 −6 4 −4

𝑥

Inputs

𝑓(𝑥)

Outputs

𝑓(𝑥)

Inputs Outputs

1 2 3 4 5 2 7 0 9 2

Lesson 22 Matching a Transformation to a Graph

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Example 8: The graph of a function 𝑓 is shown below in red, along with

another graph labeled 1. Use transformations (shifting and/or reflecting

only) to express the graph labeled 1 in terms of 𝑓.

𝑓(𝑥) 2

2: 𝑦 = 𝒇(−𝒙)

Inputs Outputs

−1 2 −3 4 −5 2 −7 0 −9 2

𝑥

Inputs

𝑓(𝑥)

Outputs

𝑓(𝑥)

Inputs Outputs

1 2 3 4 5 2 7 0 9 2

Lesson 22 Matching a Transformation to a Graph

12

Answers to Examples:

1a. 1: 𝑦 = −𝑓(𝑥)

1b. 2: 𝑦 = −𝑓(𝑥 − 5)

1c. 3: 𝑦 = −𝑓(𝑥 − 5) − 3

2. 1: 𝑦 = 𝑓(𝑥 + 2) + 4

3. 2: 𝑦 = 𝑓(−𝑥) + 1

4. 3: 𝑦 = −𝑓(𝑥 − 2) − 1

5. 1: 𝑦 = ℎ(𝑥 + 4) − 7

6. 2: 𝑦 = −ℎ(𝑥 + 4) + 7 ; 7. 1: 𝑦 = 𝑗(𝑥 + 5) − 6 ;

8. 1: 𝑦 = 𝑓(−𝑥)