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