1. 4.2 Patterns and Linear Functions: Independent Variable: The variable that is not dependent. Dependent Variable: The variable that depends upon the value of another. Input: The values of the independent variable. Output: The values of the dependent variable.

2. Linear Function: A function whose graph is a nonvertical line or part of a nonvertical line. Function  f(x): A relationship that pairs one input to exactly one output (x, y)

3. GOAL:

4. In math we use tables, words, equations, set of ordered pairs and graphs to represent a relationship between two variables. 1. Geometric Relationships, 2. Linear Functions This can be done when we are presented with the following:

5. GEOMETRIC RELATIONSHIPS: Ex: Use words, an equation, a table and a graph to represent the relationship between the number of rectangles and the perimeter of the figure.

6. Words: Look at the figure, multiply the number of rectangles by 2 to get the total lengths of the top an bottom sides of the combined figure. Then add 2 times the length of the left and right sides of the combined figure to obtain the final answer for the total perimeter of the figure.

7. Equation: Look at the figure: We must realize that the only thing that is changing is the number of the short side (width). Also, the number of the length is constant which is = 12. Using this info we see that the equation is: y or f(x) = twice the number of short + twice the length. f(x) = 2x + 12

8. Table: Looking at the figures we see that: X = number of rectangles (independent) y = Perimeter of the figure (depends on figure) Number of rectangles (x) Perimeter (y) y = 2l + 2w Ordered Pairs (x, y) 12(6) + 2(1) = 14(1, 14) 22(6) + 2(2) = 16(2, 16) 32(6) + 2(3) = 18(3, 18) 42(6) + 2(4) = 20(4, 20) 52(6) + 2(5) = 22(5, 22)

9. Graph: Perimeter Figure Ordered Pairs (x, y) (1, 14) (2, 16) (3, 18) (4, 20) (5, 22) 2 4 6 8 10 12 14 16 18 20 22 1 23 4 5

10. YOU TRY IT: Use one method to represent the relationship between the number of triangles and the perimeter. 1 1 1 1 1 1 1 1 1 1 1 1

11. Words: Triangles = 1 Perimeter = 3 Look at the figure, The perimeter is 2 more than the number of triangles. 1 1 1 1 1 1 1 Triangles = 2 Perimeter = 4 1 1 1 1 1 Triangles = 3 Perimeter = 5

12. Equation: Again, the perimeter [ y or f(x)] is 2 more than the number of triangles (x) y = x + 2 f(x) = x + 2

13. Table: Looking at the figures we see that: X = number of rectangles (independent) y = Perimeter of the figure (depends on figure) Number of rectangles (x) Perimeter (y) y = x + 2 Ordered Pairs (x, y) 11 + 2 = 3(1, 3) 22 + 2 = 4(2, 4) 33 + 2 = 5(3, 5) 44 + 2 = 6(4, 6) 55 + 2 = 7(5, 7)

14. Graph: Perimeter Figure Ordered Pairs (x, y) (1, 3) (2, 4) (3, 5) (4, 6) (5, 7) 1 2 3 4 5 6 7 8 9 10 1 23 4 5 Q: What is the value of y if x = 0?

15. LINEAR FUNCTIONS: Data from a table can be scrutinize to see if it is a linear relation. In order for us to make the final decision, we first must see how the y – function, changes for each x in the table. Ex: Is there a linear relation in this table? Number of Photos (x) 0123 Memory (y) 512509506503

16. To answer the question we must take a look at what is happening in the table. Number of Photos (x) 0123 Memory (y) 512509506503 + 1 - 3 The dependent variable y decreases by 3 The independent variable x increases by 1 The starting memory is 512 MG

17. Taking the info to consideration, we can see that the equation for the problem is: Number of Photos (x) 0123 Memory (y) 512509506503 + 1 - 3 The dependent variable y decreases by 3 y = 512 – 3x

18. YOU TRY IT: For the table, determine whether the relationship is a linear function. Then represent the relationship using words, an equation and a graph. Hours (x)Money(y) 010 118 226 334

19. YOU TRY IT: (SOLUTION) Looking at both variables, we have: Hours (x)Money(y) 010 118 226 334 +8 +1 Both, the x and y are changing at a constant rate.

20. YOU TRY IT: (Words Solution): A person had 10 dollars and then starts a job where he earns eight dollars per hour.

21. YOU TRY IT: (Equation Solution): We started with 10 dollars and earn 8 after each hour. y = 8x + 10 f(x) = 8x + 10

22. YOU TRY IT: (Graph Solution) Money $ Hours Hours (x) Dollars (y) 010 118 226 334 4? Q: What will the total money after 4 hrs?. 5 10 15 20 25 30 35 1 2 3

23. VIDEOS: Linear Functions https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/analyzing-functions- algebra/v/constructing-and-interpreting-a-linear- function https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/analyzing-functions- algebra/v/constructing-a-linear-function-word- problem

24. CLASS WORK: Pages: 243 – 245 Problems: As many as it takes to master the concept.

25. CLASSWORK: Page 243-245 Problems: 5, 7, 9, 11, 12, 13, 14, 16, Review Handout