2-6: Absolute Value Functions Absolute Value: A function in the form y = a | x - h| + k a = concavity (if a is (+) the graph opens up, if a is (-) the graph opens down h = shift in the graph opposite of the sign in the absolute value left or right k = shift in the graph up or down in the same direction as the sign

Absolute Value Function: A function in the form y = |mx + b| + c (m 0) 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) 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 1 2

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

Try these -4│x + 6 │ - 3 6. -3│x + 6 │ - 3

Direct Variation Function: A linear function in the form y = kx, where k 0. 2 4 6 –2 –4 –6 x y y=2x

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

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

Greatest Integer Function: A function in the form 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. 2 4 6 –2 –4 –6 x y y=[x]

Greatest Integer Function: A function in the form y = [x] Graph y= [x] + 2 by completing the t-table: 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 x y -3 -2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 1 2 4 6 –2 –4 –6 x y