Cornell Notes Section 1.2 Day 1 Section 1.2 Day 2 Section 1.4 Day 1 Summary (3 sentences) Section 1.2 Day 2 2-3 Level 1 Questions Section 1.4 Day 1 Highlight, Underline, ?

Cornell Notes Section 1.2 Day 2 Section 1.4 Day 1 Section 1.6 Day 1 Summary (3 sentences) Section 1.4 Day 1 2-3 Questions Section 1.6 Day 1 Highlight, Underline, ?

Cornell Notes Section 1.4 Day 1 Section 1.6 Day 1 Section 1.6 Day 2 Summary (3 sentences) Section 1.6 Day 1 2-3 Questions Section 1.6 Day 2 Highlight, Underline, ?

Section 1.6 Relations Algebra 1

Learning Targets Define the vocabulary terms: coordinate system, x-axis, y-axis, origin, ordered pair, x-coordinate, y-coordinate, relation, mapping, domain, range, independent variable, and dependent variable Verbalize the relationships between ordered pairs, tables, graphs, mappings, and relations Determine the domain and range of a relation Construct different representations of relations given one representation Identify the independent and dependent variables of a relation Describe the pattern of a relation in a context Construct a graph of a relation given a verbal description of the relation

Coordinate System Constructs the coordinate system Ordered Pair Y-Axis Each point on the coordinate plane in the form (𝑥,𝑦) Constructs the coordinate system X-Axis Origin (0,0) In (2, 3), 2 is the x-coordinate and 3 is the y-coordinate Point where axes intersect (2,3)

Relation Relation Definition: A set of pairs of input and output values Characteristics: Multiple Representations: Ordered pairs Table Graph Equation Words Mapping Diagram Relation Examples: 4, 7 , 3, 2 , 0,0 , −1,−4 Input: X 𝟎 −𝟐 𝟑 Output: Y 1 4 7

Relation Representations Ordered Pairs: 1, 2 , 3, −1 , 5, 6 , 0,−9 , −3,6 Table: X: Input Y: Output X 𝟎 −𝟐 𝟑 Y 1 4 7 Graph: X-Coordinate: Input/Horizontal Y-Coordinate: Output/Vertical Mapping Diagram: Maps inputs to outputs −2 −1 1 4 6 8 9

Example 1: Multiple Representations Express the relation 2, 5 , 5, −2 , −2, 3 , −1, −2 as a table, graph, and a mapping X Y −2 3 −1 2 5 −2 −1 2 5 3 −2 5

Domain & Range Range: The output values of a relation Domain: The input values of a relation. EX: The x-coordinate in ordered pairs, tables, and graphs. EX: The inputs of the mapping diagram. Range: The output values of a relation EX: The y-coordinate in ordered pairs, tables, and graphs EX: The outputs of the mapping diagram

Domain & Range: Example 2: Ordered Pairs Find the domain and range of the following relation { 3, 4 , −2, 1 , 6, 5 , 9, −10 , −5, −7 } Domain (Input, x-values): −5, −2, 3, 6, 9 Range (Output, y-values): {−10, −7, 1, 4, 5}

Domain & Range: Example 3: Tables Find the domain and range of the following relation X Y −4 3 5 −2 7 X Y −4 3 5 −2 7 X Y −4 3 5 −2 7 Domain (Input, x-values) {𝟎, 𝟑, 𝟓, 𝟕} Range (Output, y-values) {−𝟒, −𝟐, 𝟎, 𝟓}

Domain & Range: Example 4: Mapping Diagram Find the domain and range of the following relation Domain (Input, x-values) {−𝟕, −𝟏, 𝟒} Range (Output, y-values) {−𝟗, 𝟎, 𝟑, 𝟏𝟖} −7 −1 4 −7 −1 4 3 18 −9 3 18 −9

Domain & Range: Example 5: Graph - Level 1 Find the domain and range of the following relation Domain (Input, x-values) {−𝟓, −𝟐, 𝟎, 𝟐, 𝟒} Range (Output, y-values) {−𝟖, −𝟒, 𝟏, 𝟑, 𝟓}

Domain & Range: Example 6: Graph - Level 2 Find the domain and range of the following relation Domain (Input, x-values) { −𝟔, −𝟏 &[𝟏,𝟒]} Closed Brackets All values between and including the end points Range (Output, y-values) { −𝟒,𝟎 & 𝟑}

Domain & Range: Example 7: Graph - Level 2 Find the domain and range of the following relation Open Brackets All values between and NOT including the end points Range (Output, y-values) {[𝟏, 𝟔)} Or 𝟏≤𝒚<𝟔 Domain (Input, x-values) {(−𝟕,𝟓]} Or −𝟕<𝒙≤𝟓

Domain & Range: Example 8: Graph - Level 3 Find the domain and range of the following relation Domain (Input, x-values) (−∞,∞) Or −∞<𝒙<∞ Or All Real #s Range (Output, y-values) {[−𝟑, ∞)} Or −𝟑≤𝒚<∞

Domain & Range: Example 9: Graph - Level 3 Find the domain and range of the following relation Domain (Input, x-values) [−𝟒,∞) Or −𝟒≤𝒙<∞ Range (Output, y-values) {[𝟏, ∞)} Or 1≤𝒚<∞

Domain & Range Example 10 Come up with a relation in a Mapping Diagram form that has the following Domain: {−𝟏𝟏 𝟎, 𝟏, 𝟐𝟓} There are many possible answers The key is that the input circle must only contain the numbers {−𝟏𝟏 𝟎, 𝟏, 𝟐𝟓}. The output doesn’t matter One possible answer: −11 1 25 −4 −2

Domain & Range Example 11 Come up with a relation in an Ordered Pair form that has the following Range: {−𝟏𝟒, −𝟐, 𝟏𝟕, 𝟑𝟗} There are many possible answers The key is that all the y-values must only be the numbers {−𝟏𝟒, −𝟐, 𝟏𝟕, 𝟑𝟗} The x-values don’t matter One possible example: { 𝟎, −𝟏𝟒 , 𝟏, −𝟐 , 𝟐, 𝟏𝟕 , 𝟑, 𝟑𝟗 }

Independent & Dependent Variables - The number of pickles you can make - The amount of money raised Dependent Variable: The output values of a relation (y-values). It relies on the independent variable. Dependent variable Independent Variable: The input values of a relation (x-values). Changing the independent variables Independent Variable: - The number of cucumbers you have - The number of cookies baked Causes change in the

Example 12: Independent/Dependent Variables Identify the independent and dependent variables for each relation A)The dance committee is selling tickets to the Fall Ball. The more tickets that they sell, the greater the amount of money they can spend for decorations. B) The air pressure inside a tire increased with the temperature C) The greater the number of hours driving, the greater the total distance traveled. tickets money air pressure temperature hours distance

Example 13: Graph Analysis Independent Variable: Time Dependent Variable: Distance Origin-A: Rider is biking at a constant pace A-B: Time is increasing as distance is constant. Rider has stopped moving B-After: Rider continues to bike at a constant pace Answer the following questions for the graph: A) What is the independent and dependent variable? B) Describe what is happening during the bike ride from the origin to point A, from point A to point B, and after point B

Example 14: Graph Construction Create a graph that describes the height of a person jumping on a trampoline over a 5 minute period. Be sure to create a realistic situation and describe what is happening in your graph. There are many situations One example:

Exit Ticket – For Feedback Find the Domain and Range of the following: 1) −2, 0 , 4, −2 , 1, 1 , 5, 9 2) 3) Identify the independent and dependent variable. A teacher tested different types of teaching (reading, lecture, video, lab, etc.) to determine which helps students score higher on the tests. 1 2 7 −10 −1 12