Essential Question: What ways can the graph of the parent function y = |x| be transformed?

A parent function is a function with a certain shape that has the simplest rule for that shape.◦ Other equations will resemble the parent function in both

formula as well as shape.◦ The parent function for an absolute value graph is y = |x|

A translation is a shift of the parent function either horizontally (left/right), vertically (up/down) or both.◦ It results in a graph with the same size and shape, but in

a different location The translations we will cover here today apply to

both y = |x| as well as y = -|x|

Vertical translations◦ For a positive number k, y = |x| + k, is a vertical

translation If k is a positive number, shift the graph up k units If k is a negative number, shift the graph down k units Vertical translations work as expected

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

parent function: y = |x|

y = |x| - 3vertical shift down 3 units

y = |x| + 2vertical shift up 2 units

Your Turn: Describe the translation:

◦ y = |x| + 1

◦ y = |x| - 2

Write an equation for the translation of:◦ y = |x| up 8 units

◦ y = |x| down ½ unit

y = |x| shifted up 1 unit

y = |x| shifted down 2 units

y = |x| + 8

y = |x| - ½

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Horizontal translations◦ For a positive number k, y = |x + h|, is a horizontal

translation (horizontal change is on the inside) If h is a positive number, shift the graph left h units If h is a negative number, shift the graph right h units Horizontal translations work opposite of expected “HI HO”

parent function: y = |x|

y = |x + 3|horizontal shift left 3 units

y = |x – 2|horizontal shift right 2 units

Your Turn: Describe the translation:

◦ y = |x + 3|

◦ y = |x – 1|

Write an equation for the translation of:◦ y = |x| right 2 units

◦ y = |x| left 4 units

y = |x| shifted left 3 units

y = |x| shifted right 1 unit

y = |x – 2|

y = |x + 4|

Putting it all together◦ Describe the translation of f(x) = |x|

y = |x + 2| + 4

y = |x – 6| + 5

Horizontal shift 2 units leftVertical shift 4 units up

Horizontal shift 6 units rightVertical shift 5 units up

2 4 6 8 10–2–4–6–8–10 x

2

4

6

8

10

–2

–4

–6

–8

–10

y

2 4 6 8 10–2–4–6–8–10 x

2

4

6

8

10

–2

–4

–6

–8

–10

y

2 4 6 8 10–2–4–6–8–10 x

2

4

6

8

10

–2

–4

–6

–8

–10

y

Assignment◦ Page 97 – 98◦ Problems 1 – 14 & 29 – 34◦ All problems

Instead of graphing (1 – 4, 8 – 11, 29 – 34), simply list the transformations that occur (e.g. “vertical shift down 2 units”) like what we did earlier