Warm Up To help guide this chapter, a project (which will be explained after the warm up) will help guide Chapter 2 To help guide this chapter, a project (which will be explained after the warm up) will help guide Chapter 2 On the front page of your 3 page packet, fill out: On the front page of your 3 page packet, fill out: 1. Your name 2. A name for your business 3. Two (appropriate) products to sell 4. The price for each product (the price must be more than $1 and less than $10 You have five minutes to do this. If you finish, read through the rest of the packet You have five minutes to do this. If you finish, read through the rest of the packet

CHAPTER 2.1, 2.2 RELATIONS AND LINEAR FUNCTIONS

Cartesian Coordinate plane The Cartesian Coordinate plane is composed of the – x-axis (horizontal) and the – y-axis (vertical), – They meet at the origin (0,0) – Divide the plane into four quadrants. – Ordered pairs graphed on the plane can be represented by (x,y).

Cartesian Coordinate Plane

VOCABULARY RELATION – SET OF ORDERED PAIRS RELATION – SET OF ORDERED PAIRS – example: {(4,5), (–2,1), (5,6), (0,2)} DOMAIN – SET OF ALL X’S DOMAIN – SET OF ALL X’S – D: {4, –2, 5, 0} RANGE – SET OF ALL Y’S RANGE – SET OF ALL Y’S – R: {5, 1, 6, 2} A relation can be shown by a mapping, a graph, equations, or a list (table). A relation can be shown by a mapping, a graph, equations, or a list (table).

Mapping -shows how each member of the domain is paired with each member of the range.

Function A function is a special type of relation. – By definition, a function exists if and only if every element of the domain is paired with exactly one element from the range. – That is, for every x-coordinate there is exactly one y-coordinate. All functions are relations, but not all relations are functions.

One-to-one Mapping – example: {(4,5), (–2,1), (5,6), (0,2)}

Function Example – B: {(4,5), (–2,1), (4,6), (0,2)} – Not a function because the domain 4 is paired with two different ranges 5 & 6

Vertical Line Test The vertical line test can be applied to the graph of a relation. If every vertical line drawn on the graph of a relation passes through no more than one point of the graph, then the relation is a function.

Graphing by domain and range Y=2x+1 – Make a table to find ordered pairs that satisfy the equation – Find the domain and range – Graph the ordered pairs – Determine if the relation is a function

More Vocab. Function Notation A function is commonly denoted by f. In function notation, the symbol f (x), is read "f of x " or "a function of x." Note that f (x) does not mean "f times x." The expression y = f (x) indicates that for every value that replaces x, the function assigns only one replacement value for y. f (x) = 3x + 5, Let x = 4 also written f(4) – This indicates that the ordered pair (4, 17) is a solution of the function.

Vocab. Independent Variable – In a function, the variable whose values make up the domain – Usually X Dependent Variable – In a function, the variable whose values depend on the independent variable – Usually Y

Function Examples If f(x) = x³ - 3, evaluate: If f(x) = x³ - 3, evaluate: – f(-2) – f(3a) If g(x) = 5x 2 - 3x+7, evaluate: If g(x) = 5x 2 - 3x+7, evaluate: – g(4) – g(-3c)

2.2 Linear Relations and Functions LINEAR EQUATION One or two variables One or two variables Highest exponent is 1 Highest exponent is 1 NOT LINEAR EQUATION Exponent greater than 1 Exponent greater than 1 Variable x variable Variable x variable Square root Square root Variable in denominator Variable in denominator

Linear/not linear 2x + 3y = -5 2x + 3y = -5 f (x) = 2x – 5 f (x) = 2x – 5 g (x) = x³ + 2 g (x) = x³ + 2 h (x,y) =- 1 + xy h (x,y) =- 1 + xy

EVALUATING A LINEAR FUNCTION The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit, f, that are equivalent to a given number of degrees Celsius, C On the Celsius scale, normal body temperature is 37°C. What is the normal body temp in degrees Fahrenheit? On the Celsius scale, normal body temperature is 37°C. What is the normal body temp in degrees Fahrenheit?

STANDARD FORM Ax + By = C 1. X POSITIVE 1. X POSITIVE 2. A & B BOTH NOT ZERO 2. A & B BOTH NOT ZERO 3. NO FRACTIONS 3. NO FRACTIONS 4. GCF OF A, B, C = 1 4. GCF OF A, B, C = 1 EX: Y = 3X – 9 EX: Y = 3X – 9 8X – 6Y + 4 = 0 8X – 6Y + 4 = 0 -2/3X = 2Y – 1

USE INTERCEPTS TO GRAPH A LINE X – INTERCEPT SET Y=0 SET Y=0 Y – INTERCEPT SET X=0 SET X=0 PLOT POINTS AND DRAW LINE EX: - 2X + Y – 4 = 0

CHAPTER 2.3 SLOPE

SLOPE CHANGE IN Y OVER CHANGE IN X CHANGE IN Y OVER CHANGE IN X RISE OVER RUN RISE OVER RUN RATIO RATIO STEEPNESS STEEPNESS RATE OF CHANGE RATE OF CHANGE FORMULA FORMULA

USE SLOPE TO GRAPH A LINE 1. PLOT A GIVEN POINT 1. PLOT A GIVEN POINT 2. USE SLOPE TO FIND ANOTHER POINT 2. USE SLOPE TO FIND ANOTHER POINT 3. DRAW LINE 3. DRAW LINE EX: DRAW A LINE THRU (-1, 2) WITH SLOPE -2

RATE OF CHANGE OFTEN ASSOCIATED WITH SLOPE OFTEN ASSOCIATED WITH SLOPE MEASURES ONE QUANTITY’S CHANGE TO ANOTHER MEASURES ONE QUANTITY’S CHANGE TO ANOTHER

LINES PARALLEL SAME SLOPE SAME SLOPE VERTICAL LINES ARE PARALLEL VERTICAL LINES ARE PARALLELPERPENDICULAR OPPOSITE RECIPROCALS (flip it and change sign) OPPOSITE RECIPROCALS (flip it and change sign) VERTICAL AND HORIZONTAL LINES VERTICAL AND HORIZONTAL LINES

CHAPTER 2.4 WRITING LINEAR EQUATIONS

SLOPE-INTERCEPT FORM y = mx + b y = mx + b m IS SLOPE m IS SLOPE b IS Y-INTERCEPT b IS Y-INTERCEPT

POINT-SLOPE FORM FIND SLOPE FIND SLOPE PLUG IN PLUG IN ARRANGE IN SLOPE INTERCEPT FORM ARRANGE IN SLOPE INTERCEPT FORM

EX: WRITE AN EQUATION OF A LINE THRU (5, -2) WITH SLOPE -3/5 EX: WRITE AN EQUATION OF A LINE THRU (5, -2) WITH SLOPE -3/5 EX: WRITE AN EQUATION FOR A LINE THRU (2, -3) AND (-3, 7) EX: WRITE AN EQUATION FOR A LINE THRU (2, -3) AND (-3, 7)

INTERPRETING GRAPHS WRITE AN EQUATION IN SLOPE- INTERCEPT FORM FOR THE GRAPH WRITE AN EQUATION IN SLOPE- INTERCEPT FORM FOR THE GRAPH

REAL WORLD EXAMPLE As a part time salesperson, Dwight K. Schrute is paid a daily salary plus commission. When his sales are $100, he makes $58. When his sales are $300, he makes $78. Write a linear equation to model this. Write a linear equation to model this. What are Dwight’s daily salary and commission rate? What are Dwight’s daily salary and commission rate? How much would he make in a day if his sales were $500? How much would he make in a day if his sales were $500?

Write an equation for the line that passes through (3, -2) and is perpendicular to the line whose equation is y = -5x + 1 Write an equation for the line that passes through (3, -2) and is perpendicular to the line whose equation is y = -5x + 1 Write an equation for the line that passes through (3, -2) and is parallel to the line whose equation is y = -5x + 1 Write an equation for the line that passes through (3, -2) and is parallel to the line whose equation is y = -5x + 1

CHAPTER 2.5 LINEAR MODELS

Prediction line SCATTER PLOT – GRAPH WITH MANY ORDERS PAIRS SCATTER PLOT – GRAPH WITH MANY ORDERS PAIRS LINE OF BEST FIT – LINE DRAWN THROUGH DATA THAT BEST REPRESENTS IT LINE OF BEST FIT – LINE DRAWN THROUGH DATA THAT BEST REPRESENTS IT

MAKE A SCATTER PLOT APPROXIMATE PERCENTAGE OF STUENTS WHO SENT APPLICATIONS TO TWO COLLEGES IN VARIOUS YEARS SINCE 1985 YEARS SINCE %

LINE OF BEST FIT SELECT TWO POINTS THAT APPEAR TO BEST FIT THE DATA SELECT TWO POINTS THAT APPEAR TO BEST FIT THE DATA IGNORE OUTLIERS IGNORE OUTLIERS DRAW LINE DRAW LINE

PREDICTION LINE FIND SLOPE FIND SLOPE WRITE EQUATION IN SLOPE-INTERCEPT FORM WRITE EQUATION IN SLOPE-INTERCEPT FORM

INTERPRET WHAT DOES THE SLOPE INDICATE? WHAT DOES THE SLOPE INDICATE? WHAT DOES THE Y-INT INDICATE? WHAT DOES THE Y-INT INDICATE? PREDICT % IN THE YEAR 2010 PREDICT % IN THE YEAR 2010 HOW ACCURATE ARE PREDICTIONS? HOW ACCURATE ARE PREDICTIONS?

CHAPTER 2.6/2.7 SPECIAL FUNCTIONS and Transformations

ABSOLUTE VALUE FUNCTION V-shaped V-shaped PARENT GRAPH (Basic graph) PARENT GRAPH (Basic graph) FAMILIES OF GRAPHS (SHIFTS) FAMILIES OF GRAPHS (SHIFTS)

EXAMPLES make table and graph

EFFECTS +,- OUTSIDE +,- OUTSIDE SHIFTS UP AND DOWN +,- INSIDE +,- INSIDE SHIFTS LEFT AND RIGHT X,÷ NARROWS AND WIDENS X,÷ NARROWS AND WIDENS

Be able to use calculator to find graphs and interpret shifts Be able to use calculator to find graphs and interpret shifts Be able to identify domain and range Be able to identify domain and range

CHAPTER 2.8 GRAPHING INEQUALITIES

BOUNDARY EX: y ≤ 3x + 1 EX: y ≤ 3x + 1 THE LINE y = 3x + 1 IS THE BOUNDARY OF EACH REGION THE LINE y = 3x + 1 IS THE BOUNDARY OF EACH REGION SOLID LINE INCLUDES BOUNDARY SOLID LINE INCLUDES BOUNDARY____________________________ DASHED LINE DOESN’T INCLUDE BOUNDARY DASHED LINE DOESN’T INCLUDE BOUNDARY

GRAPHING INEQUALITIES 1. GRAPH BOUNDARY (SOLID OR DASHED) 1. GRAPH BOUNDARY (SOLID OR DASHED) 2. CHOOSE POINT NOT ON BOUNDARY AND TEST IT IN ORIGIONAL INEQUALITY 2. CHOOSE POINT NOT ON BOUNDARY AND TEST IT IN ORIGIONAL INEQUALITY 3. TRUE-SHADE REGION WITH POINT 3. TRUE-SHADE REGION WITH POINT FALSE-SHADE REGION W/O POINT

On calculator Enter slope-int form under “y=“ Enter slope-int form under “y=“ Scroll to the left to select above or below Scroll to the left to select above or below Zoom 6 Zoom 6

GRAPH THE FOLLOWING INEQUALITIES x – 2y < 4