4.3 Patterns and Non-Linear Functions:

1. Nonlinear: Is a function whose graph is not a line or part of a line. 4.3 Patterns and Non-Linear Functions: 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. 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 = 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. Sauce Cost Pizza Area

10. Graph: Pizza Area Area, A 300 200 100 6 2 4 8 10 Radius, r The graph does not produce a line. Nonlinear Function.

11. Graph: Sauce Cost Cost, C 6 4 2 6 2 4 8 10 Sauce, w The graph does produce a line. Linear Function.

12. YOU TRY IT: Classify the following function as linear or nonlinear. Cutting Paper

13. YOU TRY IT (SOLUTION): Cutting Paper 1.0 Area, A 0.8 0.6 0.4 0.2 3 2 1 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. Ex: What is the pattern we can use to complete the table?

16. To answer the question we must take a look at what is happening in the table or the figures: +1 +7 +1 +19 +1 +37 Notice: although the x increases by 1, the number of blocks is not constant anymore.

17. 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. Filling in the table: +1 +7 +1 +21 +1 +37 64 125

19. Graph: f(x) = x3 60 50 # of Total Blocks 40 30 20 10 6 2 4 8 10 # of Blocks on Edge The graph does not produce a straight line. It is part of the Cubic Functions.

20. YOU TRY IT: Provide the rule that represents the function:

21. YOU TRY IT:(SOLUTION) Looking at the data on the y values we can see that it is notlinear, thus we have the following equations to choose from: Y = x2 Y = x3 Y = |x| Notice also that the values are negative, thus: Y = -|x| Y = -x2 Y = -x3 Plugging in numbers we see that y = - x2 should be our equation.

22. Graph: f(x) = -x2 The graph does not produce a straight line. It is part of the Square Functions.

23. 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

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