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Graph Transformations: Shifts, Compressions, and Reflections, Lecture notes of Calculus

Various graph transformations including horizontal and vertical shifts, compressions and stretches, and reflections about the x-axis or y-axis. It provides examples of different types of shifts and compressions, and demonstrates how to apply multiple transformations to a base function.

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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3.5 Graph Transformations
Transformations are when we make changes to the base functions from Section
3.4 to graph more complicated functions.
Methods
- Horizontal and vertical shifts
- Compressions and stretches
- Reflections about the x-axis or y-axis
Order of Transformations
1. Honda horizontal shifts
2. Sells stretches/compressions
3. R reflections
4. Vs vertical shifts
Vertical Translations vs. Horizontal Translations
Change f(x) or y Change x
Shifts
Vertical
Up Add to f(x) i.e.
Down Subtract
from f(x)
i.e.
Horizontal
Left Add x i.e.
Right Subtract
from x
i.e.
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Transformations are when we make changes to the base functions from Section

3.4 to graph more complicated functions.

Methods

- Horizontal and vertical shifts

- Compressions and stretches

- Reflections about the x-axis or y-axis

Order of Transformations

1. Honda horizontal shifts

2. Sells stretches/compressions

3. R reflections

4. Vs vertical shifts

Vertical Translations vs. Horizontal Translations

Change f(x) or y Change x

Shifts

Vertical

Up

Add to f(x)

i.e.

Down

Subtract

from f(x)

i.e.

Horizontal

Left Add x

i.e.

Right

Subtract

from x

i.e.

Ex. What kind of shifts are these?

Horizontal shift vertical shift

right 2 up 4

horizontal shift left 13

vertical shift down 4

Compressing and Stretching 𝒗, 𝒉 > 𝟎, multiplication

Vertical

Stretch Integer

i.e.

Compression

Fraction

i.e.

Horizontal

Stretch Fraction

i.e.

3

Compression Integer

i.e.

3

For the following graphs, provide (a) the base function, (b) which transformations

were performed, (c) the equation of the function. Hint: There are not vertical or

horizontal stretched or compressions.

  1. (a)

(b) Horizontal shift left 3

Vertical shift down 2

(c)

  1. (a)

(b) Horizontal shift right 5

Vertical shift up 3

x-axis reflection

(c) −

3

3

  1. (a)

(b) Horizontal shift right 1

x-axis reflection

(c)

OR y-axis reflection