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2.6 Special Functions

Explore piecewise functions by graphing different equations over various parts of the graph. Learn how to transition smoothly between functions to represent the unique characteristics of each segment. Understand how to plot points accurately to visualize the different functions.

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2.6 Special Functions

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

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

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

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

  5. 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)

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

  7. CW 2-5 Page 104 #8-11

  8. 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

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

  10. 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.

  11. 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?

  12. 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)

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

  14. Graph the piecewise function

  15. Write the Piecewise-defined Function

  16. CW 2-6 (Cont.) Page 105 # 12-16

  17. HW 2-6 • #’s 17-19, 24-30

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