1. 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.

2. GOAL:

3. 1. Linear : Graph is nonvertical line or part of a nonveritcal line
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.

4. 1. Linear : Graph is nonvertical line or part of a nonveritcal line
Insert Parent Graphs Page 246

5. Linear Parent Functions:
y = mx + b
y = -mx + b
y = n
x = n

6. 2. Nonlinear Function: Graph whose part is not a line or part of a line.

7. Nonlinear Parent Functions:
y = x2
y = x3
y = | x |

8. Nonlinear Parent Functions:
y = 𝒙
y = ax if a> 1
y = ax 0<a<1

9. 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.
Pizza Area
Sauce Cost
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

10. The graph does not produce a line. Nonlinear Function.
Pizza Area
Radius (in.) , r
Area (in2), A 2
12.57 4
50.27 6
113.10 8
201.06 10
316.16
Area, A
300
200
100 2 4 6 8 10
Radius, r
The graph does not produce a line. Nonlinear Function.

11. The graph does produce a line. Linear Function.
Sauce Cost
Weight (oz), w
Cost, C 2
$0.80 4
$1.60 6
$2.40 8
$3.20 10
$4.00
Cost, C 6 4 2 2 4 6 8 10
Sauce, w
The graph does produce a line. Linear Function.

12. Fraction of Original Area
YOU TRY IT:
Classify the following function as linear or nonlinear.
Cutting Paper
# of cuts, n
Fraction of Original Area 1
1/2 2
1/4 3
1/8 4
1/16 5
1/32

13. The graph does not produce a line. Nonlinear Function.
YOU TRY IT (SOLUTION):
Cutting Paper
# of cuts, n
Fraction of Original Area 1
1/2 2
1/4 3
1/8 4
1/16 5
1/32
1.0
Area, A
0.8
0.6
0.4
0.2 1 2 3 4 5
Cuts, n
The graph does not produce a line. Nonlinear Function.

14. 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.

15. Number of Blocks on the edge (x) Total number of Blocks (y)
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) 2 8
(2 , 3) 3 27
(3, 27) 4 ? 5

16. Number of Blocks on the edge (x) Total number of Blocks (y)
To answer the question we must take a look at what is happening in the table or the figures:
Number of Blocks on the edge (x)
Total number of Blocks (y) 1 2 8 3 27 4 ? 5 +1 +7 +1
+19 +1
+37
Notice: although the x increases by 1, the number of blocks is not constant anymore.

17. The equation should be f(x) = x3
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 ?
The equation should be f(x) = x3
f(x) = 4  43 64
f(x) = 5  53 125

18. Number of Blocks on the edge (x) Total number of Blocks (y)
Filling in the table:
Number of Blocks on the edge (x)
Total number of Blocks (y) 1 2 8 3 27 4 ? 5 +1 +7 +1
+21 +1
+37 64
125

19. # of Total Blocks # of Blocks on Edge
Graph:
f(x) = x3
Ordered Pair (x, y)
(1 , 1)
(2 , 3)
(3, 27)
(4, 64)
(5, __) 60 50
# of Total Blocks 40 30 20 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.

20. Provide the rule that represents the function: x y 1 - 1 2 - 4 3 - 9 4
YOU TRY IT:
Provide the rule that represents the function: x y 1
- 1 2
- 4 3
- 9 4 ?

21. YOU TRY IT:(SOLUTION) x y 1
- 1 2
- 4 3
- 9 4 ?
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|
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.

22. 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.

23. VIDEOS: Non-Linear functions

24. Problems: As many as it takes to master the concept.
CLASS WORK:
Pages: 249 – 251
Problems: As many as it takes to master the concept.