BEFORE: October 16, 2017 The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation. 2. Write a possible situation for the given graph. 1. Graph C 2. The level of water in a bucket stays constant. A steady rain raises the level. The rain slows down. Someone dumps the bucket.

What are these graphs showing us? RELATIONS A “relation” is simply a set of ordered pairs. In addition, a “relation” is just a relationship between sets of information. Example: Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. (Just like the pairing in the previous graphs) What are these graphs showing us?

DURING: Identifying Functions Learning Target: I can explain the difference between… A relation and a function A domain and a range

Ways to Represent a Relation Ordered Pairs Input/Output Table Coordinate Plane Mapping Diagram

Ordered Pairs-We always write ordered pairs in curly braces with commas, in between each ordered pair. { (1,5), (-2, 7), (4,3), (0,0), (-2,-3) } Input/Output Table Coordinate Plane Mapping Diagram

Ordered Pairs Input/Output Table--The input is always x. The output is always y. In a vertical table, x always goes on the left. In a horizontal table, x always goes on top. Coordinate Plane Mapping Diagram

Ordered Pairs Input/Output Table Coordinate Plane--The x-coordinate tells us to move left/right. The y-coordinate tells us to move up/down. Mapping Diagram

Ordered Pairs Input/Output Table Coordinate Plane Mapping Diagram--In the left box/oval, write all of the inputs (x). Do not repeat any values! In the right box/oval, write all of the outputs (y). Do not repeat any values! Draw an arrow from each input value to its output value.

What do functions have to do with a relation? A function is a “well- behaved” relation. Meaning, given a starting point, we know exactly where to go. While all functions are relations, not all relations are functions. What do functions have to do with a relation?

Function Definition--A function is a relations where every input has exactly one output. Characteristics: If you can draw a vertical line through the graph that touches the graph only once, then it is a function. If the vertical line touches the graph more than once, then it is NOT A FUNCTION. We say that the relation “fails the Vertical Line Test”.

Ways to Identify a Function Examples Ordered Pairs { (1,2), (2,2), (3,2), (4,2) } Table c. Mapping i. Does each input (x-value) have exactly one output (y- value)? ii. If so, then it’s a function! iii. If all the x-values are different, then it’s a function! d. Graphing i. See next slide! X Y 1 4 2 7 3 8 -2 Demonstrate mapping to students using ordered pairs and table

“WE DO”-Identify Functions

* all the values of the dependent variable The domain of a relation is the set of first coordinates (or x -values) of the ordered pairs. * all the values of the independent variable The range of a relation is the set of second coordinates (or y -values) of the ordered pairs. * all the values of the dependent variable

Definition Example {(3, 6), (2, 8), (5, 3)} { 3, 2, 5} { 6, 8, 3} Domain All the x-coordinates in the function's ordered pairs { 3, 2, 5} Range All the y-coordinates in the function's ordered pairs { 6, 8, 3}

“WE DO” Identify the domain and range of the function below. Ordered pairs “WE DO” Identify the domain and range of the function below. { (2, 7), (4, 11), (6, 15), (8, 19)} The domain is { 2, 4, 6, 8} The range is { 7, 11, 15, 19}

Graphs “I DO” The domain of a function is the set of all the x-coordinates in the functions’ graph. Domain 3 ≤ x ≤ 12

Graphs “I DO” The range of a function is the set of all the y-coordinates in the functions’ graph. Range 6 ≤ y ≤ 12

Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5 “WE DO” What is the domain of this function? What is the range of this function? Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5 Answer: Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5

Classwork/Homework: Page 173, #3-14