The document is a tutorial on absolute value functions authored by Jeffrey Bivin of Lake Zurich High School. It covers translating and graphing absolute value functions, writing absolute value functions as compound functions using vertices and slopes, and defining absolute value functions from their definitions. Examples of various absolute value functions are shown along with their graphs and expressions as compound functions.

1. Absolute Value
Functions
Graphs
and
Compound Functions
By: Jeffrey Bivin
Lake Zurich High School
jeff.bivin@lz95.org
Last Updated: November 15, 2006
Jeff Bivin -- LZHS

2. Absolute Value Functions
Select the desired MENU option below
Graphs
1. Translations
2. Quick Graphs
3. Graphing Inequalities
Writing as Compound Functions
4. Using the vertex and slopes
5. From Definition
Jeff Bivin -- LZHS

3. Absolute Value
Functions
Translations
Jeff Bivin -- LZHS

4. y =|x|
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5. y = |x+1| + 2
Jeff Bivin -- LZHS

6. y = |2x+5| - 4
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7. y = 2 |x - 3| + 1
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8. y = 3 |2x - 3| - 4
Jeff Bivin -- LZHS

9. Absolute Value
Functions
Quick Graphs
Jeff Bivin -- LZHS

10. y =|x|
Jeff Bivin -- LZHS

11. y = |x+1| + 2
Jeff Bivin -- LZHS

12. y = |2x+5| - 4
Jeff Bivin -- LZHS

13. y = 2 |x - 3| + 1
Jeff Bivin -- LZHS

14. y = 3 |2x - 3| - 4
Jeff Bivin -- LZHS

15. Absolute Value
Functions
Graphing Inequalities
Jeff Bivin -- LZHS

16. y ≤ |2x+5| - 4
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17. y > -2 |x - 3| + 1
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18. y ≥ |x+1| + 2
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19. y < -2 |3x + 4| + 1
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20. y ≥ 3|-2x + 8| - 1
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21. Absolute Value
Functions
Writing as
Compound Functions
using
Vertex and slopes
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22. x+2=0
y=|x+2|
x = −2 Slopes of sides
x = −2 x < −2 x ≥ −2
m=±1
Vertex (-2, 0) m = −1 m =1
left side right side
( − 2, 0)
y − 0 = −1( x − ( − 2 ) ) y − 0 = 1( x − (−2) )
y = −1( x + 2 ) y = 1( x + 2 )
y = −x − 2 y=x+2
x + 2 , [ − 2,+∞)
y=
Jeff Bivin -- LZHS
− x − 2 , ( − ∞, − 2)

23. y = 3| x - 4 |
x−4=0 x=4 Slopes of sides
x=4 x<4 x≥4
m=±3
Vertex (4, 0) m = −3 m=3
left side right side
( 4, 0)
y − 0 = − 3( x − 4 ) y − 0 = 3( x − 4 )
y = − 3 x + 12 y = 3 x − 12
3 x −12 , [ 4, + ∞)
y=
Jeff Bivin -- LZHS
− 3 x + 12 , ( − ∞, 4)

24. y = -2| x + 1 |
x +1= 0 ( −1, 0) Slopes of sides
x = −1 m=±2
m=2 m = −2
Vertex ( −1, 0 )
x < −1 x ≥ −1
left side x = −1 right side
y − 0 = 2( x − (−1) ) y − 0 = − 2( x − (−1) )
y = 2( x + 1) y = − 2( x + 1)
y = 2x + 2 y = − 2x − 2
− 2 x − 2 , [ −1, + ∞)
y=
Jeff Bivin -- LZHS
 2x + 2 , ( − ∞, −1)

25. y = 2| x - 5 | +3
x −5=0 x =5 Slopes of sides
x =5 x<5 x≥5
m=±2
Vertex (5, 3) m = −2 m=2
left side right side
( 5, 3)
y − 3 = − 2( x − 5) y − 3 = 2( x − 5)
y − 3 = − 2 x + 10 y − 3 = 2 x − 10
y = − 2 x + 13 y = 2x − 7
− 2 x + 13 , ( − ∞, 5]
y=
Jeff Bivin -- LZHS
2 x − 7 , ( 5, + ∞)

26. y = -4| 2x - 5 | + 7
2x − 5 = 0 ( 5 , 7)
2 Slopes of sides
2x = 5 m=±8
x=5 2
m =8 m = −8
Vertex ( 5
2 , 7) x< 5
2
x≥ 5
2
left side x= 5 right side
2
y − 7 = 8( x − 5 )
2 y − 7 = − 8( x − 5 )
2
y − 7 = 8 x − 20 y − 7 = − 8 x + 20
y = 8 x − 13 y = − 8 x + 27
− 8 x + 27 , ( 5 , + ∞)
2
y=
Jeff Bivin -- LZHS
 8 x −13 , ( − ∞, 2 ]
5

27. Absolute Value
Functions
Writing as
Compound Functions
From Definition
Jeff Bivin -- LZHS

28. x = −2
y=|x+2|
If x > -2 If x < -2
x+2=0
y = ( x + 2) y = − ( x + 2)
x = −2
y=x+2 y = −x − 2
x + 2 , x ≥ −2
y=
− x − 2 , x < − 2
Jeff Bivin -- LZHS

29. x=4
y = 3| x - 4 |
If x > 4 If x < 4
x−4=0
y = 3( x − 4 ) y = − 3( x − 4 )
x=4
y = 3 x − 12 y = − 3 x + 12
3 x − 12 , x≥4
y=
− 3 x + 12 , x < 4
Jeff Bivin -- LZHS

30. y = -2| x + 1 |
x = −2
x +1= 0 If x > -1 If x < -1
x = −1 y = − 2( x + 1) y = 2( x + 1)
y = − 2x − 2 y = 2x + 2
− 2 x − 2 , x ≥ −1
y=
2 x + 2 , x< 4
Jeff Bivin -- LZHS

31. y = -3| 2x + 3 | + 1
x= −3
2
If x > −3 If x < −3
2
2x + 3 = 0 2
2x = − 3 y = − 3( 2 x + 3) +1 y = − (−3)( 2 x + 3) +1
x = −3 y = − 6x − 9 + 1 y = 3( 2 x + 3) +1
2
y = − 6x − 8 y = 6 x + 9 +1
y = 6 x + 10
− 6 x − 8 , x ≥ −3
2
y=
 6 x + 10 , x <
−3
2
Jeff Bivin -- LZHS