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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.
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.
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
Insert Parent GraphsPage 246
Linear Parent Functions:
y = n
y = mx + b 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 =
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.
Radius (in.) , r Area (in2), A
2 12.57
4 50.27
6 113.10
8 201.06
10 316.16
Weight (oz), w Cost, C
2 $0.80
4 $1.60
6 $2.40
8 $3.20
10 $4.00
Pizza Area Sauce Cost
Graph:Ar
ea, A
Radius, rThe graph does not produce a line.
Nonlinear Function.
100
200
300
2 6 104 8
Radius (in.) , r
Area (in2), A
2 12.574 50.276 113.108 201.06
10 316.16
Pizza Area
Graph:Co
st, C
Sauce, wThe graph does produce a line.
Linear Function.
2
4
6
2 6 104 8
Sauce CostWeight (oz), w
Cost, C
2 $0.80
4 $1.60
6 $2.40
8 $3.20
10 $4.00
YOU TRY IT:
Classify the following function as linear or nonlinear.
# of cuts, n Fraction of Original Area
1 1/2
2 1/4
3 1/8
4 1/16
5 1/32
Cutting Paper
YOU TRY IT (SOLUTION):Ar
ea, A
Cuts, nThe graph does not produce a line.
Nonlinear Function.
0.2
0.4
0.6
1 3 5 2 4
# of cuts, n
Fraction of
Original Area
1 1/22 1/43 1/84 1/165 1/32
0.8
1.0
Cutting Paper
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:
+7
+19
+37
+1
+1
+1
Number of Blocks on
the edge (x)
Total number of Blocks (y)
1 1
2 8
3 27
4 ?
5 ?
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):
The equation should be f(x) = x3
f(x) = 1 1 2 8 3 27 x ?
f(x) = 4 43 64
f(x) = 5 53 125
Filling in the table:
+7
+21
+37
+1
+1
+1
Number of Blocks on
the edge (x)
Total number of Blocks (y)
1 1
2 8
3 27
4 ?
5 ?64
125
Graph:#
of T
otal
Blo
cks
# of Blocks on EdgeThe graph does not produce a straight line. It is part of the Cubic Functions.
10
20
30
2 6 104 8
40
50
60
Ordered Pair (x, y)
(1 , 1)
(2 , 3)
(3, 27)
(4, 64)
(5, __)
f(x) = x3
YOU TRY IT:
Provide the rule that represents the function:
x y0 01 - 12 - 43 - 94 ?
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:
x y0 01 - 12 - 43 - 94 ?Y = x2 Y = x3 Y = |x|
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:
The graph does not produce a straight line. It is part of the Square Functions.
Ordered Pair (x, y)
(0 , 0)
(1 , -1)
(2, -4)
(3, -9)
(4, -?)
f(x) = -x2
VIDEOS: Non-Linear functions
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/linear-nonlinear-functions-tut/v/linear-and-nonlinear-functions-example-1
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/linear-nonlinear-functions-tut/v/linear-and-nonlinear-functions-example-3
CLASS WORK:
Pages: 249 – 251
Problems: As many as it takes to master the concept.
