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Grade&Level/Course:&& Algebra(1 &
Lesson/Unit&Plan&Name:& (Graphing(Piecewise(Functions &
Rationale/Lesson&Abstract:&& Students(will(graph(piecewise(defined(functions(using(three(
different(methods. &
Timeframe:&&1&to&2&Days&(60&minute&periods)&
Common&Core&Standard(s):&&FHIF.7b& Graph(square(root,(cube(root,(and(piece>wise(defined(
functions,(including(step(functions(and(absolute(value(functions.(
( Notes: ((( The(Warm>Up(is(on(page(8.(
&&&&&&&&&&&& ((( A(graphing(handout(is(provided(specifically(for(example(1((method(1)(on(page(6.(
Instructional&Resources/Materials:&& Graph(paper(or(coordinate(plane(handout,(rulers. &
Page 2 of 9 MCC@WCCUSD 03/ 25 /
Lesson:
Think-Pair-Share: Describe the graph below. Then compare it to other graphs we have seen in this class.
Example 1: Graph the piecewise function ( )
4 ,if 0
2
3 1 ,if 0
x x
x x
f x .(
Think-Pair: Predict what the graph will look like.
โ5 โ4 โ3 โ2 โ1 1 2 3 4 5 x
1
2
3
4
5
y
A piecewise function is a function represented by two or more functions, each corresponding to a part of the
domain.
A piecewise function is called piecewise because it acts differently on different โpiecesโ of the number line.
Possible Descriptions:
- The graph is a function.
- The graph is composed of part of a line
and a part of a parabola.
- The graph is not continuous, there is a
break in the graph at x = 1.
Further Discussion:
Show the equation of the graph and discuss how
it relates to the graph.
( )
6 8 ,if 1
1 ,if 1
2 x x x
x x
f x
*Notice the structure of the function, after each
function you see a restricted domain.
Method 1 (Uses 3 coordinate planes- see p. 6):
(Complete use of method 1 shown on page 7)
- Identify the two functions that create the
piecewise function.
y x
y x
- Graph each function separately.
- Identify the break between each function as given
by the domain of the piecewise function.
- Use a different color to highlight the piece of the
graph that is given by the domain of the piecewise
function.
- On the third graph, graph the piecewise function.
To the left of x = 0 and including x = 0 , graph
y = 3 x โ 1. To the right of^ x = 0 and excluding
x = 0 , graph 4
2
y = x +.
Solution:
โ5 โ4โ3 โ2 โ1 1 2 3 4 5 x
1
2
3
4
5
y
Try: Graph the function (^) ( )
( )
4 ,if 2
2
1 ,if 2
2
x x
x x
f x .(
Think-Pair: What kind of function is this? Explain.
Predict what the graph will look like.
Which method did you use?
How is this graph different from the previous graphs?
Example 3 : The function below describes the price of a movie ticket (in dollars) depending on the age of the
person (in years). Graph (^) p ( x ).
( )
8 ,if 55
11 ,if 16 55
8 ,if 0 16
x
x
x
p x
Discuss the meaning of the function:
People under 16 years of age pay $8 per ticket
People who are at least 16 year of age, but younger than 55 years old pay $11 per ticket.
People who are 55 years old or older pay $8 per ticket.
What kind of functions are y = 8 and y = 11?
Think-Pair-Share: Predict what the graph is going to look like.
- The graph is going to be in the first quadrant.
- The graph will consist of three linear functions, which are all pieces of horizontal lines.
- The horizontal axis is labeled age.
- The vertical axis is labeled price.
- The axes will not be labeled by oneโs.
Solution:
โ5 โ4 โ3โ2โ1 1 2 3 4 5 x
1
2
3
4
5
y
5 10 15 20 25 30 35 40 45 50 age( yrs)
2
4
6
8
10
12
14
16
18
20
Price( $)
Method 3 (Direct Approach):
- Graph the horizontal line y = 8 from x = 8 to
x = 16. The point ( 16 , 8 )is open.
- Graph the horizontal line y = 11 from x = 16 to
x = 55. The point (^) ( 16 , 11 )is closed and
( 55 , 11 )is open.
- Graph the horizontal line (^) y = 8 from x = 55 to
infinity. The point ( (^55) , 8 )is closed.
Try: Graph the function (^) ( )
15 ,if 100 200
10 ,if 50 100
6 ,if 0 50
x
x
x
f x.
( ( Write a scenario represented by this function.
Possible scenario:
The function describes the cost to ship packages given the weight
of the package. It cost $6 to ship packages weighing 50 pounds or less,
$10 to ship packages weighing over 50 pounds up to 100 pounds, and
$15 to ship packages weighing over 100 pounds up to 200 pounds.
Think-Pair-Share: The functions in example 3 and Try 3 are a specific type of piecewise function called a step
function. Why do you think they are called step functions?
SPECIAL STEP FUNCTIONS:
The Greatest Integer Function,
or The Floor Function
f ( x ) =! x "
The Ceiling Function
f ( x ) =! x "
Describes the largest integer not greater than x , or the
largest integer less than or equal to x.
Describes the smallest integer not less than x.
โ2 โ1 1 2 3 4 5 6 x
1
2
3
4
5
6
y
โ2 โ1 1 2 3 4 5 6 x
1
2
3
4
5
6
y
An example of a floor function is a personโs age. If
someone is 15 years and 4 months old, the person
would simply say that they are 15 years old.
An example of a ceiling function is a cell phone service.
Suppose the company charges by the number of minutes.
If you are on the phone for 2.7 minutes, the company
will charge for 3 minutes.
Solution: (
20 40 60 80 100 120 140 160 180 200 220 weight
2
4
6
8
10
12
14
16
18
20
Price
A step function is a piecewise function whose graph resembles a staircase or steps.
โ8 โ7 โ6โ5 โ4 โ3 โ2โ1 1 2 3 4 5 6 7 8 x
1
2
3
4
5
6
7
8
y
โ8 โ7 โ6โ5 โ4 โ3โ2 โ1 1 2 3 4 5 6 7 8 x
1
2
3
4
5
6
7
8
y
โ8 โ7 โ6 โ5 โ4 โ3 โ2 โ1 1 2 3 4 5 6 7 8 x
1
2
3
4
5
6
7
8
y
Graphing Piecewise Functions โ Ex. 1 Sample
Warm-Up
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y
x
Graph(the(function(
f ( x ) = x
2 (for(
x โฅ 0 .(
Given(the(function(
f ( x ) = x
2 โ 2 x + 5 ,(
find(the(following(function(values:(
a)(((
f ( 0 )(
b)((
f (โ 3 )(
c)((
f ( 4 ) (
For(the(function(graphed(to(the(left(in(
quadrant(III,(write(the(rule(for(the(
function.(
f ( x ) =(
Find(the(domain(of(the(function(shown(
in(the(graph(below.(
Domain:(
