piecewise functions. piecewise function objectives: 1.evaluate piecewise functions 2.graph piecewise...
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PIECEWISE FUNCTIONS
PIECEWISE FUNCTION
Objectives:1. Evaluate piecewise functions2. Graph Piecewise Functions3. Graph Step Functions
Vocabulary: Piecewise Functions, Step Functions
• Up to now, we’ve been looking at functions represented by a single equation.
• In real life, however, functions may be represented by a combination of equations, each corresponding to a part of the domain.
• These are called piecewise functions.
PIECEWISE FUNCTION
PIECEWISE FUNCTION
All piecewise functions start with:
f x
PIECEWISE FUNCTION
Since this one is in three parts, it will have three lines within f(x)
,
,
,
if x
f x if x
if x
PIECEWISE FUNCTION
This graph already tells us, the equation for each branch.The part we need to focus on is x = 2, where the graph splits.
2 , 2
6 , 2
10 , 2 6
y x if x
f x if x
x if x and x
EVALUATE F(X) WHEN X=0, X=2, X=4
2 ,12
2 ,2)(
xifx
xifxxf
First you have to figure out which equation to use.You NEVER use both!Then evaluate using that equation. Now graph your equation.
STEP FUNCTIONS
Step Function: Type of Piecewise function. The output remains constant with in each branch and changes in value from one interval to the next.
43,432,321,210,1
)(
xifxifxifxif
xf
STEP FUNCTIONS
GRAPH :
01,412,323,234,1
)(
xifxifxifxif
xf
GRAPH:
312 2
2
, 1( )
( 2) 1, 1
x if xf x
x if x
1. Make a table for several values of each function2. Graph those values3. Remember your endpoints! Open or closed?
GRAPH:
2 ,1
2 ,3
2
3
2
)(xifx
xifxxf
WORKBOOK PAGE 12 #1-14
Pg. 13 shows how to put piecewise functions into your graphing calculator (TI-83 and TI-84)
EXAMPLE WITH DIFFERENT BOUNDARY POINTS
2 ,1
2 ,3
2
3
2
)(xifx
xifxxf
1. Make a table with a few points for each function.
2. ALWAYS plug in the boundary point(s) into BOTH functions (show on your table)
3. If the y-value for the boundary point(s) is the same in both functions, then just plot the point and keep going
4. If the y-value for the boundary point(s) is different in each function, then you plot both points- one should be open circled, one should be close circled
EXAMPLE WHERE THE BOUNDARY POINTS ARE THE SAME
1.) 2.)
GRAPH : STEP FUNCTION
It costs $1.40 for the first minute of a phone call to Paris, France, and $0.80 for each additional minute or fraction thereof.
Draw a graph of a step function that models this cost. MAKE A TABLE FIRST!
WORKBOOK PAGE 14-15
Practice Sess20 minutes
BELLRINGER:WRITE THE PIECEWISE FUNCTION FOR EACH PICTURE
WRITE AN EQUATION FOR THE GIVEN GRAPH
Look for at least 2 points on each part of the graph
• For linear functions, find the slope of the line using those 2 points
Put into point-slope form
• Keep in mind that horizontal lines are f(x)= #
WHER ARE YOUR BOUNDARIES AT??
WRITE AN EQUATION FOR THE GIVEN GRAPH
(2,2)(-3,0)
HOMEWORK!
• Workbook pg. 16-19 all