4.3 Patterns and Non-Linear Functions:
Nonlinear: Is a function whose graph is
not a line or part of a line.
Family of Functions: is a group of
functions with common characteristics.
Parent Function: is the simplest function
of a family of functions.
GOAL:
One of the ways to classify a function is to see a
visual representation of it.
A graph would show the function’s characteristics
and would provide us with info to classify it as:
1. Linear : Graph is nonvertical line or
part of a nonveritcal line
2. Nonlinear Function: Graph whose part
is not a line or part of a line.
1. Linear : Graph is nonvertical line or
part of a nonveritcal line
Insert Parent Graphs
Page 246
Linear Parent Functions:
y = mx + b
y=n
y = -mx + b
x=n
2. Nonlinear Function: Graph whose part
is not a line or part of a line.
Nonlinear Parent Functions:
y = x2
y = x3
y=|x|
Nonlinear Parent Functions:
y= 𝒙
y = ax if a> 1
y = ax 0<a<1
LINEAR OR NONLINEAR?:
The area A, of a pizza is a function of its radius r,
in inches. The cost C, in dollars, of the sauce for a
pizza is a function of the weight w, in ounces, of
sauce used. Graph this functions and classify as
linear on nonlinear.
Sauce Cost
Pizza Area
Radius (in.) , r
Area (in2), A
Weight (oz), w
Cost, C
2
12.57
2
$0.80
4
50.27
4
$1.60
6
113.10
6
$2.40
8
201.06
8
$3.20
10
316.16
10
$4.00
Area, A
Graph:
Pizza Area
300
200
Radius
(in.) , r
2
4
6
Area
(in2), A
12.57
50.27
113.10
8
201.06
10
316.16
100
2
4
6
8
10
Radius, r
The graph does not produce a line.
Nonlinear Function.
Cost, C
Graph:
Sauce Cost
6
4
Weight
(oz), w
2
Cost, C
4
$1.60
6
$2.40
8
$3.20
10
$4.00
2
2
4
6
8
10
Sauce, w
The graph does produce a line.
Linear Function.
$0.80
YOU TRY IT:
Classify the following function as
linear or nonlinear.
Cutting Paper
# of cuts, n
1
Fraction of
Original Area
1/2
2
1/4
3
1/8
4
1/16
5
1/32
YOU TRY IT (SOLUTION):
# of cuts, Fraction
n
of
Original
Area
1
1/2
2
1/4
1.0
Area, A
Cutting Paper
0.8
0.6
3
4
5
0.4
0.2
1/8
1/16
1/32
1 2 3 4 5
Cuts, n
The graph does not produce a line.
Nonlinear Function.
REPRESENTING PATTERNS AND NONLINEAR
FUNCTIONS:
Data from a table can be scrutinize to
see if there is any relation or pattern
that can help us find what is missing,
or complete a table.
Ex: What is the pattern we can use to
complete the table?
Number of
Blocks on the
edge (x)
Total number
of Blocks (y)
Ordered Pair
(x, y)
1
1
(1 , 1)
2
8
(2 , 3)
3
27
(3, 27)
4
?
?
5
?
?
To answer the question we must take a look
at what is happening in the table or the
figures:
Number of
Total
Blocks on number of
the edge (x) Blocks (y)
+1
+1
+1
1
1
2
8
3
27
4
?
5
?
+7
+19
+37
Notice: although the x increases by 1, the
number of blocks is not constant anymore.
Taking the info to consideration, we can
concentrate on how we manipulate the x to
get f(x):
f(x) = 1  1
28
3  27
x ?
3
The equation should be f(x) = x
f(x) = 4  43 64
f(x) = 5  53 125
Filling in the table:
Number of
Total
Blocks on number of
the edge (x) Blocks (y)
+1
+1
+1
1
1
2
8
3
27
4
?
64
5
?
125
+7
+21
+37
# of Total Blocks
Graph:
f(x) = x3
Ordered Pair
(x, y)
(1 , 1)
60
50
(2 , 3)
40
(3, 27)
30
(4, 64)
20
(5, __)
10
2
4
6
8
10
# of Blocks on Edge
The graph does not produce a straight
line. It is part of the Cubic Functions.
YOU TRY IT:
Provide the rule that represents
the function:
x
0
1
2
3
4
y
0
-1
-4
-9
?
YOU TRY IT:(SOLUTION)
Looking at the data on the y
values we can see that it is not
linear, thus we have the
following equations to choose
from:
Y = x2
Y = x3 Y = |x|
x
0
1
2
3
4
y
0
-1
-4
-9
?
Notice also that the values are negative, thus:
Y = -x2 Y = -x3 Y = -|x|
Plugging in numbers we see that y = - x2
should be our equation.
Graph:
Ordered Pair
(x, y)
(0 , 0)
(1 , -1)
(2, -4)
(3, -9)
(4, -?)
f(x) = -x2
The graph does not produce a straight
line. It is part of the Square Functions.
VIDEOS:
Non-Linear
functions
https://www.khanacademy.org/math/algebra/line
ar-equations-and-inequalitie/linear-nonlinearfunctions-tut/v/linear-and-nonlinear-functionsexample-1
https://www.khanacademy.org/math/algebra/line
ar-equations-and-inequalitie/linear-nonlinearfunctions-tut/v/linear-and-nonlinear-functionsexample-3
CLASS WORK:
Pages: 249 – 251
Problems: As many as it takes
to master the
concept.