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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
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.
|][|)( xxf
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) =
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
Now lets look at 0.5,1.5, -0.5, -1.5
f(-1.5) = -2 It is the same as f( - 2) = -2f(-0.5) = - 1 f( - 1) = -1f(0.5) = 0 f(0) = 0f(1.5) = 1 f(1) = 1
So between 0 and almost 1 it equal 0f(0.999999999999999999999) = 0
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?
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)
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.
Absolute Value function: f(x) = | x |
Let plot some pointsx f(x)
0 01 1
-1 - 12 2
-2 - 2Shape V for victory
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)
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
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xxxf
Piecewise Functions
2 is where the graph changes.
Less then 2 uses 3x + 2
Greater then 2 uses x - 3
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xxxf
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.
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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?
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xxxf
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 = -1Graphs a filled in point
at (2, -1)
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xxxf
Piecewise Functions
So put in an x where the domain changes and one point higher
and lower (2, 8)
(2, -1)