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.

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)

GOAL:

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:

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.

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.

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

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)

1 2(6) + 2(1) = 14 (1, 14)

2 2(6) + 2(2) = 16 (2, 16)

3 2(6) + 2(3) = 18 (3, 18)

4 2(6) + 2(4) = 20 (4, 20)

5 2(6) + 2(5) = 22 (5, 22)

Graph:Pe

rimet

er

Figure

Ordered Pairs(x, y)(1, 14)

(2, 16)

(3, 18)

(4, 20)

(5, 22)

24

68

10 12 14 16 18 20 22

1 2 3 4 5

YOU TRY IT:

Use one method to represent the relationship between the number of triangles and the perimeter.

1

1

1 1

1

1

11

1

1

1 1

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

11

Triangles = 3 Perimeter = 5

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

y = x + 2

f(x) = x + 2

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)

1 1 + 2 = 3 (1, 3)

2 2 + 2 = 4 (2, 4)

3 3 + 2 = 5 (3, 5)

4 4 + 2 = 6 (4, 6)

5 5 + 2 = 7 (5, 7)

Graph:Pe

rimet

er

Figure

Ordered Pairs(x, y)(1, 3)

(2, 4)

(3, 5)

(4, 6)

(5, 7)

12

34

5 6 7 8 9 10

1 2 3 4 5

Q: What is the value of y if x = 0?

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)

0 1 2 3

Memory (y)

512 509 506 503

To answer the question we must take a look at what is happening in the table.

Number of Photos (x)

0 1 2 3

Memory (y)

512 509 506 503

+ 1 + 1 + 1

- 3 - 3 - 3

The dependent variable y decreases by 3The independent variable x increases by 1The starting memory is 512 MG

Taking the info to consideration, we can see that the equation for the problem is:

Number of Photos (x)

0 1 2 3

Memory (y)

512 509 506 503

+ 1 + 1 + 1

- 3 - 3 - 3

The dependent variable y decreases by 3

y = 512 – 3x

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)

0 10

1 18

2 26

3 34

YOU TRY IT: (SOLUTION)

Looking at both variables, we have:Hours (x) Money(y)

0 10

1 18

2 26

3 34

+8

+8

+8

+1

+1

+1

Both, the x and y are changing at a constant rate.

YOU TRY IT: (Words Solution):

A person had 10 dollars and then starts a job where he earns eight dollars per hour.

YOU TRY IT: (Equation Solution):

We started with 10 dollars and earn 8 after each hour.

y = 8x + 10

f(x) = 8x + 10

YOU TRY IT: (Graph Solution)M

oney

$

Hours

Hours(x)

Dollars(y)

0 10

1 18

2 26

3 34

4 ?

Q: What will the total money after 4 hrs?.

5

10

15

20

25

30

35

1 2 3

VIDEOS: Linear Functions

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/analyzing-functions-algebra/v/constructing-and-interpreting-a-linear-function

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/analyzing-functions-algebra/v/constructing-a-linear-function-word-problem

CLASS WORK:

Pages: 243 – 245

Problems: As many as it takes to master the concept.

CLASSWORK:

Page 243-245

Problems: 5, 7, 9, 11, 12, 13, 14,

16, Review Handout