2-6: Special Functions

• Direct Variation: A linear function in the form y = kx, where k 0

• Constant: A linear function in the form y = b

• Identity: A linear function in the form y = x• Absolute Value: A function in the form

y = |mx + b| + c (m 0)• Greatest Integer: A function in the form y = [x]

Direct Variation Function: A linear function in the form y = kx, where k 0.

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=2x

Constant Function: A linear function in the form y = b.

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y = 3

Identity Function: A linear function in the form y = x.

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=x

Absolute Value Function: A function in the form y = |mx + b| + c (m 0)

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=|x - 2|-1

Example #1

The vertex, or minimum point, is (2, -1).

Absolute Value Function: A function of the form y = |mx + b| +c (m 0)

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y = -|x + 1|

Example #2

The vertex, or maximum point, is (-1, 0).

Absolute Value Functions

Graph y = |x| - 3 by completing the t-table:

x y-2 -1 0 1 2

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

Absolute Value Functions

Graph y = |x| - 3 by completing the t-table:

x y-2 y =|-2| -3= -1

-1 y =|-1| -3= -2 0 y =|0| -3= -3 1 y =|1| -3= -2 2 y =|2| -3= -1

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

The vertex, or minimum point, is (0, -3).

Greatest Integer Function: A function in the form y = [x]

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=[x]

Note: [x] means the greatest integer less than or equal to x. For example,

the largest integer less than or equal to -3.5 is -4.

Greatest Integer Function: A function in the form y = [x]

Graph y= [x] + 2 by completing the t-table:

x y -3

-2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 0 1

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y x y -3 y= [-3]+2=-1

-2.75 y= [-2.75]+2=-1 -2.5 y= [-2.5]+2=-1 -2.25 y= [-2.25]+2=-1 -2 y= [-2]+2 =0 -1.75 y= [-1.75]+2=0 -1.5 y= [-1.5]+2=0 -1.25 y= [-1.25]+2=0 -1 y= [-1]+2=1 0 y= [0]+2=2 1 y= [1]+2=3