January 9, 2015Today:The Coordinate Plane: Graphing in Two VariablesRelations & FunctionsClass Work 2.11Reminder: Khan Academy due Sunday

The Coordinate Plane: Graphing in Two VariablesReview:1. The 4 quadrants labeled:2. The axes (plural of axis) labeled:3. Ordered pairs and the quadrants4. The Intercepts:5. Independent & Dependent Variables6. Graphs & Equations:7. Three ways to display data: a) Table b) equation c) graph

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Vocabulary ReviewRelation: A set of order pairs.Domain: The x-values in the relation.Range: The y-values in the relation.Function: A relation where each x-value is assigned (maps to) on one y-value.

Vertical Line Test: Using a vertical straightedge to see if the relation is a function.Mapping: A diagram used to see if the relation is a function.

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VocabularyPlease turn to practice portion of notebook

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Graphing LINEar EquationsThis unit is about lines. Straight lines, mostly. Just as "A picture is worth a 1000 words", a line contains lots of information too.. if you know where to look and how to find itAs we progress through the unit, keep these main points in mind always:1. There is enough information in an equation (y = 2x + 4) to be able to graph a line.2. There is enough information in a line to be able to write an equation. 3. The purpose, the entire purpose, of linear equations is to see how something (let's call it variable "y") changes when something else (let's call it variable "x") changes.GIVE ME PICTURES

Breaking Down the Graph...is to see how something (let's call it variable "y") changes when something else (let's call it variable "x") changes.

xy

1-1

(0,4)

(4,0)-2Since 'y' only changes when 'x' changes, the y variable is the dependent variable..That is, it depends on x. The x variable is the independent variable. It changes before y does.Looking at the graph above, we see how as x increases, y decreases

Our x variable represents time,Time

Height/ft.

and the y is height from the groundThe graph, then is showing what?Most likely, the time it takes an object to fall from what height?Notice: A.) As x gets larger, y becomes smaller. This is a negative relationship, or negative slope. B.) The line is straight, telling us that the change is the same throughout. The speed of fall is the same, or constant.

The Cost of a Phone CallIT&E charges $0.16/minute for calls to the nation of Mali. Answer the following based on the above information:A. What 2 pieces of data will form our variables?B. Which one is the independent? The dependent?C. Write the equation showing the cost of a call based on the number of minutes.D. Complete a table based on phone calls of 1, 2,3, and 4 minutes of length.E. Write the data for each minute as an ordered pairThe ordered pairs are: (1,16), (2,32), (3,48), (4,64)F. This set of ordered pairs is called a relation and is written in the following way:

y = .16x

What is a Relation? A relation is a set of ordered pairs.When you group two or more points in a set, it is referred to as a relation. When you want to show that a set of points is a relation you list the points in braces.For example, if I want to show that the points (-3,1); (0, 2); (3, 3); & (6, 4) are a relation, it would be written like this:{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}

.G. Rewrite your set of ordered pairs as a relation.{ (1,16) ; (2,32) ; (3,48) ; (4,64) }

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Domain and RangeEach ordered pair has two parts, an x-value and a y-value. The x-values of a given relation are called the Domain.The y-values of the relation are called the Range.When you list the domain and range of relation, place each value in a separate set of braces. For Example, the relation is: {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}

Domain: { -3, 0, 3, 6} Range: {1, 2, 3, 4}

.H. Separate the domain and the range from your relation.Let's look at some domain and range examples before continuing with our data.Write the domain and range of the above relation

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Example 1:1. List the domain and the range of the relation {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}

Domain: { -3, 0, 3, 6} Range: {1, 2, 3, 4}

2. List the domain and the range of the relation {(-3,3) ; (0, 2) ; (3, 3) ; (6, 4) ; ( 7, 7)}

Domain: {-3, 0, 3, 6, 7} Range: {3, 2, 4, 7}Notice! Even though the number 3 is listed twice in the relation, you only note the number once when you list the domain or range!.

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What is a Function?A function is a relation that assigns each x-value only one y-value.What does that mean? It means, in order for the relation to be considered a function, each domain value can point or be associated with only one range value.

There are two ways to see if a relation is a function:MappingsVertical Line Test .

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Functions: A Short Video

MappingsIf the relation is not graphed, it is easier to use what is called a mapping.

When you are creating a mapping of a relation, you draw two ovals. In one oval, list all the domain values. In the other oval, list all the range values.Draw a line connecting the pairs of domain and range values.If any domain value maps to two different range values, the relation is not a function.

I. Follow the steps above to map your domain and range values. Determine whether your relation is a function.

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Example of a MappingCreate a mapping of the following relation and state whether or not it is a function.{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}StepsDraw ovalsList domainList rangeDraw lines to connect

-3036

1234

The relation is a function because each x-value maps to only one y-value.

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One More Example:Create a mapping of the following relation and state whether or not it is a function.{(-4,-1) ; (-4, 0) ; (5, 1) ; (3, 9)}

-453-1019

This relation is NOT a function because the (-4) maps to the (-1) & the (0).It is NOT a function if one x-value goes to two different y-values.Make sure to list the (-4) only once!

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Using the Vertical Line TestUse the vertical line test to check if the relation is a function only ifthe relation is already graphed.

Hold a straightedge (pen, ruler, etc) vertical to your graph. Drag the straightedge from left to right on the graph. 3.If the straightedge intersectsthe graph once in each spot , then it is a function. If the straightedge intersects thegraph more than once in any spot, it is not a function.

A function!

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Examples of the Vertical Line Test

functionfunctionNot a functionNot a function.

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Function Rules f(x)..g(x)..h(x)The equation that represents a function is called a function rule.

A function rule is written with two variables, x and y.

It can also be written in function notation, f(x). The f(x), or g(x), is simply the notation for the dependent variable.

When you are given a function rule, you can evaluate the function at a given domain value to find the corresponding range value..

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How to Evaluate a Function RuleTo evaluate a function rule, substitute the value in for x and solve for y.Examples: Evaluate the given function rules for x = 2 f(x)= x + 5 g(x)= 2x -1 f(x)= -x + 2

f(x)=(2)+ 5 y= 7g(x) = 2(2)-1= 4 1 y= 3f(x = -(2)+2= -2 + 2 y= 0.

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Evaluating for a Given DomainYou can also be asked to find the range values for a given domain.

This is the same as before, but now youre evaluating the same function rule for more than one number.

The values that you are substituting are x values, so they are a part of the domain.

The values you are generating are y-values, so they are a part of the range..

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ExamplesFind the range values of the function for the given domain.y = -3x + 2 ; {-1, 0, 1, 2}

The range values for the given domain are { 5, 2, -1, -4}.

StepsSub in each domain value in one @ a time.Solve for y in eachList y values in braces. y = -3x + 2 y = -3(-1) + 2 y = 3 + 2 y = 5y = -3x + 2y = -3(0) + 2y = 0 + 2y = 2y = -3x + 2 y = -3(1) + 2y = -3 + 2y = -1y = -3x + 2 y = -3(2) + 2y = -6 +2y = -4

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PracticeComplete the following questions:

Identify the domain and range of the following relations: a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)}b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)}

Graph the following relations and use the vertical line test to see if the relation is a function. Connect the pairs in the order given.a. {(-3,-3) ; (0, 6) ; (3, -3)}b. {(0,6) ; (3, 3) ; (0, 0)}Use a mapping to see if the following relations are functions: a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)}b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)}

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Answers:

1a. Domain: {-4, -2, 3, 4} Range: {-2, 2, 1}1b. Domain: {0, 1, 7} Range: {-6, 2, -4, 4}2a. 2b.

3a. 3b.

FunctionNot a Function

-4-234-121

017

-62-44

FunctionNot a Function

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Find the range values of the function for the given domain.

y = 5x - 7 ; {-3, -2, 4}

y = 5x -7 y = 5x -7 y = 5x - 7 y = 5(-3) - 7 y= 5(-2) -7 y = 5(4) - 7 y = -15 - 7 y= -10 - 7 y= 20 - 7 y= -22 y= -17 y= 13

The range values for the given domain are: { -22, -17, 13}

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Practice1. Find the range values of the function for the given domain.f(x) = 3x + 1 ; {-4, 0, 2}

2. Find the range values of the function for the given domain.g(x) = -2x + 3 ; {-5, -2, 6}StepsSub in each domain value in one @ a time.Solve for y in eachList y values in braces.

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Answers y = 3x + 1 y = 3(-4) + 1 y = -12 + 1y = -11y = 3x + 1 y = 3(0) + 1 y = 0 + 1y = 1Ans. { -11, 1, 7}y = 3x + 1 y = 3(2) + 1y = 6 + 1y = 7y = -2x + 3y = -2(-5) + 3y = 10 + 3y = 13y = -2x + 3y = -2(-2) + 3y = 4 +3y = 7Ans. { 13, 7, -9}y = -2x + 3y = -2(6) + 3y = -12 +3y = -9

1.

2.

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Class Work:2.10

Another MappingCreate a mapping of the following relation and state whether or not it is a function.{(-1,2) ; (1, 2) ; (5, 3) ; (6, 8)}

Notice that even though there are two 2s in the range, you only list the 2 once.

-1156

238

This relation is a function because each x-value maps to only one y-value. It is still a function if two x-values go to the same y-value.

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