4.3 Patterns and Non-Linear Functions: Nonlinear: Is a function whose graph is not a line or part of...

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.