6.3 Piecewise Functions
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities , and interpret solutions as viable or nonviable options in a
modeling context.
For the Board: You will be able to write and graph piecewise functions and use piecewise functions to
describe real-world situations.
Anticipatory Set:
A piecewise function is a function that is a combination of one or more functions.
The rule for a piecewise function is different for each part or piece of the function.
Instruction:
A piecewise function that is constant for each piece is called a step function.
The cost of a movie ticket is an example of a step function.
1. If you are under 13 years old you pay $5 per ticket.
2. If you are 13 years old, up to 55 years old you pay $9 per ticket.
3. If you are over 55 years old you pay $6 per ticket.
The following would be a rule for describing the price of the tickets based on ages.
5 if 0 < x < 13
12
f(x) =
9 if 13 ≤ x ≤ 55
6 if x > 55
10
Steps to graph:
1. Draw vertical lines through the end x-values.
0, 13, 55
2. Draw the lines y = 5, y = 9, and y = 55 within
their intervals.
3. Use closed dots for ≤ and open dots for <.
8
6
4
2
0
10
20
30
The domain of the function is the x-values and comes from the “if” part of the rule.
It is different for each “piece” of the function. The range is the y values. {5, 9, 6}.
40
50
60
Graphing Activity:
Practice: Graph the function.
f(x) = 4 if
-2 if
6
x ≤ -1
x > -1
4
2
-6
-4
-2
0
2
4
6
-2
-4
-6
To evaluate a piece-wise function:
1. Use the x-value to determine in which part of the function you to work.
2. Apply the rule for that part of the function to the x-value.
Open the book to page 423 and read example 2.
Example: Evaluate each piecewise function for x = -1 and x = 4. Find f(-1) and f(4).
a. f(x) = 2x + 1 if x ≤ 2
x2 – 4 if x > 2
f(-1) = 2(-1) + 1 = -1
f(4) = 42 – 4 = 12
b. g(x) = 2x if x ≤ -1
5x if x > -1
f(-1) = 2-1 = ½
f(4) = 5(4) = 20
White Board Activity:
Practice: Evaluate each piecewise function for x = -1 and x = 3. Find f(-1) and f(3).
a. f(x) = 12 if x < -3
15 if -3 ≤ x < 6
20 if x ≥ 6
f(-1) = 15
f(3) = 15
b. g(x) =
3x2 + 1 if x < 0
5x – 2 if x ≥ 0
g(-1) = 3(-1)2 + 1 = 4
g(3) = 5(3) – 2 = 13
To graph a piecewise function graph each piece of the function separately.
Steps: 1. Use vertical lines to help divide the “pieces” of the graph.
2. Taking the pieces one at a time, determine what type of function you have.
Linear, quadratic, polynomial, etc.
3. Use appropriate graphing methods based on the type of function.
Open the book to page 423 – 424 and read example 3.
Example: Graph each function.
a. g(x) = ¼ x + 3 if x < 0
-2x + 3
if x ≥ 0
1. Plot a y-intercept of 3,
then move up 1 and
right 3.
2. Plot a y-intercept of 3,
then move down 2 and
right 1.
6
4
2
-6
-4
-2
0
2
4
6
-2
-4
-6
b. f(x) =
x2 – 3
½x–3
(x – 4)2 – 1
if x < 0
if 0 ≤ x < 4
if x ≥ 4
6
4
2
1. Plot y = x2, then translate
down 3.
2. Plot a y-intercept of -3, then
move up 1 and right 2.
3. Plot y = x2, then move right 4
and down 1.
-6
-4
-2
0
2
4
6
-2
-4
-6
Graphing Activity:
Practice: Graph the function.
g(x) =
-3x
x+3
if x < 2
if x ≥ 2
6
4
1. Plot a y-intercept of 0,
then move down 3 and
right 1.
2. Plot a y-intercept of 3,
Then move up 1 right 1.
2
-6
-4
-2
0
-2
-4
-6
2
4
6
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 426 – 427 prob. 4 – 8, 11 – 19, 25, 26.
For a Grade:
Text: pgs. 426 – 427 prob. 12, 14, 16.