1. 2.6 Special Functions Step functions Greatest integer functions Piecewise functions

2. Step functions: A range of values give a certain outcome. Your grades are based on a step function. Grade Scale Letter grades have the following percentage equivalents: A+ 99-100 B+ 91-92 C+ 83-84 D+ 75-76 F 0- 69 A 96-98 B 88-90 C 80-82 D 72-74 A- 93-95 B- 85-87 C- 77-79 D- 70-71

3. Greatest Integer Function is a step function The function is written as It is not an absolute value. The function rounds down to the last integer.

4. Find the value of a number in the Greatest Integer function f(x) =[| x |] f(2.7) = 2 f(0.8) = 0 f(- 3.4) = - 4 It rounds down to the last integer Find the value f( 5.8) = f(⅛) = f(- ⅜) =

5. A step function graph

6. How to graph a step function; f(x)= [| x |] Find the values of x =.., -2, -1, 0, 1, 2, …… f(-2) = -2 f(-1) = - 1 f(0) = 0 f(1) = 1 f(2) = 2

7. Now lets look at 0.5,1.5, -0.5, -1.5 f(-1.5) = -2 It is the same as f( - 2) = -2 f(-0.5) = - 1 f( - 1) = -1 f(0.5) = 0 f(0) = 0 f(1.5) = 1 f(1) = 1 So between 0 and almost 1 it equal 0 f(0.999999999999999999999) = 0

8. How to show all those number equal 0 A close circle at (0, 0) and an open circle at (1, 0). (1, 0) What happens when x = 1?

9. How to show all those number equal 0 A close circle at (0, 0) and an open circle at (1, 0). (1,1) (2,1) (1, 0) What happens when x = 1? It jumps to (1,1)

10. Is the step only one unit long? It will be in f(x) = [| x |]. Here is how I graph them. Find the fill in circles. Draw line segments ending in a open circle.

11. The Constant Function Here f(x) is equal to one number. f(x) = 3. Have we seen this before?

12. Absolute Value function: f(x) = | x | Let plot some points xf(x) 0 0 1 -1 1 2 -2 2

13. Absolute Value function: f(x) = | x | Let plot some points xf(x) 0 0 1 -1 - 1 2 -2 - 2 Shape V for victory

14. Lets graph f(x) = - | x – 3| x- | x – 3|f(x) 0- | 0 – 3| = - | - 3| - 3(0, - 3) 1 - | 1 – 3| = - | - 2|- 2(1, - 2) 2 - | 2 – 3| = - | - 1|- 1(2, - 1) 3 - | 3 – 3| = - | - 0| 0 (3, 0) 4 - | 4 – 3| = - | 1 | - 1(4, - 1) 5 - | 5 – 3| = - | 2 | - 2(5, - 2)

15. Lets graph f(x) = - | x – 3| (0, - 3) (1, - 2) (2, - 1) (3, 0) (4, - 1) (5, - 2)

16. Homework part 1 of section 2.6 Page 94 #24 – 35

17. Piecewise Functions Graphing different functions over different parts of the graph. One part tells you what to graph, then where to graph it. What to graph Where to graph

18. Piecewise Functions 2 is where the graph changes. Less then 2 uses 3x + 2 Greater then 2 uses x - 3

19. We can and should put in a few x into the function If f(0) we use 3x + 2, then 3(0) + 2 = 2 If f(3) we use x – 3, then (3) – 3 = 0 The input tell us what function to use.

20. We can and should put in a few x into the function If we want to find out what f(2) = we use both equations, but leaving an open space on the graph for the point in the function 3x + 2. Why?

21. We can and should put in a few x into the function f(2) in 3x + 2; 3(2) + 2 = 8 Graph an open point at (2,8). f(2) in x – 3 (2) – 3 = -1 Graphs a filled in point at (2, -1)

22. Piecewise Functions So put in an x where the domain changes and one point higher and lower (2, 8) (2, -1)

23. Graph the piecewise function

24. Homework Page 93 – 94 # 15 – 20 # 36 – 41, 44